Thought of Search: Planning with Language Models Through The Lens of Efficiency
Authors: Michael Katz
Harsha Kokel
Kavitha Srinivas
Shirin Sohrabi
Venue: NeurIPS 2024 poster (2024)
Keywords: planning
large language models
search
Primary area: generative_models
TL;DR: We find that recent trends in using LLMs for planning are profoundly uneconomical, unsound and incomplete. We propose a significantly more efficient approach that is sound and complete and argue for a responsible use of compute resources.

Abstract: Among the most important properties of algorithms investigated in computer science are soundness, completeness, and complexity. These properties, however, are rarely analyzed for the vast collection of recently proposed methods for planning with large language models. In this work, we alleviate this gap. We analyse these properties of using LLMs for planning and highlight that recent trends abandon both soundness and completeness for the sake of inefficiency. We propose a significantly more efficient approach that can, at the same time, maintain both soundness and completeness. We exemplify on four representative search problems, comparing to the LLM-based solutions from the literature that attempt to solve these problems. We show that by using LLMs to produce the code for the search components we can solve the entire datasets with 100% accuracy with only a few calls to the LLM. In contrast, the compared approaches require hundreds of thousands of calls and achieve significantly lower accuracy. We argue for a responsible use of compute resources; urging research community to investigate sound and complete LLM-based approaches that uphold efficiency.

Thought of Search: Planning with Language Models Through The Lens of Efficiency
Michael Katz
Harsha Kokel
IBM T. J. Watson Research Center Yorktown Heights, NY 10598
IBM Almaden Research Center San Jose, CA 95120 harsha. kokel @i @imm..m......mm........e.. com
michaed katz1@ibn com
Kavitha Srinivas
Shirin Sohrabi
IBM T. J. Watson Research Center Yorktown Heights, NY 10598 kavitha. srinivas@ibm. com
IBM T. Watson Research Center Yorktown Heights, NY 10598 ssohrabdus. ibm com
Abstract
Among the most important properties of algorithms investigated in computer science are soundness, completeness, and complexity. These properties, however, are rarely analyzed for the vast collection of recently proposed methods for planning with large language models. this work, we alleviate this gap. We analyse these properties of using LLMs for planning and highlight that recent trends abandon both soundness and completeness for the sake of inefficiency. We propose a significantly more efficient approach that can, at the same time, maintain both soundness and completeness. We exemplify on four representative search problems, comparing to the L.---b solutions from the literature that attempt solve these problems. We show that by using LLMs to produce the code for the search components we can solve the entire datasets with 100% accuracy with only a few calls the LLM. In contrast, the compared approaches require hundreds of thousands of calls and achieve significantly lower accuracy. We argue for responsible use of compute resources; urging research community to investigate sound and complete LLM LLM-based approaches that uphold efficiency.
Introduction
Recent work has addressed the issue of planning in Large Language Models (LLMs), spurred by their abilities in natural language tasks. The approaches vary widely from giving planning problem output entire plan to asking LLM plan step by step, including
LLM and asking backtracking. Roughly, these approaches can be partitioned into two sets. The first exemplifies yet another capability of language models, while the second aims at presenting practical tools for solving planning problems. In the latter case, however, since planning problems are often computationally hard, crucial to understand the properties and the complexity of the algorithms proposed.
The purpose of our work is precisely that. Going over a collection of recent methods for planning with large language models, we analyse the most important properties of the proposed methods such as soundness and completeness. We find all these methods to be neither sound nor complete. We further investigate the computational efficiency of these methods in terms of the number of invocations of their most expensive routine I.LM evaluation. From a pragmatic perspective, the cost of L.M evaluations is significant, either in terms of GPU resources, expensive API calls to hosted L.Mss. We find that abandoning the soundness and completeness does not provide any benefit
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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in computational efficiency, as the proposed methods are prohibitively inefficient, expensive, and most importantly harming the environment.
We propose an alternative named Thought of Search: thinking before searching, an approach that exploits the strengths of LLM,, while mitigates the weaknesses of the existing approaches, doing in an efficient manner. We propose using the language models for deriving the symbolically represented search components that allow for performing the search itself without calling LLMs. That way, the search space correctness can be checked before the search is performed, allowing for soundness and completeness (and sometimes optimality) of the search algorithms imply these properties of the overall solution. Specifically, we focus on the two essential components of any search, successor generator and goal test, using the large language models to obtain their implementation in Python. We exemplify our proposed approach on four representative search problems, comparing to the LLM- based solutions from the literature that attempt to solve these problems and show these approaches to be prohibitively expensive. We show that by using L.MMs, possibly with human feedback, to produce the code for the search components we can solve the entire datasets with 100% accuracy with only few calls to the LLM. We argue for responsible use of compute resources; urging research community to investigate sound and complete LLM-based approaches that uphold efficiency.
2 Properties and Complexity of Planning using LLMs
In this section, we review the existing methods for planning with language models from the point of view of the soundness and completeness properties of the proposed algorithms, as well as their LM Evaluation Complexity. We quantify the number of LLM requests required by each algorithm. An algorithm sound if it produces only valid solutions and is complete if it guaranteed to produce a solution before terminating successfully. For consistency, we unify the notation here. In what follows, we denote the bound on the number of successors generated per state by b, the number of rollouts by T, and the length of rollout/path by L.
I0 The Input-Output (IO) prompting is the straightforward use of L.Ms to generate an output for a given input. Some examples that employ IO for planning include querying pre-trained model (21 or fine-tune model [7 []. Here, the model is usually evaluated once, generating an output that may include a sequence of steps, making the complexity of this approach O(1). The method is neither sound nor complete for planning, as it can generate incorrect solutions and not guaranteed to provide a solution if one exists. Further, the methods that fine-tune a model can have computationally intensive step of data generation.
CoT The Chain-of-Thoughe approach 22 prompts the model with a predefined example of a chain of steps (thoughts) to resolve the question, in attempt to make the model generate similar chains in its output. Here, still the model is evaluated once, so the complexity is O(1) and the approach is still neither sound nor complete for planning, for the same reasons as before.
The ReAct approach [25] aims at interleaving CoT with acting steps. Each acting step may result in a separate evaluation of the language model. Therefore, the number of evaluations is worst case linear in the number of steps L in the expected output, OL)). This approach is akin to re-planning at each step; or treating LLM as a policy, referenced for each state in the trajectory. Still, no guarantees of soundness or completeness can be obtained for this approach.
ReAct
output
ReWOO ReWOO 223 aims at tackling the inefficiency in the number of LLM evaluations, re- questing the first model evaluation to plan all future steps, then each step is executed without model evaluation if possible (and with, if not), finalizing by model evaluation with the concatenated output of previous states as the imput. The best case complexity is therefore (1). with only two evaluations performed and when external tools are not LLM. The worst case complexity is however still O O(L), where L is the plan's 's length, since each of the L external tool calls can be to an LLM. Same as before, no guarantees of soundness or completeness can be obtained.
RAP Reasoning via Planning (RAP) approach performs a search (e.g., MCTS) using the LLM for expansion (generate successors) and for heuristic or reward prediction of state [7]. Here, the complexity is the worst among the approaches explored so far, being linear in the size of the search

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space. MCTS is an incomplete algorithm, where the search tree size can be controlled by bounding the number of successors generated per state b, number of rollouts T, and their depth L. The overall complexity is O(T x b x T.). Since the L.M is used for generating successors, it can generate incorrect successors, making the approach also not sound.
ToT Tree of Thoughts [44] approach is similar to that of RAP, where the "thought generator" is expanding a state by calling an LLM and the state evaluator is calling an LLM to evaluate the generated states. The evaluation can be done per state (one call per state), or across states (one call across all the current generated states). Then, search is performed on the search tree. The worse case overall complexity is ((N), where N is the number of states generated by the search algorithm. The authors use bounded incomplete versions of the well-known Breadth-First Search (BFS) and Depth-First Search (DFS) algorithms, with the depth bound L, branching bound b, and beam size m, restricting the complexity to O x m x L). Here as well, the successor generation by LLM eliminates soundness of the otherwise sound algorithms.
Graph of Thoughts (GoT) [1] is similar the tree of thought approach, except that it can connect the thought nodes in the tree above into a graph for more flexible representations through (a) aggregation of thoughts a, (b) improvement of thought to refine it further i, (c) generation of a thought based on existing ones In GoT, in addition, repeat operation k defines how often a particular operation repeated. The complexity is similar to that of the ToT, O(N), where N is the number of nodes in the graph generated, in practice bounded by the branching bound b and the depth bound L,, resulting in the overall complexity of O(bx L). Here as well, there are guarantees of soundness or completeness, as the graph is constructed with the language model.
GoT
Reflexion Reflexion [9] is a feedback incorporated approach where in addition to so-called actor model, which can be either CoT or ReAct approach, it has an internal and external feedback component, named an evaluator model and a self-reflection model. The feedback is given within a bounded number of trials and a bounded memory that provides additional context to the agent. The number of evaluations is dependent on which actor model is used plus a constant (two evaluations, one for the evaluator model and one for the self-reflection model) times the number of trials, T. So its either (T) if CoT is used, or O x T) if ReAct is used. While the authors report 100% accuracy given large enough T in some domains, the approach is not complete approach. Similar to the other cases, the approach is also not sound.
LATS Similarly to 77. Language Agent Tree Search (LATTS) 28 performs a Monte-Carlo Tree Search (MCTS), using language models as algorithm components. Using notation of b for the number of successors generated per state, T for the number of rollouts, and L for their depth, the algorithm complexity here as well is O x b X L). The authors note that in practice, however, their method produce more calls to the LLM than RAP, exploring larger portions of the search space and producing somewhat higher accuracy solutions. As mentioned before, MCTS is already an incomplete algorithm by itself, and LATS limits it further. Since the successors here as well are generated by language model, the approach is not sound.
AoT Algorithm of Thoughts (AoT) [18] combines the CoT and the ToT approach in a following
sense: rather than a standard CoT prompting, incorporates the search strategy (DFS or BFS) within the prompt through the in-context examples. That is they use standard DFS or BFS to produce the in-context examples. The approach neither sound nor complete, as it relies completely on the language model. Further, it assumes an existence of search component that can solve the in-context examples. If such component already cxists, the only possible reason to use this approach would be if the search component is not efficient, which would imply that the problems are too large to be handled by BFS/DFS (c.g., above 10 states). But then, these search spaces would certainly be beyond any imaginable future capabilities of LLMs.
While technically AoT makes one call to the LLM per problem instance, the input and output number of tokens grows exponentially with the instance size. Therefore it would be misleading to say that the complexity of this approach is (1). From the cost perspective, when charged based on the number of input and output tokens, it might well be more expensive than the other approaches. Since we do not have an adequate uniform way of transforming this single call to multiple calls of similar size to other approaches, we skip this approach in our analysis.
3

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LLM-Modulo Framework The framework suggests to perform planning by so-called Generate- Test-Critique loop [TO]. The LLM is generating candidate plans, which are validated/critiqque by external means, providing feedhack to the I.I.M These means are assumed to be sound, thus the overall framework is claimed to inherit soundness guarantee. It is not mentioned, however, how to obtain such means of sound validation or critique for cases where no symbolic model already exists. Once critiques deemed a plan valid, the algorithm stops. Otherwise, next candidate is generated. There is no guarantee of convergence towards valid plan and therefore the algorithm is not guaranteed to terminate. This can be fixed by employing a bound on the number of iterations. This, however, does not provide completeness guarantees.
Policy generation with LLMs Focusing on problems where no search is required (e.g.. polynomial cases) and general policy (or generalized plan) exists, the authors propose querying LLMs to provide a policy implementation in python directly [20]. The policy is then checked on a few held out problem instances from the domain, providing feedback on problems that are not solved by the policy. The number of calls to the LLM is therefore O(1) per domain, and the result can be amortised among all the problem instances of that domain. This approach, while tackling the simpler case of problems where no search is needed, has served as an inspiration for our proposed approach.
3 Proposed Approach
In this work, we argue for an efficient and responsible use of compute resources. First and foremost, this means using the computationally expensive large language models efficiently and appropriately. In the case of solving search problems, as in the focus of this work, we argue against using an LLM at each expansion and evaluation. Such implementation of search algorithms is inefficient, but equally importantly, it also sacrifices important properties that search algorithms possess, such as soundness and completeness. While completeness is often sacrificed knowingly, in order to limit the number of possible calls to the language model, soundness can be lost unintentionally. If the state successors are generated by large language model, there is no guarantee that the produced successors are valid. An additional mechanism that validates the produced output would be required to render the algorithm sound. Such a mechanism would be symbolic by nature, since it must guarantee correctness. However, if such mechanism exists, it may be possible to use that mechanism to produce successors, without the need for performing the calls to a large language model each evaluation. The large language models would be a prime candidate for producing such a mechanism. The mechanism can be a symbolic planning model, e-g.. Planning Domain Definition Language (PDDL), from which all the search components can be computed, as is done by classical planners that perform heuristic search 9 Alternatively, large language models can also directly produce the search components code: a successor function, a goal test, and even heuristic function
[8] reward. While the former approach has been explored in the literature 6 16. here we focus on the latter. We propose to use large language models for obtaining Python implementation of two critical search components, successor functions and goal test. We query the language model for each component separately, using textual description of the problem at hand. We assume the process to be iterative and user-guided, by a person capable of validating the code obtained, leaving automation of the process for future work. It is worth noting that if the expensive evaluation of large language models is not performed during search, there is no need to artificially restrict the algorithms to their incomplete variants. Still, the code must be validated for soundness, ensuring that all produced successors correct, as well as completeness, ensuring that all possible immediate successors are produced.
4 Experiments
We exemplify the proposed approach with a variety of popular search problems, demonstrating the use of large language models to obtain the search components code for representative set of the same search problems that the vast recent work on planning using LLMs used. Our intention is to exemplify the efficiency and accuracy gap created by the unsupervised use of large language models at every search step. For each search problem, we count the number of interactions with the language model to obtain valid") successor function and goal test implementations. We then implementation of either a BFS or a DFS algorithm with these two functions on a set of instances
standard
Valid from the perspective of software developer who understands the problem solved.

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User: The Game is a mathematical card game in which the objective to way to manipulate four integers so the end result 24. The game played with list of four numbers, and the player must use all four numbers exactly once, using any combination of addition. subtraction. multipiccation. or division, to arrive at the number 24. state delined by list of numbers (4 or less), how are successor states defined! Provide only the pythom code that returns list successor stales for state.
CharGP: Below the Pythen code that defines how to generate the successor states for given state the 24 Game. state the 24 Game list of numbers. The successur stales generated by laking two numbers from the state, applying operation (additioon, subtraction, multiplication, or division if the divisor isn't 't zemo), and then forming new list with the result and the remaining unused numbers.
import itertcols
def senerate__uccosaor.. aucceasor atatesssttta
succassora = [] (anatt)
for (i, 3) in centtoole..ombiintiionnn
(a), 2):
a, state(i), state[j] remaining [state for k in ranga(n) if & t= i and lt !-
8 Generate all possible new states
auceessors.appand (romattngg + [a + bl)
auccessore wappoorreeppppeeeeeeeeeaaa if a b:
seeessorr.apeedddemaiiinnng [a b])
else:
siccessore..ppenddeee
+ [b- al) nuccesaoss.apenddrennna + [a / bl)
if !-
successore. successore.appendcreeee appendennnnnn + [b / a])
return SucCesSors
This function takes state list of numbers), iterates all pairs of numbers, applics the operations to these pairs. and then forms lists with the result of the operation and the other numbers that were not used in the operation. This emsures that each successor state is valid step towards finding solution reach using all four numbers exactly along with any of the allowed operations.
User: Provide only the python code that lests whether a state is goal state.
Here's 's the Python code that tests whether given state is goal state in the 24 Game. goal state is achieved if the list contains exactly one number, and that number is 24.
CharGP:
def is is_gool__e etttttttee (state): return (etttt)
state
24
This function checks the longth of the state list and whether the sole element the list the number 24.
Listing 1: 24 game example correspondence to obtain the successor function and the goal test.
and report the accuracy and the total time solve the instances. We repeat the experiment 5 times, obtaining 5 implementation variants and compare the efficiency of these variants in terms of total time. Note that if the implementation of successor function and goal test obtained from the large language model is correct, the accuracy is guaranteed to be 100% (if sufficient resources are given to the search). The opposite is not necessarily true, as the set of tested instances might not include some corner cases that make the code perform incorrectly. However, we see that as one of the strengths of our approach, as such cases can be used for testing and debugging the obtained code. Under the assumption of correctness of the successor function and goal test implementations, our approach inherits the soundness and completeness guarantees of the search algorithms used [4]. In all our experiments we use the GPT-4 model [14], in chat mode. The code obtained from GPT-4 in all 5 correspondences provided in the appendix. The search with the obtained code was run locally, on a machine with 2.3 GHz 8-Core Intel Core 19 CPU, no GPUs were used.
24 game The 24 game (4 takes 4 integers as input that can be manipulated through the four most common arithmetic operations: addition, subtraction, multiplication, and division. The goal of the game is to produce formula (also representable by sequence of operations) that evaluates to 24, if one exists. As mentioned before, (4 achieve that with an incomplete algorithm that performs se 100 calls to the LLM, according to their report. The authors report a success rate of 75% subset of 100 medium difficulty problems out of a collection of 1362 problems (24).
As mentioned before, we propose instead to use the LLM to produce the code for two functions: successor function and goal test. We leave the heuristic function to future work, noting that even without it, the search space of the 24 game is rather small, up to + 36 648 + 3888 = 4573 states. I Listing T describes an example correspondence with the L.L.M We observe that the model required 1.2 interactions on average for the successor function and 1 interaction for the goal test. We see
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that, with minor help from the user, the model is able to produce correct successor and goal test functions for this problem. We further tested all 5 successor functions and goal test within a simple BFS implementation on the entire suite of 1362 instances from ToT 24 For each of the 5 tested successor functions, BFS found a solution the 1361 cases where a solution exists and report that no solution exists in the only one unsolvable case, a 100% success rate. The total time to solve all 1362 problems varies over these 5 cases from 1.92s to 6.83s in our naive BFS implementation, hinting that some successor functions can be more efficient than other. This is comparable to single LLM evaluation time; which is R 78 for GPT-4 Chat [15]. Note that the generated successor functions are generic enough to be able solve the generalized version of the 24game, Countdown [3], with only minimal adaptation to the goal test.
Mini crosswords The mini crosswords 24 is a 5x5 crosswords dataset that includes 20 games, where the input describes the horizontal and 5 vertical clues and the output is the full 25 letters board. We used GPT-4 to produce the two functions: successor function and the goal test and repeated the experiment 5 times. The correspondence with the model can be found in the appendix. We observe that the model required 2.4 interactions on average to produce a valid successor function, and 1.4 interactions on average to produce the goal test, with errors primarily related to not handling the corner cases of incorrect input. In all cases, after providing the exact error to the model, it added safeguards which fixed the issues. We tested the obtained functions within a standard implementation of a DFS with a closed list on the entire suite of 20 games [24]. As our focus in this work is on the search aspects, we assume that each clue in each crossword has 10 variants of possible answers, including the correct one, All 20 games were solved by all five generated pairs of implementation of successor function and goal test (100% accuracy), with a total time for all 20 games varying from 5.5s to 346s, yet again signifying the importance of efficient implementation of the successor function.
To compare to the ToT approach, with the bound of 100 on state expansions, it calls the LLM = 200 times in the worst case for each of the 20 games. To be fair the ToT approach, it does not assume the availability of possible answers, deriving the candidates, at each step, with the help of the language model. This, however is mixed blessing, since the language model does not provide a guarantee of including the correct answer among produced variants.
BlocksWorld Probably the most famous planning domain is BlocksWorld, where the blocks can be picked up from the table, put down on the table, unstacked from other blocks or stacked on other blocks to transform the initial configuration planning model, described in PDDL and it is one of the tasks considered by the reasoning-as-ppa saatinggss-laaane asllanning approach (7]. As in the previous cases, we use the GPT-4 model to obtain the successor function and the goal test implementation in Python. We use the same textual description of the domain as 77 A correct successor function and goal test are obtained after 2.8 and iterations on average, respectively.
given goal configuration. The domain has known
The mistakes GPT-4 makes when producing the code repeat from one experiment to another, and are often easy to fix inline, without additional iterations (e.g., using shallow copy when deep copy is needed). In our experiments, however, did count these as additional iterations. In order to evaluate the obtained functions, we used them within a standard implementation of a BFS and experimented with the collection of 502 instances from 77]. All 502 tasks were solved by all five generated pairs of successor function and goal test (100% accuracy), with the total time for all 502 tasks varying from 0.56s 9.7s. The more time efficient approaches represented state as a set of strings representing boolean variables (or logical predicates), while the less efficient representation used dictionaries of lists of objects or pairs of objects with predicates as keys. The simpler state representation also resulted in more efficient duplicate detection in our rather naive implementation.
Note that the accumulated number of expande and generated states in the most efficient case (with duplicate detection) was 50143 and 129408, respectively. If we needed to call the GPT-4 model on each expansion and generation, by the most conservative estimation, would take approximately 14 days and cost over $ $1000, while not guaranteeing correctness of the outcome. Note that the instances considered are quite small- they have only to 5 blocks. Larger Blocks instances would require significantly more resources. This is true for the proposed approach as well, and larger instances mean larger state spaces, becoming too large for an uninformed search such as BFS. In such cases, a heuristic function may be of help, either, similarly, implemented in Python or automatically derived
A state is expanded if its successors are generated.

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User: Logical deduction is performed by applying rales with each rule application. A rule if X then Y is applicable if we know X. If state is defined current knowledge of the world, bow are successor states delined, given a collection rules? Provide only the Python implementation of successor states lanction. CharGP:
then our current knowledge world, extending our knowledge
def succassor states (carrent__tttatttttee rules):
Generate all successor states from the current state applying applicable rules.
Arge:
current, staae (set): A set of facts known in the current state. each tuple represents rules tuples): uhere conditioon" can be aingle fact or a set of facta.
form (conditions, conclusion),
Beturne:
1ist of sets: 1ist of all possible successor states. n**
aev_states -
for conditions, conclusion
A Check sf the rule's condstions net the current state ionntaanoeeooiiiiiio set):
if conditions. iesubeet
Apply the ruler create new stale with the conclusion added new_state curreat_stater strreattetatt.snion.. state.uuioo(ccaccuusio nes_states. www.=aatoc.appomd(eeccccass
If conditions in aurrent_atate:
sea_state current seteent__tttt_eeeeeee www.aattosssppendeere..tttt
return bev_states
User: Provide python cade that target fact, whether state goal ChutGP:
def poal_stateestattee targot_fact):
u**
Check if the state contains the target fact, indicating it 1a goal state.
Arge:
state (set): The current state repreeented as a aet of facts, target_fact (str):
defines the goal state.
Raturns:
bool:
True the target fact is in the state, False othervise.
return sarget_fact in state
Listing 2: PrOntoQA example successor states function and goal test (non-code text removed).
from PDDL representation, which in turn may be obtained with the help of large language model [6]] It is also worth noting that BFS not only guarantees that the obtained solution correct, but also guarantees that it optimal, while LLMs can never provide such guarantee checking whether a given solution is optimal is as hard as finding an optimal solution.
PrOntoQA Logical reasoning can be viewed as a search problem of finding a sequence of logical rules that when applied to the known facts, derive or disprove the target hypothesis. Previous work applies Monte-Carlo Tree Search (MCTS) with successor function and rewards obtained by calling an LLM, to examples from the PrOntoQA dataset to derive the answer but also the proof, a sequence of reasoning steps (7]. The authors report performing 20 iterations for MCTS and 20 samples for self-consistency, resulting in 94.2% correct answer rate and 78.8% proof accuracy.
Similarly to the previous cases, we have generated the successor function and the goal test with the help of GPT-4 model and obtained the answer and the proof by running BFS most twice per question: once trying to prove the positive hypothesis and, if not successful, once more trying to prove the negation of the hypothesis. The search, if successful, returns path from the initial state to a found goal state, which corresponds precisely to the sequence of applications of the reasoning rules the proof. We performed the experiment times, with language model being able to produce a correct successor function and goal test after 1.6 and iterations on average, respectively. We tested the obtained pairs of functions on the entire collection of 4000 questions generated by [7]. All 4000 questions were answered correctly, with all generated proofs guaranteed to be valid, resulting in
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100% accuracy for both, with the total time for all 4000 questions varying between 2.16s and 2.53s. Listing[2] shows an example correspondence with the model, with non-code text removed for brevity.
5
Discussion
It is hard to overstate the importance of the ability to solve search problems and it is natural to solve these problems by exploring some portion of their state space. All the methods we discuss here do that in one way or another. The difference between those approaches is how big is the portion of the explored state space and what is the cost of exploring that portion. As all the discussed approaches are greedy, unsound, and incomplete, the accuracy of the solutions they provide can only be evaluated experimentally for particular problem and dataset, and the results do not reflect other problems or even datasets within the same problem.
In this section we perform thought experiment. We ignore the accuracy of the approaches and only compare their estimated cost and the portion of the search space they have the potential to explore. For each approach we estimate two values. First, we estimate the number of calls to a language model performed if the approach was run on the four datasets from the previous section, as proxy for the cost of using the approach. Second, we estimate the number of states explored during search, as a proxy for a portion of the state space size. The actual state space size for these datasets can be calculated precisely, and we provide these values. Note that the number of states explored is an over-estimation for the portion of the state space explored, since the same state can be explored several times. This is true for sound and complete search algorithms as well. These algorithms often employ a duplicate detection mechanism to prevent re- re-evaluating and re-expanding such states, when such re-expansions admissible but inconsistent heuristics are used). We also provide the actual number of calls to the language model and the number of states explored using our proposed approach. The actual number of calls for our proposed approach is the number of calls that was sufficient for obtaining soundness and completeness (and sometimes optimality). The number of states explored indicated upper bound on the portion of the states space necessary for exploration.
not needed to guarantee algorithm properties (e.g., optimality when
Datasets size We use the same four representative search problem datasets from previous section. The size of each dataset is denoted by D. For the 24 game, D is 1362, for Crossword it is 20, for Blocks World it is 502, while for PrOntoQA it is 4000. We used the datasets provided by the approaches we compare to, but these numbers could also have been significantly larger. For these datasets, have also computed the size of the search problems in terms of the summed number of states over the instances in the dataset. For the 24 game, the number of states per instance is 4, 573 and therefore the summed number of states is 6, 228, 426. For Crossword, the number 10
of states is -i= (?) 10 (10 + 1)10 per puzzle, and therefore the summed number of states our dataset includes 447 instances with 4 blocks and 55 instances with 5 blocks. The instances with 4 blocks have 125 states each and the instances with 5 blocks have 866 states each, giving us the total of 103, 505 states. Finally, for PrOntoQA, the number of states per task varies between 12 and 54, with the total summed number being 97, 608.
is 518, 748, 492, 020. For Blocks
Uniform restriction of the search space Each approach sets limitations to restrict the number calls the language model, which makes difficult to compare. In this thought experiment, we choose to use only the bounds on the breadth/depth of the search and the number of trials, ignoring the absolute constant bounds on the number of explored states used by some approaches. For fair comparison across the approaches, we will use the same parameter value for all, even if the original work used different value. We aimed smaller values among used across the different approaches for each parameter. This results in under ander-approximation of the number of calls in most cases, and possibly would have resulted in an accuracy that is lower than the one reported by the approaches. In this experiment, however, we ignore the accuracy of the approaches. We use the branching bound b = 5, the number of trials T = 10, and the beam size m = 5. The only parameter we set according to the dataset is L. the bound on the sequence length. This is due to the fact that if L is not sufficiently large, the algorithms will not be able to find a solution. In the 24 game L 3 is sufficient, while in Mini crosswords it must be at least 10. In both Blocks and PrOntoQA the length of a plan/proof varies, with the maximal length over the instances in the datasets in Blocks being 16 and in PrOntoQA being 6.
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24Game
Crossword
Blocks
PrOntoQA
Approach
Complexity
States
Calls
States
Calls
States
Calls
States
Calls
10
O(D)
0.02%
1362
4c-9%
20
0.5%
502
4%
4000
CoT
O())
0.02%
1362
4e-9%
20
0.5%
502
4%
4000
ReAct
O(LD)
0.07%
4086
4e-8%
200
7.8%
8032
24.6%
24K
ReWOO
O(LD)
0.07%
4086
4e-8%
200
7.8%
8032
24.6%
24K
RAP
O(TbLD)
3.3%
245K
2c-6%
12K
388% %
482K
1229%
1.44M
ToT
O(bmLD)
1.6%
102K
1e-6%
5K
194%
201
615%
600K
GoT
O(bLD)
0.3%
20K
2c-7%
1K
39%
40K
122%
120K
Reflection
O(LTD)
0.7%
68K
4e-7%
2.4K
77.6%
90K
245%
320K
LATS
O(TbLD)
3.3%
286K
2c-6%
14K
388% %
562K
1229%
1.68M
ToS (ours)
0(1)
27.0%
2.2
3e-4%
3.8
125%
3.8
175%
2.6
Table I: Comparison of various approaches on evaluated tasks. The number of tasks is D, the length bound of the search/rollout/plan I., number of rollouts T, beam size is m, and branching bound is b. The summed number of states over the D) tasks and the projected number of L.M evaluations are given per approach and problem.
Analysis The complexity analysis performed in the previous section does not tell the whole story. It hides the constant multiplier, which in some cases could be significant. We also need additional information about the number of states traversed. Let us take another look at the explored approaches.
Both I0 and CoT only call language model once per instance. Assuming that the output is plan, it traverses L states. ReAct and ReOOO (worst case) make exactly calls to the LLM, exploring L states. Both RAP and LATS perform MCTS, but the actual number of calls to the LLM varies. RAP performs LT + bLT calls (see Algorithm in (7]). while LATS performs 2LT + bLT calls (see Algorithm 1 in 28)). The number of states explored in both cases is bLT. Reflection with ReAct performs (2 + L)T calls, exploring LT states. ToT introduced their own variants of known BFS and DFS algorithms with bounds the branching factor, search depth, but also a limit on the open list size (beam size). While their performance can vary greatly on individual tasks, with DFS typically being greedier and therefore faster, in the worst case they both call the LLM dmL times and explore bm states. The GoT approach calls the L.M b. times, exploring bL states.
Table D shows the estimated numbers according to the parameters values and the analysis above. For comparison, the last row of the table depicts our proposed approach. &States columns denotes the search effort in terms of the number of states explored, possibly with repetitions, described as a percentage of the overall number of states. For our approach, this is the actual value from our experiments. Recall, in the case of PrOntoQA the BFS search is performed once or twice, until the hypothesis or its opposite is proven. For the other approaches, this is an estimate, under the assumptions that the search does not explore the same states multiple times. Clearly, this assumption does not hold in practice, with the methods exploring the same state many times. This is also true for sound and complete search, albeit to lower extent, it can generate the same state multiple times. An important property of sound and complete search is that it generates states systematically. and expands them most once. The number of times the same state is generated tends be higher in MCTS than in BFS/DFS. Non-systematic successor generation is also major contributor to that inefficiency. Having in mind that these numbers are very crude over-approximations. we observe that the investigated approaches explore only very small portion of the search space. This is one of the major sources of their low accuracy - if the solution is not completely included in that explored portion, then there is no way for that solution to be found. In 24 game and crossword, the portion explored tends to be very low, one or two orders of magnitude smaller compared to our method. In Blocks World it is comparable to our method, and in PrOntoQA it is often much higher, order of magnitude larger than the size of the state space, indicating that the same states are explored over and over again.
Looking at the overall number of calls, some methods are more expensive than other. Assuming an average of 500 tokens per input and 50 tokens per output, according to the current pricing of
We do not consider in analysis the length of the input, which is crucial factor of the evaluation cost.

---

GPT-4-turbo (the cheaper of GPT-4 models), the overall cost varies from $40 for 10/CoT and $200 for ReAct/ReWOO, to $14, 000 for RAP and $16, 000 for LATS.
Related work
As the related work on the problem of planning with language models is detailed in Section here review the work related to proposed method.
Generation of planning representation Our work proposes to use LLM to generate python code for successor and goal functions. By using these two functions, along with a search algorithm, one can thoroughly explore the entire state space of problem and compute plan. Recently, LLMs have also been used to generate PDDL [6
II 30J, which can be used with existing planners to search for a plan. Of these work, Guan et al. 66] is quite closely connected to our work as they also use corrective human feedback to generate the PDDL descriptions. While related, our work differs from theirs in three major ways. First, they assume the goal state is provided in the PDDL form. Our work does not make this assumption; rather asks LLM to generate goal function. Second, in their approach action descriptions are obtained one by one (separate calls to LLMs) along with predicates to use. We provide entire description of the problem and asks for a single successor function. Third, their feedback on the generated PDDL is explicit and requires nuance understanding of modeling in PDDL (e.g., "unnecessary precondition "no other object stacked on object ?x").. While ours is mostly generic python code feedback (e.g., issues with shallow copy of dictionary with list values, error trace) and generic logic (e.g., 'two of the operations are not symmetric, division and subtraction').
16
Code generation Our work proposes to use LLMs to generate code that helps solve the search problem, instead of using LLMs solve the search problem directly. Similar idea has been explored by Zhou et al [29] for mathematical reasoning problems. In that, they illustrate that LLMS can be used to generate, execute (verify), and self-refine the python code that helps solve the math problem. Their study corroborate our findings that LLMS can be used for generating verifiable code with some feedback. Additional evidence supporting the feasibility of our approach of using LLMs to generate code comes from the recent advancements in LLM-based program synthesis literature
27 26
7 Conclusions and Future Work
The current trends planning with large language models focus on performing a search when the search components are realized through the large language models. We analyze the existing approaches and show them to be unsound, incomplete, and quite expensive. We propose to use the large language models to instead generate code for these search components, to be reused throughout the entire dataset. The significant differences in the use of computational resources and the performance measures clearly demonstrate that LLMs can be effectively employed for planning purposes without compromising on soundness, completeness, efficiency or accuracy.
For future work, we would like to explore the use of large language models to obtain the code for search guidance or search pruning techniques. But more importantly, we would like to relax the need for human feedback in coming up with valid implementations of the search components. This could be done in ways similar to the way a policy is obtained in generalized planning (20). The first attempt at automating the process was recently made with SutoTooS (2). which replaced the human experts in the loop of feedbacking the language model with an automated feedback, based instead on
unit tests. These unit tests attempt to find evidence of non-soundness and incompleteness and may in turn be automated in the future.
References
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for- paw.llaeeessseeeeeeeaaaaaaareeaa
into-account acc0unt/628388 Feb 2024.
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[20] Tom Silver, Soham Dan, Kavitha Srinivas, Josh Tenenbaum, Leslie Pack Kaelbling, and Michael Katz. Generalized planning in PDDL domains with pretrained large language models. In Jennifer Dy and Sriraam Natarajan, editors, Proceedings of the Thirty-Eighth AAAI Conference on Artificial Intelligence (AAAI 2024). AAAI Press, 2024.
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[21] Karthik Valmeekam, Matthew Marquez. Sarath Sreedharan, and Subbarao Kambhampati. On the planning abilities of large language models- critical investigation. In Proceedings of the Thirty-Seventh Annual Conference on Nearal Information Processing Systems (NeurIPS 2023). 2023.
[22] Jason Wei, Xuezhi Wang, Dale Schuurmans, Maarten Bosma, Brian Ichter, Fei Xia, Ed H. Chi, Quoe V. Le. and Denny Zhou. ofain-of-thought prompting elicits reasoning large language models. Proceedings of the Thirty-Sithh Annual Conference on Neural Information Processing Systems (NeurIPS 2022), pages 24824- 24837, 2022.
[23] Binfeng Xu, Zhiyuan Peng, Bowen Lei, Subhabrata Mukherjee, Yuchen Liu, and Dongkuan Xu. ReWOO: Decoupling reasoning from observations for efficient augmented language models. arXiv:2305.18325 [cs.CL], 2023.
[24] Shunyu Yao, Dian Yu, Jeffrey Zhao, Izhak Shafran, Tom Griffiths, Yuan Cao, and Karthik Narasimhan. Tree of thoughts: Deliberate problem solving with large language models. In Proceedings of the Thirty-Seventh Annual Conference on Neural Information Processing Systems (NeurIPS 2023), 2023.
[25] Shunyu Yao, Jeffrey Zhao, Dian Yu, Nan Du, Izhak Shafran, Karthik Narasimban, and Yuan Cao. ReAct: Synergizing reasoning and acting language models. In Proceedings of the Eleventh International Conference on Learning Representations (ICLR 2023). OpenReview.net, 2023.
[26] Kechi Zhang, Zhuo Li. Jia Li, Ge Li, and Zhi Jin. Self-edit: Fault-aware code editor for code generation. In ACL. (I), pages 769-787. Association for Computational Linguistics, 2023.
[27] Li Zhong, Zilong Wang, and Jingbo Shang. Debug like human: large language model debugger via verifying runtime exccution step by step. ACL,(indiingg (Findings), pages 851-877777 Association for Computational Linguistics, 2024.
[28] Andy Zhou, Kai Yan, Michal Shlapentokh- Rothman, Haohan Wang, and Yu-Xiong Wang. Language agent tree search unifies reasoning acting and planning in language models. CoRR, abs/23 10./2/30004406,, 2023.
[29] Aojun Zhou, Ke Wang. Zimu Lu, Weikang Shi, Sichun Luo, Zipeng Qin, Shaoging Lu, Anya Jia, Linqi Song, Mingjie Zhan, and Hongsheng Li. Solving challenging math word problems using GPT-4 code interpreter with code-based self-verification. In ICLR. OpenReview.net, 2024.
[30] Max Zuo, Francisco Picdrahita Velez, Xiaochen Li, Michael L. Littman, and Stephen H. Bach, Planctarium: A rigorous benchmark for translating text to structured planning languages. CoRR, abs/2407.03322 2024.
12

---

A Search implementation
The search components successor_stacee and goal_test test are plugged into the search algorithms. In this work we used BFS and DFS algorithms, implemented as follows.
def _str(state))
if (alssaance(state dict): return str(state) return (soon (i ((str(s) for s in state))))
reconstruct.plante. Closed)
plan = []
current =
while current None:
_sttr (surrent)
current Closed[c]
return plan[::-1]
def bfs (state, auccessor_atates is_goal):
expanded 0
generated
state
Q [tuple((s, None))]
Closed dict()
while len(Q) > 0:
Get the top from the queue s, parent Q[0] [o], Q[o] [1]
del q[o]
_str(s)
if с in Closed:
continue
Closed[c]] parent if ds_goal(s):
return reconstruut.pplanee, Closed), expanded, generated expanded +- for successor_states((:
Q. append (tupen((tuul(( generated +=
return None, expanded, generated
def dfs (state, auccessor_states is_goal): expanded = 0
generated
state
Q [tuple((s, None))]
Closed dict()
while lQ)(Q) > 0:
Get the top from the queue s, parent Q[-1] [0], [1]
del Q[-1]
str(s)
if с in Closed: continue
Closed[c] parent if is)
return reconstrruc._planGe Closed), expanded, generated expanded for successor_statesse
Q. (tuple((tt)))) generated +- 1 return None, expanded, generated
13

---

B Model correspondences
B.1 24 game
The prompt
successor function as follows.
The 24 Game is mathematical card game in which the objective is to find way to manipulaee four integers that the end result is 24. The gume is played with a list of four numbers, and the player must use all four numbers exactly once, using any combination of addition, subtraction, multiplication, division, arrive the number 24, If state is defined by a list of numbers (4 or less), how are successor states defined? Provide only the python code that returns a list of successor states for a state.
prompt goal test as follows. Provide only the python code that tests whether state is a goal state.
B.1.1 Experiment Run 1.
User: The 24 Game is mathemattcal card game in which the objective is to find way manipulate four integers that the end result is 24. The game is played with list of four numbers, and the player must all four numbers exactly once, using any combination addition, subtraction, multiplication, division, to arrive at the number 24. If'a state is defined by a list of numbers (4 or less), how are successor states defined? Provide only the python code that returns a list of successor states for state.
ChatGPT:
import itertools
def satrasoorsstttesttttaesees if len (statte == 1:
return 0
successors = 0
for i, j in itertools. combinations (range (1en (lentstee)), 2): remaining [state[k] for & in range (len(state) if not in (1, })]
pairs [
state [i] + state [j]
state [i] state [j],
state [j] state [i],
state sjate[j],
]
if state[j] != 0: gairs.appent (state [i] / state(j])
if state[i] 1= 0: pairs. append (state [j] / state [i])
for result pairs: successors. append (remaining [result])
return successors
User: Provide only the python code that tests whether a state goal state.
ChatGPT:
def is is_goallstateeeeeeeeeellelesesee goal_statt(sttate) return len (state) and abs (state [0] 24) 1e-9
B.1.2 Experiment Run 2.
User: The 24 Game is mathematical card game in which the objective is to find a way to manipulate four integers that the end result is 24. The game is played with a list of four numbers, and the player must use all four numbers exactly once, using any combination of addition. subtraction. multiplication, division, to arrive
14

---

the number 24. If state is defined by a list of numbers (4 or less). how successor states defined?? Provide only the python code that returns list of successor states for state.
ChalGPT:
from itertools import permutations, product
def apply_op(a, b, op):
if op = '+': return elif op 1.'; return a
elif op = 1s1:
return
elif op 1/1:
if b != 0:
return /
else: return None
def generate, generate_succesor_.oo auccessor_statessstttaaa if len (state) = 1:
return 0
successors 0
operators ['+', 1-1, ',, 1/']
for a, b in permutations (atate, 2) remaining for in state
x b] eaclude and
for op in operators:
result apply_op(a, b, op)
result not None: successors. sapcessors.appedieee (remaiinng + [result])
return successors
User: Provide only the python code that tests whether a state goal state.
ChatGPT:
def Sa_goal_statesstttte
return len (state) = 1 and abs (state [o] 24) 1e-6
User: the generate_successorr successor states function, two the operations are not symmetric, division and subtraction. Please fix the code
ChatGPT: Here's 's the updated Python code generating successor states that considers the non-symmetry of subtraction and division operations:
from itertools import permutations, product
def apply_op(a. b, op):
if op == '+1 return a +
elif op 1.':
return a b
elif op = 's';
return a * b
elif op 1/1:
if b !- 0:
return a / b
else:
return None
15

---

def Snnerate_successor_sstee (statt);
if len (state)
return 0
successors - 0
operators ['+1, '+', '/']
Iterate over all unique paärs of indices to handle non-comuutative operations len (state)
for in range
for in range(n): if i != j:
a, b state[il] state[j]
remaining = [state[k] for k in range if k != and != j]
for op in operators:
result apply_op(a, b, op)
if result is not None:
new_state remaining [result]
successors. appenedeeeesnnetttttt
# Add the reverse operation if it is 1.' or '
if op in {'-', 1/1}:
result apply_op(b, a, op)
result not None:
new_state = remaining + [result]
successors. append auccessors.apeenddees...cc..
return successors
This code correctly handles the non non-commonee commutativity of subtraction and division by explicitly computing the result for both a and oo whenever the operator is 1-' or '/"
B.1.3 Experiment Run 3.
User: The 24 Game is a mathematical card game in which the objective is to find a way to manipulate four integers that the end result is 24. The game is played with a list of four numbers, and the player must use all four numbers exactly once, using any combination of addition, subtraction, multiplication, division, to arrive the number 24. If state is defined by a list of numbers only the python code that returns a list of successor states for state.
less), how successor states defined? Provide
CharGPT: Below is the Python code that defines how generate the successor for given state the 24 Game, A state in the 24 Game a list of numbers. The successor states are generated by taking two numbers from the state, applying an operation (addition, subtraction, multiplication, or division if the divisor isn't t zero), and then forming new list with the result and the remaining unused numbers.
import itertools
def htt.aaaccccccrrtttteeetttteee. successors = 0
len (staate)
for (i, j) in itertools. comiiattionnsraagee 2):
a, b stateli], state [j] remaining = [k] for & in (n)ge(n) if & 1= i and t=
Generate possible new states
successors. append (remaining + [a b])
successors aapend (remaining + [a bl)
if a b:
successors. successors.appendd append else:
[a bl)
successors. append (remaining + [b a))
if b !- 0: successors appeaadddeppddddeieenennnnnn + [a b])
16

---

if 1= 0:
successors.appenddreeme + [b a])
return successors
This function takes state (a list of numbers), iterates over all pairs of numbers, applies the operations to these pairs, and then forms lists with the result of the operation and the other numbers that not used in the operation. This ensures that each successor state is valid step towards finding solution reach 24, using all four numbers exactly once along with any of the allowed operations.
User: Provide only the python code that tests whether a state goal state.
ChatGPT: Here's the Python code that tests whether a given state is goal state in the 24 Game. A goal state achieved list contains exactly one number, and that number 24.
def is goal_state return len (state) = 1 and state [o] = 24
This function checks if the length of state list is and whether sole element the list the number
B.1.4 Experiment Run
User: The 24 Game is mathematical card game in which the objective is to find way manipulate four integers that end result is 24. The game is played with a list four numbers, and the player must use all four numbers exactly once, using any combination of addition, subtraction, multiplication, or division, to arrive at the number 24. If a state is defined by a list of numbers (4 or less), how are successor states defined? Provide only the python code that returns list of successor states for state.
CharGP:: In the 24 Game, the concept of "state" can be represented by list of numbers, with the game beginning with four numbers. The "successor states" are generated by applying binary operation (addition, subtraction, multiplication, division) any pair of numbers the current state, then replacing these two numbers with the result the operation, which gives a new state with one fewer number. Here's Python code that defines a function to generate all possible successor states from a given state:
from itertools import combinations, product
def generate_successor_r def aaplyyoperation(s (x, y):
(numbers):
results fx + y. y. x, x yy
if != 0:
results. add (y x)
if y != 0: results. add / y)
return results
successors 0
num_len len (numbers)
for (i, j) in combinations (range (num_lem) 2):
for result in opply-operation.e (numbers numbers [j]):
new_state [numbers [k] for k in range (num_len) if k !- and j] new new_state (result)
anw.
return successors
This code leverages combinations from the itertools module to select each pair of numbers from the current state, and then applies every possible operation to these numbers. The amply_operatie function computes possible outcomes for the operations hetween two numbers, including division only when the divisor is not zero. Fach result used create
state, which is added the list of successor states.
User: Provide only the python code that tests whether state is goal state.
ChatGPT: Here's Python code that defines function to check whether a given state is goal state in the 24 Game. The goal state is when the list contains only the number 24.
17

---

def (s__gal__statesstate)) return len(state) and state [o] 24
This function returns True if the list state contains exactly one element and that element is 24, which would indicate that the goal of achieving 24 has been successfully reached.
B.1.5 Experiment Run 5.
User: The 24 Game is mathematical card game in which the objective is to find a way to manipulate four integers that the end result is 24. The game is played with list of four numbers, and the player must use all four numbers exactly once, using any combination of addition, subtraction, multiplication, division, to arrive the number
state is defined by a list of numbers (4 less). how are successor states defined? Provide only the python code that returns a list of successor states for state.
Game, we need consider pairs numbers from the current state and all possible operations between them. After selecting a pair of numbers and an operation, we replace the pair in the current state with the result of the operation to create new state. This process is repeated for every possible pair and operation.
CharGPT: generate successor states for a given state
Here's Python function that returns all successor states for given state:
from itertools import combinations
def successor_ otooessso_sttatesssttta
def apply_operation b):
Generate all possible results of operations on a and b resulta [a + b, a b, a, a * b] if b != 0:
appunda.append
b)
if a !- 0: results. append / a)
return results
successors 0
n (en(staee)
for i, j in combinations (range(n) 2):
# Get all possible results of operations between state[i] and state[j] for result in [pppy_operation(statelll, state(jl): Form new state rep replacing state(t) and state(f) with result
new_state [state [k] for k in range (n) if k !- and !- j] new_state. append (result)
successors (ww.rrnesssppeddeee_eeeeseeeeeee
return successors
This function works follows:
It iterates over all combinations two indices from state.
For each pair, it applies all possible arithmetic operations.
It forms new states by replacing chosen pair with the result of operation and includes other numbers from the original state.
It collects and returns all these states as potential successors.
User: Provide only the python code tests whether a state goal state.
ChatGPT: Here's Python function to check if given state goal in Game. The goal achieved the state contains exactly one number which is 24;
def is_goal_staee stae__sttt(sstaee) return len(state)
and state[O] [0]
This function checks whether the list state has only one element and that element is 24, indicating that the operations the initial numbers have successfully resulted in 24.
18

---

B.2 Mini crossword
The prompt for the successor function is as follows.
The task is 5x5 mini crosswords. Given input of possible answers horizontal clues and vertical clues, come up with successor state function written in python, where state 2D array representing the current puzzle state (initially all empty).
The prompt
goal test as follows.
Can you come up with a goal test function ensuring that all cells are not None and that all words horizontally and vertically match at least one of the answers?
B.2.1 Experiment Run 1.
The task a 5x5 mini erosswords. Given an input possible answers to horizontal clues and vertical clues, come up with successor state function written in python, where a state a 2D array representing the current puzzle state (initially all None). The possible clues for each row and each column are given separately. Here is example:
User:
horizontal. Lanszers
["taaka", "goala", "plana", "agend", "chores", "works", "deeds", "itens", "lists", "brief"], ('notor', "power", "drive", "diesel', "stean", "punpe", "crank", "geare", "turbn", "motor"],
['grand", "artay", "abowy", "ornate", "fancy", "vain", "proud", "vogue", "swank", "Iuxus"], ["venue", "salle", "forun", "atria", "lobby", "parle", "court", "malle", "mall", "lobby"], ["jeer", "scoff", "sleer", "deris", "sneer", "scorn", "derid", 'gibes", "gibed", "flout"]
]
vertical, anewers -
["anass', "stack", "hoard", "pile", "store", "heape", "massy", 'gathe", "lumpe", 'mound"], ["nilga", "goral", "eland", "lepus", "gazal", "Kudu", "oryx", "gau", "iups", "carb"), ("scheme", "deeige", "ettle", "nettle", "eting", "viles", "plans", "ideas", "plots", "cocks"), ('spout", "nosle", "snout", "nouth", "nostr", "ports", "inlet", "vents", "outlt", "beake"), ['driar', "arid", "sare", "parch", 'dryar", "vring", "drear', "sear", "pall", "lack"]
CharGPT: create successor state function 5x5 mini crossword puzzle Python, we'll consider following:
State Representation: A state will be represented by Sx5 grid, where each cell either contain letter or empty (represented by None or an emply string).
2. Input: The inputs the function will be a list of possible answers for horizontal and vertical clues. Each answer exactly 5 characters long since it's 5x5 grid.
3. Successor State Function:
This function generates all possible successor states from the current state. For each empty row or column, the function will try to place every suitable word from the clues into the grid and generate state for each valid placement.
Here is a Python implementation the successor state function for the 5x5 mini crossword puzzle:
def is_ sraliddpplccmmet (grid, word, start_row, start_col, is horizontaa)): an.
Checks if word can be placed at the specified starting row and column.
for in range(5):
if is horizontal:
Check horizontal placement
if grid[start_rov] [start_col i] not in (None, word(i]):
return False
else:
Check vertical placement if grid[start_ron + i] [start_col] not in (None, word(i]): return False
return True
def (orde_dord word, start_row, start_col, horoiizzzna))
Places a word on the grid at the specified position.
19

---

new_grid = [row[:] for row in grid] # Make copy of the grid for in range(5):
if is_horizontal:
new new_grid (startrssttt_rror]] [start_col + word
else:
new now_grid[start___
[start_col] word[i]
new_grid
def successor_ satrssorr_ttttteeeeeee (grid, horizontal, anavers, vertical_answersy)
Generates all successor states for the current state of the grid. a.. successors
Attempt to place words from horizontal, answers in each for r5nge(5):
if all(cell. None for cell grid[rov]): # If row completely emp empty for word in horizontal answers [row]
if us_valid_placemeee (grid, word, row, 0, True):
new_grid place_word(gria word, row, 0, True) successors append (nev_grid))
Attempt to place words from vertical, answers in each co column for col in r5))
if all (gridd [col] is None for in range If lomn is completely empty for word in vertical_answery [col]: if new_grid (aace_orddgrrid word, 0, col, False)
(grid, word, 0, col, False)
successors append
return successors
Emamp le usage
initial_grid [[None]+5 for in range [
borizontal_ansaen -
('tacke", "goals", "plans', "agend", "chores",
"deeds", "itens", "lists", "brief"),
('motor", "power", "drive", "diesel', "stean", "pumps" "crank", "gears", "turba", "notor"), "grand", "artay", "ahouy", "ornate", "fancy" "vain", "proud", "vogus", "swank" "luxus"], ['venue", "salle", "Torun", "atria", "lobby", "parls", "court", "malls", "mall", "Iobby ['jeer", "scoff", "sleer", "derie", "sneer", "scorn", "derid" "gibee", "gibed", "flout")
]
cortical_ansvar.
['anase", "stack", "hoard', "pile", "store", "heape", "massy", 'gathe", "lumpe", 'mound"), ['nilga", "goral", "eland", "lopus", "gazal", "kudu", "oryx", "gnu", "imps", "carb"], ('schene", "deaige", "attle", "nattie", "sting", "kiles", "plans", "Ideas", "plata", "cocks"], ['spout", "boele", "snout', "mouth", "noetr", "ports", "inlet", "vente", "outle", "beake"], ['drier", "arid", "sere", "parch", "dryer", "wring", "drear", "sear', "pall", "lack"]
]
Generate successors from the instial state
successors suceesor_stateeeinitiiio. grid, horizontal, answers, vertical_anavert print("Number ("Number of successor states:", len(successors) for state in successors:
for row in state: print (row) print() ()
Explanation
is plaaemee_lla Checks if the word can be placed on the grid without conflicting with existing characters.
place_word word:
Places the word on the grid and returns the new grid state.
succossor_states: Iterates over each row and column to find empty spaces and attempts to place cach cluc word, generating new successor states.
This function will give you all the potential successor states by attempting to fill each empty row and column with every possible word from the clues. Note that this example assumes the initial state has conflicting
20

---

pre-filled characters and does not handle partial word placements or overlaps, which could be extensions to this basic successor function.
User: Can you come up with goal test function Come with goal test function ensuring that all cells are not None and that all words horizontally and vertically match at least of the answers?
CharGPT: Certainly! To create goal test function for the 5x5 mini crossword puzzle, need to ensure that:
No Empty Cells: There should be no cell the grid that contains None.
2. Valid Words Horizontally: Each row must contain word that is present in the list of possible horizontal answers for that specific row.
Valid Words Vertically: Each column must contain a word that is present the of possible vertical answers for that specific column.
Here is how you could implement such function in Python:
def goal_test (grid, porizontal_answert vertical_answers) ...
Check if the current grid state is goal state.
Check for any None values in the grid if any (None row for row in grid): return False
Check horizontal words against possible answers for row range(5):
row_word "ooiingrrd
if row_vord not in anarees [row] return False
Check vertical words agaénst possible answers for col in range(5):
col_word " ''jjon [col] for row in range(5)) if word not in vertical, aaaiccallaaaeersaaaa [col] return False
return True
Enamp le of usage
horizontal_ sesiiontal__esveee
L'tasks", "goals", 'plans', "agend", "chores", "sorks", "deeds", "itens', "lists", "brief"), I'notor, "power", "drive', "diesel', "stean", "pumpa" "crank", "gears", "turbn", "notor"], "grand", "artay", "shouy", "ornate", "Tancy", "vain", "proud", "vogue", "suank", "Iuxus"], ['venue", "salle", "forun', "atria", "lobby" "parls", "court", "salle", "mall", "lobby"], ['jeer" "scoff', "sleer", "deris", "sneer", "scorn", "derid" "gibes", "gibed", "flout"]
vertical_anewen
['amass", "stack", 'hoard', "pile", "store", "hasps", "mansy", 'gathe", "lumpe", 'mound") ['nilga", "goral", "eland", "lepus", "gazal", "kudu", "oryx", "gnu", "inpa", "carb"], ['schene", "design", "ettle", "nettle", "sting", "wiles", "plans", "ideas", "plots", "cocks"), ['spout", "nosle", "enout', "mouth", "noetr", "ports", "inlet", "vente", "outle", "beake"), ('drier", "arid", "sere", "parch", "dryer", "vring", "drear", "sear', "pall", "lack"]
Assume a sample grid which is a valid goal state sample_grid [[ ["t", "a", "s", "k" "s"], ["m", "o", "t", "o" "x"]. ["s", "s", "a", "n", "k"] ["m", "a", "1", "1", "s"], ["d", "e", "r", "i", "s"]
1
Test the goal test function is_goal (asteellesampl__e horizontal anstess)_answers) print ("Is this goal state?", is_goal)
21

---

Function Explanation:
The goal_test function iterates over rows and columns. For rows, concatenates the elements form a word and checks against list possible answers for that row. It repeats similar process for columns. If any word doesn't match the possible answers or if there's s any None value in the grid, the function returns False.
If all checks pass, it returns True, indicating that grid in goal state.
User:
For the successor state generation am getting the following error:
Traceback (most recent call last):
File ///gpt4_successors..ppp"" line 70, <module> successors
sttprssoor_sattteeeiieeee .grid,, porizontal_ansvers vertical_answers File ". "./ggtt__auceesoorsi.y
line 38, in succesaor_stater
if is_valisd_placcment (grid, word, row, 0, True):
File " ///ppp4_succeesoorss..p
1ine 8, in is_valid.placement
if grid[start_row/ [start_col i] not in (None, word(i)): string index out of range
IndexError:
ChatGPT:
The error message "IndexError: string index out of range" typically indicates an attempt to access an index that does not exist in string. This issue often arises when working with grid-related operations and is usually due of two reasons your crossword context:
1. Mismatch in Word Length: Since grid cells are designed for 5x5 crosswords. every word used as clue should be exactly 5 characters long. If any word is shorter longer than this, you'll Illl encounter this eтror when you try place it the grid.
2. Out-of-Bounde Access: This happens if the code attempts to place a word in position where it doesn't 't fit entirely within the grid's boundaries, although based your function setup (starting placements row 0 and column horizontal and vertical placements respectively) and the checks you seem have intended, this shouldn't normally occur unless the word length is incorrect.
How to Fix:
Verify Word Lengths: Ensure that all words provided in the horizontal answers and vertical answers lists are exactly characters long. You can add validation step or filter out incorrect entries.
Debugging Additions: Include checks your valid placement function to ensure you don't run into out-of-bounds emors. This more of a safeguard than solution to the current issue but is good practice.
Here's how you might modify your code include these fixes:
def is_ ss_valid_ppl www.liidaaccmeettteem (grid, word, start_row, start_col, is horiooozzotl)
Checks word can be placed at specified starting row column.
if len (word) != 5:
Ensure the word is ezactly 5 characters as expected for 525 grid return False
try:
for in range(5):
if is horizontal:
Check horizontal placement
if grid[start_ron [start_col i] not in (None, word (i]):
return False
else:
Check vertical placement grid[start_rov [start_col] not in (None, word [i]): return False
return True except IndexError:
22

---

return False # Safeguard against vat-of-bounde access
Additionally, a check when preparing data
horizontal_ansve answers [word for word in 1st if len (word) -- 5] for 1st in horizontal_ansver
vertical [word for word in 1st len (word) for 1st vertical_ answers
]
Verify the Data: Before running your grid operations, it's beneficial confirm that your input data as expected:
Emample le of verifying the word lengths
(len (word) 5 for row in horizontal, annwers for word row) and all (worr)) 5 for col in vertical_ vertical_answers answers for word in col)
This code snippet ensures that every word both the horizontal and vertical clues exactly 5 characters long. If this assertion fails, you will need to revise your input data to meet this requirement. This helps prevent runtime emrors related to string length mismatches in fixed-size grid context like this.
B.2.2
Experiment Run
User: The task is 5x5 mini crosswords. Given an input of possible answers to horizontal clues and vertical clues, come up with successor function written in python, where state 2D array representing the current puzzle state (initially all None). The possible clues for each row and cach column are given separately, Here is an example:
horizontal, enswers
["tacks", "goals", "plans", "agend", "chores", "works" , "deede", "iteme", "lists", "brief"), ["notor", "pover", "drive", "diesel', "stean", "pumpa" "crank", "gears", "tarbn", "motor"], ['grand", "artay", "showy", "oreate", "fancy" "vain", "proud", "vogue", "ewank", "luzue"), ["venue" "salle", "forun" "atria", "lobby". "parls", "coure", "malle" "mall", "lobby"] ['jerr" "acoft", "slear", "daris", "sneer", "scorn", "darid", "gibes", "gibed", *flout"]
]
vertical Lanserrs
["anana", "stack", "hoard", "pile", "store", "heaps", "massy", 'gathe", "lumps", "nound"], ('nilga', "goral", "eland", "lepus", "gazal", "kudu", "oryz', "gnu", "inpe", "carb"), ("schemo", "design", "ettle", "nettle", "sting", "wiles", "plans", "ideas', "plots", "cocks"), ['spout", "nosle", "anout", "nouth", "nostr", "porta", "inlet", "vents", "outit", "beaks"], ["drier", "arid", "sere", "parch", "dryer", "vring", "drear', "sear", "pall", "lack"]
]
Note that some of the possible answers are not exactly 5 character long.
ChatGPT:
To approach creation of successor state function for this mini crossword problem in Python, let's consider a few key aspects:
State Representation: We'll represent the state the puzzle as 2D list characters. Fach cell in the list can either be specific character or empty (denoted by a space or some placeholder).
Initial State: The imitial state will have all cells empty.
3. Successor State Function: This function will generate all possible states that be reached from the current state by making single move. "move" in this context would involve placing a word in row or column that fits with the already placed characters.
4. Constraints:
Words must match the allowed words in horizontal answers for rows and vertical answers for columns.
Any placed word must conflict with characters already placed in intersecting rows or columns.
Given the constraints and specifications, we'll proceed follows:
Successor State Function: a given state, iterate over each row and column, trying to place each possible word from the respective lists. Generate a new state for each valid placement.
Here's simplified Python implementation of this successor state function:
23

---

def we_d (grid, word, row-Nooe,, col=Ilone):
Check if a word can placed the specified row or column.
unn
len (grid)
if is not None:
for range (n):
grid[row] [j] not None grid[row] [j] word[j]:
return False
elif col is not None:
for in range(n):
grid[i] [col] not None grid[i] [col]
word[i];
return False
return True
def place_ (grdd,, word, row=lione, col=None)): aun
Place word at the specified row or column.
new_grid - [row[:] for in grid] Copy the grid
if row not None:
new_grid[rov]
(ist(word)
elif col is not None:
for in range (len(word)): new_grid[i] [col] word
return new_grid
def generate, successors (state, horizontal, answers, pertical_answergg
Generate possible successor states from current state. n.n
n len (state)) successors 0
Try placing each word each row for in range(n):
for word in horizontal, [ns_errss]
if ds_valid_ word_ paarddeeaammenttttm (state, word, row=i); new_state place, word(state, word, row=i) successors.o wetssors.appere append (newsstaee) (ttta
Try placing each word in each co culmm for in range(n):
for word vertical vertical_answers[]]]
if is is_valida sordidd_vrd__aaceeeeee (state, word, col-j): new_state = woree word, col-j)
successors append
return successors
Écample le usage
initial_state [[None]+5 for in range (5)]
generate_ succrasoos (initial, state, horizontal, ansvers,, pertical_answera
Print first few successor states for demonstration for in successors[:3].
for in print .join (row))
print("-..")
This code assumes answers 5 characters long to match the 5x5 grid exactly. words that not characters, additional preprocessing would be necessary to either filter them out or adjust the grid size accordingly. Also, this code only illustrates the initial placement of words without handling intersections; further logic is needed ensure words do not contradict each other at intersecting cells.

---

User: Can you write goal test well?
ChatGPT: Cortainly! A goal test function for the 5x5 mini crossword puzzle would determine a complete and valid crossword has been formed. Here are the criteria for complete and valid crossword:
1. Full Grid: All cells in the grid must be filled with valid letters (no empty spaces allowed)..
2. Valid Words: All words formed in rows and columns must match one of the possible answers given for that specific row or column.
Let's write Python function that checks if given state meets these criteria. This goal test function will:
Verify that grid cells are filled.
Check that every word in each row is in the respective list from horizontal answers.
Check that every word in each column is in the respective list from
Here the implementation:
def (sccompete((staae))
Check if cells in the state are filled with non-space characters.
return all (stateel [j] is not None for in range(len(state)) for in range(len(state (i])))
def is soluuiion (state, horizontal answers, vertical_answers)) aun
Check if the current state is a complete and valid solution.
(en(stt
Check if all rows contain valid words for in ra)ge row_word " [i))
if row_word not aorizontal_ansverss return False
Check all columns contein valid words for in range (n) col_word foiin(statt(e]]]] for ra)g
if col_word not in ansver]ll return False
return True
def geal_test (state, horizontal_ vertical ansvers)))
Determine if the given state is goal state.
if not is_ comppeeee
return False return is_valid_eollneee saaliddsohutioonetaate horizontal_ answers, vertical_ansvers,
Example le usage
Assume "final_state" is a hypothetical fully filled grid you want to test f final_state
['t', lal, 'sl, ik', 's'), ['m', 'o' 't', '0', 'r'], ['g', 'x', aa. 'n', 'd']. ['v', 'e', 'n', lu', 'e']. D'j', le', tel, 'r', 's']
1
Check if the final_ state is a goal state is_goal goal_test wwwtteefftnnttteeeeeeette horizontal_ansvee answers, aesticsl_ansvers) print ("Is the final state goal?", is_goal)
25

---

This goal test function relies on final state being fully filled and accurately matching the corresponding answer lists. This function be used in puzzle-solving algorithm to check the current state of the board is the solution.
User: am getting
"new [new_grid[i] [col] word IndexError: 1ist index out of range"
The error you' re experiencing suggests that there's a mismatch between the indices being accessed and the size of the grid or word. This usually happens if the grid is initialized incorrectly, or if the word doesn't t fit exactly into the intended per word), any deviation from this can cause an index out of range emoor.
CharGPT:
column. Since the code assumes each word perfectly fits 5x5 grid letters
Let's address by ensuring that:
The grid and the words are correctly aligned in size.
2.
The placement functions are handling indexing safely.
I'll revise the place_worl word function
it correctly handles the dimensions of both and columns:
def (are_word word, row=None, col col=None):
Place word the specified uun
column.
new new_grid = [list (r) if r is not None else [None]*5 for r in grid] Safely copy the grid
len (grid) if row is not None:
if (word) !- n:
Return original grid if word length does not match, preventing indez error return grid
new new_grid[row] [row] = list(word)
elif col is not None:
if len (word) !- n:
# Return original gród if word length does not match, preventing indez error return grid
for in (n)g
new nev_grid(i] [col] word[i]
return new_grid
This version of place_wooe word adds checks ensure that words are only placed if they match the expected length, preventing indexing errors.
Additionally, ensure that the initial grid is correctly populated with None values in each cell:
initial_state [[None] +5 for in r5)]
you are still encountering issues, please double-check that:
All words used in Norizontal_answer answers and vertical answers are exactly characters long.
The grid dimensions are strictly 5x5.
If you need to handle words that aren't characters long, you'd 'd need additional logic to either skip those words adjust the grid accordingly. If you't working with dynamic sizes irregular word lengths, the implementation would require more complex adjustments manage these variations.
User: am now getting this emor:
File " "//gpt__suuu successors2 py"" 1ine 13, is_valid word_placemen if grid [col] is not None and grid[i] [col] !- word[i); IndexError: string index out of range
ChaiGPT: The error "IndexError: string index out of range" in the is valid word. placemeet function suggests that there's attempt to access index in the word that beyond its length. This occur if:
26

---

1. The length of the word being checked does not match the expected column height (which should be 5 for your 5x5 grid).
2. The column index exceeds the length of the word.
To resolve this issue, we should verify that:
All words used for placement rows and columns exactly characters long (as your grid 5x5 grid).
The checks within is_ ss_vvlid____ word placement are safely accessing characters in words.
Here's an updated version the is_ valid___w ___vaidd_wor__ppppp word_placement function with added safety checks:
def is is_valid_word wordd vorlicmeord (grid, word, row=None, col=Nonee):
Check if word be placed the specified column.
Enaure the word the correct length and fits the existing grid values. uun
n len (grid) if len (word) != n:
return False # The word does not füt the expected dimension
if row is not None:
for in range (n)
if grid[rov) [row] [j] is not None and grid[row] [j] word[j]:
return False
elif col is not None:
for in :n): if gri [col] is not None and grid[i] [col] != word[i]:
return False
return True
This modified function includes a check to ensure the word length matches the grid dimension (5 for your 5x5 grid) before proceeding with any further logic. This should help prevent the "IndexError" you encountered by immediately returning False if the word length is incorrect.
Please ensure that:
All words horizontal annsers and vertical answers are exactly 5 characters before using them in your code.
The place_ word function is also safely handling word placements, particularly in the columns.
These changes should help climinate the index errors by ensuring accessed indices are within the valid range for both the grid and the words. If you have any additional needs or run into further issues, feel free to ask for more assistance.
B.2.3 Experiment Run 3.
The task is a 5x5 mini crosswords. Given an input of possible answers to horizontal clues and vertical clues, come with successor state function written in python, where a state is a 2D array representing the current puzzle state (initially all None). The possible clues for each row and each column are given separately. Here is example:
User:
horizontal.
["tacks", "goals", "plans", "agend", "chores", "sorks", "deeds" "itene", "lists", "brief"], ['notor', "pover", "drive". "diesel', "stean", "pumps" "crank", "gears", "carbn" "motor"]
['grand", "artay", "shouy", "ornate", "fancy", "vain", "proud", "vogue", "suank". "luxus"], ["venue", "salle", "forun", "atria", "lobby", "parls", "court", "malls", "nall", "lobby" ["jeer", "scoff", "sleer", "derie", "eneer", "ecorn", "derid", "gibes", "gibed", "flout"]
]
vertical ansxers
["anass", "stack", "hoard", "pile", "store", 'heaps", "massy", 'gathe", "lumps", "mound"], I'ilga"" "goral", "eland", "lepus", "gazal", "kudu", "oryx', "gau", "impa", "carb"], ["scheme", "deeign", "ettle", "nettle", "sting", "viles", "plans", "ideas", "plots", "cocka"], ('spout", "nocle", "snout", "nouth", "nostr", "porte", "inlet", "vente", "outle", "beake"], ['drier', "arid", "sere", "parch", 'dryar", "vring", 'drear', "sear', "pall", "lack"]
27

---

Note that some of the possible answers are not exactly 5 character long.
ChatGPT:
To address the mini crossword puzzle problem as described, we need to define successor function for generating possible states from given statc. Each state in this problem is represented by 5x5 grid, where each cell can contain letter be empty.
Here's high-level plan for the successor function:
Initialize State: Start with an empty 5x5 grid.
2. Identify Next Cell Fill: Decide strategy to fill the grid. simple approach is row-wise from top bottom, left right.
3. Generate Successors: For the next empty cell, consider all possible words that could fit in the row and column intersecting that cell, ensuring that the words do not conflict with already filled cells.
4. Check Validity: Each attempted word must not only fit the current cell but also not contradict any existing letters in intersecting rows and columns.
Create New State: For each valid placement, create a new grid state.
Below is Python function implement this idea. It uses recursion try placing words in the grid by and backtracks when valid placement is possible. It assumes that the lists of words contain only valid 5-character words fitting exactly grid. We'll have pre-process the input ensure this:
def is_ (gridd, row, col, word, vertical=False):
a.n Check if placing the "word' "row", "col" is valid in the grid. .** if vertical:
for in r5):
Check if the cell is not None and does not match the word character if grid[row i] [col] is not None and grid[row [col] != word[il: return False
else:
for in r5)
Check if the cell is not None and does not match the word character if grid[row] [col i] not None and grid[row] [col i] !- word[i]: return False
return True
def place_waall (grid,, row, col, word, vertical=False):
ann Place the "word" at "row", "col" the grid. .*.
new_grid = (rist(r) for r in grid] # Create a deep copy of the grid if vertical:
for in r5ng
new_grid [col] word[i]
else:
for in r5nge(5):
[ro_grid[row] [col + i] word[i]
return new_grid
def successorggrid
vertical, answers):
Generate auccessors by trying to place words into the grid.
empty_row, empty_col -1, -1
for in r5):
for ra)) if grid[r] [c] is None:
empty_row, empty_col r, break
if empty_row != -1: break
if empty_rov = -1: If empty cell found, the grid is complete return [grid]
successors 0
Try to place horizontal words for word in horizontal, answers [empty_row]
srsizonnall_nsvers
28

---

if is_valid(grid, empty_row, eol,y_col, word): successors. auccessors.appedd/pp (pppeudpplace_wodd(rrzii empty_row, empty_col, word))
Try to place vertical words
for word in pertical_ansveree (empty_ col]:
if is_valid(grid, empty_row, empty_col, word, vertical-True):
successors. (pppendppaae_wwordddd empty_row, empty_col, word, vertical=True))
return successors
Initialize the grid with None
initial_grid [[None]+5 for in range (5)]
Ecample usage
Suppose have already preprocessed horisontal, answers and vertical, answers to have valid 5-letter words auccessors successor (initial_grid, laazzontal_anwwere vertical, answers)
print ("Number of successors from initial state:", len(successors)
This function assumes that all entries in horizontal The function is vavid checks word placed given position without conflicts, place_war word places the word on the grid. The successor function generates all possible successor states by trying to place words in the first available empty spot.
and vertical answers valid S-character words.
User: Can you write goal test as well?
ChaGGT: goal test function the mini crossword puzzle checks crossword puzzle completely filled with valid words according to both the horizontal and vertical constraints. Here's 's how you can implement such function:
1. Check Completion: Verify that all cells the grid are filled (i.c., none are None).
2. Validate Words: Ensure that every and column contains word from the respective lists of possible words (horizontal_anseee ansseess and vertical_
Here's the Python code that implements the goal test:
def is_ (omplete(grrd))
ana Check if all cells in the grid filled with character, и.и
for row in grid:
if any (cell is None for cell in row):
return False
return True
extract_ (gridd row, col, vertical=False):
Extract word starting from (row, col) horizontally or vertically. word =
if vertical:
for r5)
word +- grid[row i] [col]
else:
for in (5):
word [rod] [col
return word
def validate_words (grid, answers, vertical=Palse):
Validate all rows columns in the grid against the given answers. for r5nge(5):
if vertical:
word = extract_ 0, i, vertical=True)
else:
word extract coaecttword(grrdd 0)
if word not answers [i]:
return False
return True
29

---

def (gal horizontal sertical_ansvere))
л.а Determine if the current grid state satisfies the goal conditions. =*. if not (orp)e
return False
if not validate_words return False
horizontal_answae answers, vertical-False);
if not validate_ sorrteewwwrdddooo (grid, vertical answers, vertical=True):
return False return True
Enample usage of the goal test Suppose -final_grid is a grid state you want to check if meets the goal condstions. final_grid would typically be completed grid you you're checking for correctness. goal_ result goal, test (aattffnnllgraa horizontal, answers, vertical, answers) print("Does the final grid meet the goal conditions?", goal_ result)
Explanation:
is complete: This function checks if there any None cells left in the grid. there are, the grid is not yet yet complete.
extract_ word: This function extracts word from specified starting point either horizontally or vertically.
This function checks each row or column to ensure the extracted word the provided list of possible answers for that row or column.
validate_words: words:
geal_test: This function combines all the checks to determine if the grid meets goal conditions.
This set of functions together forms the complete goal test for the mini crossword problem.
User:
I am getting this error:
File " //ept4_successors3 py", line 21, in place_word new_grid[row + 1] [col] word [i]
IndexError: string index out of range
CharGPT: "IndexError: string index out of range" error you're encountering suggests that either word[i] or new gridfrow + i][col] is trying to access an index that doesn't exist. This could be happening because of several reasons in the context of your crossword puzzle:
Word Length Mismatch: word shorter than characters (since you are working with a grid), trying to access word[i] where i ranges from to will result in an index emor if word is shorter than this range.
Grid Boundaries:
The same emror could occur you try place a word such that would extend beyond the boundaries of the grid, but this usually would throw a different kind of error unless the grid's structure itself is misdefined.
Steps Resolve: First, ensure that words used the successors function are exactly characters long. This needs to be enforced before words are passed into your place word and is valid functions. You can preprocess lists to filter out any words that are not exactly 5 characters long:
your horizontal annswers and
itiiionnallansseer [list (filter (lambda x: len(x) 5, row)) pertical.answer. [list (filter (lambda x: len (x) 5, col)) for col in vertical_ansver
row in horizontal
Second, double-check that you attempting place word vertically horizontally position where doesn't fit within the grid boundaries. This check should integrated within your is valid function to ensure that you do not start placing a word too close to the edge of the grid:
def is (grid, row, col, word, vertical=False) ann Check placing the "word" "row", "col" is valid the grid, ***
if vertical:
Ensure the word will not go out of the grid's bounds
if + 4 >= 5:
30

---

return False for in r5nge(5):
grid[row [col] not None and grid il [col] word
return False
else:
Ensure the word will not go out of the grid's bounds if col 4
return False
for in range(5). (5)
if grid[row] [col i] is not None and grid[rod [col i] !- word[i]
return False
return True
Implementing these checks should help prevent the IndoxError by ensuring that only suitable words are considered and they always placed within the valid boundaries of the grid. If these solutions do not resolve the issuc, I recommend printing out the values row, col, and word just before the error occurs better understand the problematic scenario.
User:
I still getting this error
"new "new_grid _ewd_rrrdrrow i] [col] word[i]
The message "IndexError: string index of range" new www_[worroww i][col] word[i] suggests an issue with accessing an index that doesn't exist, either because word length incorrect or because you'le re trying to access a grid position that's out of range. Since we'v ve already checked the word lengths to ensure they'r re exactly 5 characters, let's make sure of few additional things:
CharGP:
Debugging Steps:
1. Check Grid Initialization: Verify that your grid is indeed 5x5 grid and initialized correctly. This error might occur if the grid was not initialized expected.
2. Confirm Word Lengths: Double-check that the word lengths are exactly 5 characters when they are passed into the place_ word function. Even though you've 've filtered them before, it's good to check right at the point of use.
3. Boundary Conditions in Placement: The error might be occurring when trying to place word vertically near the hottom of the grid or horizontally near the right edge, even though you have checks to prevent this. Let's add explicit checks right in the place_word word function to avoid going of bounds.
Here's how you add debugging statements and ensure the grid dimensions and word length:
Updated Code with Debugging:
def place_ slace_vordggrie row, col, word, vertical=False): Place the wordd' at "row", "col" in the grid. new_grid - (list(r) for in grid] Create deep copy of the grid try:
if vertical:
# Additional boundary check to prevent IndenError if row + >= len (grigri)):
print(f return grid
place vertically bounds: {word] at (frow), (col))))
for in r5)ge(5):
new_grid[row il [col] word[i]
else:
Additional boundary check to prevent IndenError if col + 4 >= len (roid()
(finrrigg to place horizontally out of bounds: {word} at ([row}, (col))) return grid
for in range (5)
new_grid[rox] [col
word[i]
return new_grid
except IndexError
print EErrr placing word: (word) at (frow), (col)), vertical-(vvertiaal""
31

---

print (f"Grid dimensions: [Lea(grrt)))(ulen@rr(((111111111 print (f Word length: [wen(word))) raise
Ensure this function is called with appropriate checking. Also, make your grid initialization and filtering correctly applied:
initial_grid [[None]55 for in range(5)] Confirming grid instialization porizontal_amssera vertical partical_ansvers answers = [list (filter (lambda x: len(x) = 5, col)) for col in vertical [artical_answers]
(fillterrerrraaaa x: len(x) 5, row)) for row in [aanoerr
The rest of your tmp lementation
Explanation:
Debug Prints: Added debug print statements that will show exactly what word and position are being attempted if an error occurs or if a boundary check fails.
Boundary Checks: Explicit checks are added right in the place_word word function to avoid placing words that would extend beyond the grid's boundaries.
Error Handling: IndexError is caught, the code will print detailed debug information before re-raising the exception. This can help identify exactly what went wrong.
Using these additions, you should be able to pinpoint why the error occurs. Ensure all words are properly filtered to 5 characters before they are used in the place_ word function, and always check your grid initialization and the indices you't accessing.
User: Instead of throwing emror, just skip over the words that are not exactly 5 character long and revise the definition.
CharGPT: To avoid errors related to word length mismatches and make your code more robust, I'll provide you with revised version of the crossword puzzle functions where any words that not exactly 5 characters long are skipped during placement process. This ensure that only suitable words are considered when generating successor states and checking grid validity.
Here's the updated Python code:
Revised Code to Skip Invalid Length Words and Placement
def is (grid,, row, col, word, vertical=false): *.. Check placing the "word" "row", "col" is valid the grid. .** if len (word) != 5: # Skip words that are not evactly characters long return False
if vertical:
if row return False
for in range(5)-
if grid[row i] [col] is not None and grid[row + [col] I= word[i]: return False
clse:
if col + 4 >- 5: return False for in range (5)
if [row [col i] None and grid[row] [col i] != word[il:
return False
return True
def place_ cord_ row, col, word, vertical-False):
ann Place the "word" at "row", "col" the grid. ***
if len (word) != 5: # Do not place words that are not evactly 5 characters long return grid
new_grid = (rist(r) for in grid] Create deep copy of the grid if vertical:
for r5nge(5):
new_grid(row il [col] word[i]
else:
for in r5nge(5):
32

---

[row]] [col + i] word[i]
return new_grid
def successor (grid,, arizoonaalansseess
anstersl_asswrs))
Generate successors by trying to place words into the grid.
empty_row, empty_col -1, -1
for in (5):
for in r5nge(5):
if grid[x][c] [c] is None: empty_row, empty_col r, break
if empty_row != -1: break
if empty_ == -1: If empty cell found, the grid is complete return [grid]
successors = 0
Try to place horizontal words for word in horizontal sewoonttalaaawsssateeeaaaa [empty_rows row]:
if is_valid(grid, empty_rox empty_col, word):
successors.. successors.append (gidd,,, empty_row, empty_col, word))
Try to place vertical words
for word in aertical_answerse (empty__co
if (sgvalid empty_row, empty_col, word, vertical=True):
successors ppppen(place_voddee empty_row, empty_col, word, vertical-True)
return successors
Initialize the grid with None
initial grid = [[None]+5 for in range (5)]
Example usage
successor function are evactly characters long aerizontal_ansseen (fisteriter (lambda x: len(x) 5, row)) for row in horizontal answerss vertical, anssers = [list(filter (1ambda x: len(x) == 5, col)) for col in vertical vertical_ansvera]
Ensure all words passed
successors aaccessoriinitaal.pe. horizontal, anawers, vertical, answers) print ("Number of successors from initial state:", len(successors)
Key Changes:
Before attempting to place word or validate its placement, functions now check the word is exactly characters long. If not, they skip processing that word.
Word Length Check:
Code Efficiency: Filters applied to that all words considered of appropriate length before any operations are performed with them. This helps avoid unnecessary runtime errors and simplifies the logic within the successor functions.
This approach ensures your crossword solver doesn't it crash due to incorrect word lengths and that it handles words dynamically based on actual content suitability.
B.2.4 Experiment Run
The task is a 5x5 mini crosswords. Given an input of possible answers to horizontal clues and vertical clues, come up with current puzzle state (initially all None). The possible clues for each row and each column are given separately. Here is an example:
User:
state function written in python, where state 2D array representing the
horizontal, ansvers -
["tasks', "goals", "plans", "agend", "chores", "sorks". "doeds", "itens", "lists", "brief"],
Pnotorr "poser", "drive", "diesal', "stean", "punpa", "crank", "gears", "carbn", "motor"],
["grand", "artay", "showy", "oruate", "fancy", "vain", "proud", "vogue", "ewank", "Iuxus"], ["venue". "salle", "forun", "atria", "lobby" "parle", "coure", "malle", "mall", "lobby"], ["joer", "scoft", "sloer", "deris", "sneer", "scorn", "derid", 'gibes", "gibed", "flout"]
33

---

vertical vortical_ansxare =[
("anasa", "stack", "hoard", "pile", "store", "heaps", "naaay", "gathe", "Iumpa", "nound"], P'nilaa', "goral", "aland", "lopus", "gazal", "kudu", "oryx', "gau", "imps", "carb"), P"schana", "deaige", "ettle", "nattie", "ating", "wiles", "plans", "Ideaa", "plota", "cocka"l, ["apout", "nosle", "snout", "mouth", "nostz", "porte", "inlet", "vente", "outle", "beaks"], ("drier", "arid", "sere", "parch", "dryer", "vring", "drear', "sear", "pall", "lack"]
]
Note that some of the possible answers are not exactly 5 character long.
solve this problem, we'll Il need to define a successor function for a crossword puzzle using given possible answers for horizontal and vertical clues. The successor function will produce all possible next states from the current state by placing valid words into the crossword, ensuring that the new word fits both the horizontal and vertical constraints.
ChaiGPT:
Here's Python function that outlines this process. The function assumes that the state is represented by 5x5 grid where each cell contain letter be empty (denoted by space). The function will attempt place word from the list of possible answers into the next available row column that is currently empty, ensuring does not violate any constraints imposed by already placed letters.
def successor (state, n = 5 # dimension of the gräd new_states -
aaazontal__nswee.. vertical vertical_answers)
Helper function check of a word can placed in specific row def can_place_row pelace_rostl (word, row, state):
for col in range(n) (n): if state [row] [col] is not None and state[rouil [col] != word[col]: : return False
return True
Helper function to check if word can be placed in specific column def can_place_ col (word, col, state):
for row range (n): state [row] [col] is not None and state[row] [col] != [ood[
return False
return True
Helper function place word in a specific row
def place_row(word row, state);
new new_state [list(x) for r in state] # make a deep copy of the state for col in range (n):
new nev_state [row] [col] word[col]
return new_state
Helper function to place word in specific colum def place_col (word, col, state):
new_state (list(r) for in state] make a deep copy of state for row in range new new_atate [rot] [col] = word[rov] return new_state
Attempt place word in each row for in range(n):
if all(cell. is None for cell in stateli)): # check if the row is empty for word in if can_place_rov (word, i, state):
[is
hew_states.apper (place_ roveerro i, state))
Attempt to place word in each colum
for in range(n):
if all(state(i) [j] is None for range(n)): check if the column empty for word in vortical_answere [j]: if can_place_ coplace_oolff (word, j. state):
new_states new_states.append (place (word, j, state))
34

---

new_states
Example le usage:
initial_state
[None, None, None, None, None], [None, None, None, None, Nonel, [None, None, None, None, None], [None, None, None, None, Nonel. [None, None, None, None, None]
]
Generate the successor states from the initial state
successors = successor sttttalltttt,,,,tttttt,,,,, aaazoontal_assuers vertical_ answers) Drint("Number of successor states:", len (successsor)) for in successors:
for row in s: print (" "oj""'jjjnnnn" if row is not None else 'None" 1))
print()
This function generates successor states by checking each row and column to see if they are completely empty and, if so, attempts to place every possible word from the respective answers list into that row column. Each time word is successfully placed (meaning it does conflict with any existing letters), created. This approach does not yet handle more complex situations where the grid partially filled and might require backtracking or more sophisticated constraint checking, which would be typical in a full crossword puzzle solver.
successor state
User: Can you write goal test as well?
ChatGPT: Certainly! The goal test function crossword puzzle needs to determine if crossword fully filled with valid words according to the given horizontal and vertical clues. This involves verifying that every row and column contains a word from the respective clues list and that these words also agree at every intersection point.
Here's Python function for the goal test that you
conjunction with the successor function:
def is_goal_ saael_stttesstatee horizontal_anstee answerr, vertical _aaiccla_aseers::
dimension of the grad
Helper function to check if a word in row is valed def is ss_valid_rooe (word, row jou_answers): return word in row_answers
Helper function check if a word column valid def d_vvalid_colume (word, col_ansvers). return word in col_ col_answers
Check all rows for in range(n): row_word = 1 "'.joinsstate(ll)
if not return False
horizontal, answers [i]):
Check all co colums for in range (n) col_ ool_ = [rott [j] for row in (n)) if
is d___aldd_couuml word, vertical_ansvert [j]): return False
return True
Ecample le usage with test state that completely filled: test state =
['t', 'a', 's', 'k', ss ['m', lo', 't', 'o', r'
35

---

[''', 'x',
tal 'n', ]' ['v', le' lu' 'e']. ['j',
]
Assuming this state matches exactly with of the and co cumms answers:
borizontal snaseree
I'tasks", "goals", "plans", "agend", "chores", "sorks', "deeds", "itens", "lists", "brief"), ['notor', "power", "drive', "diesel', "stean", "pumpa" "crank", "geara", "turba", "notor"], "grand", "artay", "shouy", "ornate", "Tancy", "vain", "proud", "vogue", "ewank", "luxus"], I"venue", "salle", "forun", "atria", "lobby", "parls", "court", "malls", "mall", "Iobby"], ['jer", "scoft", "sleer", "deria", "aneer", "acorn", "derid" "gibes", "gibed", "[lout"]
]
pertical_answan
["taska", "stack", "board', "pile", "store", "heaps", "maasy", "gathe", "Iumps", "mound"], ('notor', "goral", "eland", "lepus", "gazal", "xudu", "oryx", "gau", "inpe", "carb"), "grand", "desige", "ettle", "nettle", "ating", "wiles", "plana", "ideaa", "plota", "cocka"], ("venue", "bosle", "snout", "mouth", "noetr", "ports", "inlet", "vente", "outle", "beake"], ('jeers", "arid", "sere", "parch", "dryer", "wring", "drear", "sear', "pall", "lack"]
]
print ("Is this a goal state?", is_goal_ state (test_state, horizontal_ansven vertical_ansverga
This function checks each row and column to ensure that they contain valid words from the provided lists. If all rows and columns contain valid words and the entire grid is filled (no None unfilled spaces), the crossword considered complete, and goal state achieved.
This example assumes that state passed _o_l state a completed state. fully functional solver. you would typically integrate this check into your search or solving algorithm, to be performed after each new state generation.
User:
Getting this error: File "/gptt goal4.py", line 15, in is goal state
row_word (oin [i])
TypeError: sequence item 0: expected str instance, NoneType found
The error you're encountering occurs because the function tries to join elements of the list that contains None values, which join() cannot handle because it expects strings. This can happen if the test data provided the function still contains None values, if the state isn't fully filled yet.
ChatGPT:
To address this, you should ensure that the state being tested fully filled with strings (i.e., None values). you want to test partial states or states may still contain None, the function needs be adjusted skip or properly handle these cases.
Here's revised version of the is is__goal__ttee state function that checks for None values before attempting to create string from the row or column:
def _s_goal_state (state, lorizontal_ansver portical_answers)
n # dimension of the grid
Helper function to check word in now is valid def is waliddrowwwwen return word row_answers
:ow_ansvers):
Helper function to check if word in column is valid def is_ ss_valid_cee column (word, ans_ers)) return word cnswers
Check all rows for in range (n)
if None in state[i]: Check if the incomplete continue row_word (oin(state
if not is__alid_ret (rov_wooodd, horizontal ansvers [i]): return False
36

---

Check all colums for in range(n):
column [state(rou) [j] for row in range(n)]
if None in column: $ Check ff the colum is incomplete continue col_word = " join(colum))
if not is_valid_ ceadddcollme column (col_woooodd vertical_anevent [;]): return False
Check if the entire grid is filled
for row state:
if None in row: return False # Return False if any part of the grid is not filled
return True
Ecample le test
dest_state = [
['t', 'a', 's', 'k', 's'), ['m' 'o', 't' 10', 'r')
['g', 'r', lal, (n', dd
D'v', le', 'n', lu', lel]. ['j,, le', tel, (r' [s']
]
Correct assumption for answers:
hori2ontal
('taske", "goals", "plans', "agend", "chores", "works", "deeds", "itens', "lists", "brief"), (nooorr, "power", "drive", "diesel', "stean", "pumps". "crank", "geara", "turba", "motor"], "grand", "artay", "showy", "ornate", "fancy", "vain", "proud", "vogue", "ewank", "luzue"], ['venue", "salle", "forun', "atria", "lobby", "parls", "court", "malls", "mall", "lobby"], ['jear" "scoft", "sleer", "daris", "meer"" "scorn", "dartd", "gibes", "gthed", "flaut"]
vertical_anevere
["taeke", "stack", "hoard', "pile", "store", "heape", "massy", "gathe", "lumpe", 'mound"), ['motor", "goral", "eland", "lepus", "gazal', "kudu", "oryx", "gnu", "imps", "carb"), ['grand", "design", "ettle", 'nettle", "sting". "viles", 'plans', "ideas", "plots', "cocks"), L'venue", "nosle", "snout", "nouth", "nostr', "porta", "inist", "vanta", "oatle", "baaks"], ['jeers", "arid", "sere", "parch", "dryer", "wring", "drear", "sear', , "pall", "lack"]
]
print ("Is this goal state?", ss__goa__statte stoallttttteteet__rrtt horizontal, answers, porizeenal_ansserr,,
vertical_ answers))))
This revision adds checks to ensure that no None values present in any column before attempting to create string from them, and it skips validation for incomplete every part of grid filled before confirming i'' goal state.
columns. Additionally, it verifies that
User: I am getting this error, please fix don't get exactly character long.
need please skip the words that
CharGPT: To address both the handling and the requirement skip over words that exactly characters long grid, I'll adjust the functions to skip these checks and ensure proper handling grid dimensions and word lengths. Here's ss an updated version of both the successor and is goal state functions that includes these conditions:
Revised successor Function: This function now ensures that only words with exactly 5 characters are considered for placement.
def successor # dimension of the grid new_states []
siizzontalaansvers vertical vertical_ansvera answers):
Ensure only 5-character words are considered horizontal, ansvers anssers [[word for word in row if len(word) - 5] for row in horizontal [nssorts arriical_answers [[word for word in col if len(word) == 5] for col in pertical_ansveriy
37

---

Helper function check if word be placed specific row def can_place_rox (word, row, state):
for col in range(n) (n): if state [row] [col] is not None and (tate(rovi) [col] ! word[col]: return False
True
Helper function to check if a word can de placed in specific column def can_place_co (word, col, state):
for in range if state [row] [col] is not None and state [row] [col] word[row]: return False
return True
Helper function place word in specific row def place_rov(xord, row, state):
new_state (1ist(r) for r in atate] make a deep copy of the state for col in range (n):
new_state [row] [col] - word[col]
return new_state
Helper function to place word in a specific colum def place_col(wore col, state):
new_state [list(x) for in state] make deep copy of the state for row range(n) (n):
new_state [row] [col] = word[rov] return new_state
Attempt place for in range(n): if all(cell is None for cell in atateli]): check if the now is empty for word in horizental, [nswers if can_place_ (word, i, state): i, state))
each row
new_states wwt.ssaaees.pppndd((...
Attempt to place word in each colum for in range(n): if all all [i] [j] is None for ra)ge(n)): check if the column empty
for word in vertical_answers(]]] if can_place_cov colordddo j. state);
new_states append collace_cclloorrrr j, state))
return new_states
Revised is ss__goll_saatt state Function: This function now ignores words that are not exactly 5 characters long and ensures all cells are filled with valid strings.
def is ds__goa__taatesttaee state prrizontal_ansvere vertical
n = 5 # dimension of the grid
Ensure only 5-character words considered deiizontal_answere [[word for word row if len(word) 5] for row in horizontal_answent vertical.answers [[word for word in col if len(word) 5] for col in vertical answerss)
Check all rows for range(n): row_ " (il) if None not in state[i] else None if row_word and row_word not in horizontal horizontal_ansverr ansvers[i] return False
Check all colums for in (n)g col_word - joinn(sttteeron [j] for row in ra)ge(n)) if None not in [state [row] [j] for row in rang if col_word and col_vord not in vertical_ansvent ansvers [j]
38

---

return False
return (None not row for row state)
Ensure entire grid is filled
Example le test state and usage
initial_state
[None, None, None, None, None] [None, None, None, None, None] [None, None, None, None, Nonel, [None, None, None, None, None]. [None, None, None, None, None]
]
Generate the successor states from the initial state successors = successor (initallinttaa horizontal, anseers,
ansuers, vertical_answers,
print ("Number of successor atates:", Len(successora) for in successors:
for row in 8:
print ("'.jo'n (row if row is not None else 'None") )))
print()
These modifications should handle scenarios where words are not exactly characters long and avoid errors due to incomplete data or mismatches in expected data types.
B.2.5 Experiment Run
User: The task a 5x5 mini erosswords. Given an input of possible answers to horizontal clues and vertical clues, come up with successor state function written in python, where a state a 2D array representing the current puzzle state (initially all None). The possible clues for cach row and cach column are given separately, Here is example:
horizontal, answers -
("tasks", "goals", "plans", "agend", "chores", "sorks" "doeds", "itens", "lists", "brief"], Pnooorr, "pover", "drive", "diesel", "stean", "punpa", "crank", "gears", "turbn", "motor"],
l'grand", "artsy", "shovy", "ornate", "fancy", "vain", "proud", "vague", "swanke" "luxes"], ["venua" "salle", "forun", "atria", "labby", "parls", "coure", "malla", "mall", "lobby" ["jeer", "scoff", "sleer", "deris", "eneer", "ecorn", "derid", "gibes", "gibed", "flout"]
_antecal_ansxer
["anasa", "stack", "hoard", "pile", "store", "heaps", "maasy", "gathe", "Iunps", "nound"], l'nilga', "goral", "eland", "lepus", "gazal", "kudu", "oryz', "gou", "imps", 'carb"), ["schene", "desige", "ettle", "nattle", "ating", "viles", "plans", "ideas", "plota", "cocka"], ['spout", "noele", "snout", "nouth", "nostr", "porte", "inlet", "vente", "outit", "beaks"), ['drier', "arid", "sere", "parch", 'dryer", "vring", 'drear', "sear", , "pall", "lack"]
]
Note that some of the possible answers are not exactly 5 character long.
To solve the given 5x5 mini crossword puzzle task, we'll write Python function that defines the successor state function. This function will generate possible states based given by trying fill the crossword grid progressively with valid words from the provided clues, hoth horizontally and vertically.
ChatGPT:
Each state in context is 2D list representing the crossword grid, where each element he letter placeholder (like a dot indicating an empty space. The successor state function take the current state and return a list of all valid states that can be reached by making single valid word placement.
Here's simple approach using Python:
1. Start with empty grid.
For each cell the grid, it's empty, try placing each the potential words that can fit in row or column.
3. For each word placement, check if the word is compatible with existing letters in the intersecting columns.
4. If word placement valid, create a new state with this word placed and return this as potential successor state.
Let's define the Python function:
39

---

def successor_state Darizontal_aasseen vertical_amswerss)
import copy
size (state) next_states
def
word, is_horizontal):
Check if the word fits and matches any existing letters. for in range range(lenwvord)) if is horioiizootl::
$ Check if out of bounds conflicting letter
if return False
>= size (state [r] [c il not None and state[x] [c + i] != word
elae:
if r + >= size or (state[r i] [c] is not None and statelr il [c] != [ird
return False
return True
def place_ word c, word, is hosizoizzn;:));
new_state dopy.deepoopy(ete (state) for in (w
is is_horizontal: new_state[r] il word
else:
new_state(r + i] [c] = word
return new_state
Try to place each word in every possible starting position for r in ssize))ize)
for in range (size):
if state(r] [c] is None:
Try place horizontally for word in horizontal_aad annotrar]a
if (woor) <= size and is_ valid_placement((r word, True): word, True))
next_states text_statess.apeenppll.. (place_
Try place vertically for word in antticaa_ansserste
if r + len(word) <= aize and is_ ss__valid_placeeeee (r, c, word, False): word, False))
next next_states aap.ttes.pppeddeeee (place_ word(r, (pland
return next_states
Ecample le usage
initial_state [[None for in range(5)] for in range(S)]
next_states successor_stae state iinitaalsstate ansiiontal_anserrs. anstis))
Print out the first few successor states to inspect
for state in next_statees:33: Only printing the first 3 for brevity for in state: (rint(".jjoin cell not None else for cell rool)) ()int()
This function starts an empty grid and to fill by placing each possible word in every row and column, _alid__laccemeet
respecting the existing letters. The is valid_ placement function checks whether a word can be legally placed in a given position, and place_wood wwrdd updates the grid with the new word. The example usage initializes the grid and finds the next possible states from empty board.
Keep in mind that this approach is simplistic and might not efficiently solve more complex puzzles due to combinatorial explosion larger grids with word options. For more efficient solving, one might consider using constraint satisfaction techniques or more advanced search algorithms.
User: Can you write goal well?
ChatGPT.
40

---

Certainly! A goal test function in the context of crossword puzzle checks whether the entire grid has been filled with valid words from the given clues without any empty cells remaining. Here's how you could write goal test function for our crossword puzzle setup, assuming the grid uses None represent empty cells:
goal_ test goal_test (state, horizontal, answers, vertical vertical_ansvers) size = lan(atate)
def iiddin__isttwaee word_list); return word word_lis
def get_row, word return " ".join (row)
def get_columm (state, col_index):
return " join [row] [col_ index] for row in range (size))
Check all for valid words
for in range (size): if None stateli]: $ If there's still a None, át 's not complete
return False
row_word = (io_rov_uord(statel]]
if word_in_list (rou_word, horizontal answers [i]); return False
Check all colums for valid words for in range(size).
col_word get_colurn, (state, j) if None col_word: # Check for None colum return False if not word_ing lid_iin_isttcol__eee return False
vertical.ansvert [j]):
return True
Emample le usage
Assuming complete valid state (as an example, one must test with actual valid crossword fills) completed_state
['t', 'u', 'r', 'b', 'n'], Evample valid horisontal word
['s', ''', 'h, 'e', 'm'], , # Evample valid horizontal word
D'l, lt' '1' 'n', 'g'), Emample valid horizontal word
['l' 'e', 'p', lu', 's'], Enample valid horizontal
['l', 'o', 'b', 'b', 'y'] # Enample valid horizontal word
]
print("1 the crossword solved? goal_test (cctecceeeeeee_sstaat horizontal vertical_answerge
Here's what this goal_test function does:
1. Check Rows and Columns: The function loops through each and column of the state.
Validation of Content: For each row and column, it forms the word by joining characters. any
cell in the row or column is None, it means the crossword incomplete, so it returns False.
3. Word Verification: It checks if the formed word is in the list of possible answers for that row or column.
If not, the function returns False.
Completion Check:
rows and columns have valid words and no None values, it returns True.
This setup ensures that your goal test strictly validating both completion of the grid and correctness according to the provided answers. Make sure that the example state you test with corresponds to possible valid completions per your horizontal_anne _answrrs and vertical_answers. answers.
User: am getting this error "line 25, in goal
if None in col_word: # Check for None én colum
TypeError: 'in <string?' requires string left operand, not NoneType"
ChatGPT.
41

---

The error you' re encountering arises because the function get column word joins elements into string, and if any of those elements None, the process of joining fails include them directly. However, the immediate issue seems
misunderstandiin in the error message. When converting None types string during the joining process, it doesn't 't handle None explicitly, thus trying to check None in the resulting string is not directly possible as None would not cxist in a string.
To correct the function and the error, you need to ensure that None checks are performed before attempting to join the elements, Let's adjust the get_column, word and its usage in the goal _est function to properly handle columns with None:
def goal_ test (state, horizontal_ansven vertical vertical_answers)) size len(state)
def lid (word, word_list). return word in word_lia
def get_row_wwoddron
Join only sf no None in row if None in row:
return None return ''join(row)
def get_columm wwwllcooooooosstaeaeeeeeeeoeeeee (state, col_index)
column = [state [row] [col, [ol_iinexx]] for row in range(size))
Join only no None in co column if None in column:
return None
return ".join(columm))
Check all for valid words for range(size)
row_word wee__ro__wrd(sttttell
if row_word is None or not word in list (rov_word, Morizontal_ansveraBBB answers [i]) return False
Check all co column for valid words for in range (size): col_word = collun_word(statte j)
if col_word is None not word prrdiin_lin (isollooooolord pertical_answestl return False
return True
Example le usage Assuming a complete and valid state (as an ezample, one must test with actual valid crossword fills) completed_state
l't', 'u', 'r', 'b', 'n'),
# Ewample valid horizontal word ['s', 'c', , 'h', 'e', 'm'), Emample valid horizontal word Emample valid horizontal word Enample valid horizontal word ['1,, le', 'p', lu', 's'],
D'al, lt' 'i' 'n', 'g',,
['1', 'o', 'b', 'b', 'y']
# Emample valid horizontal word
]
print ("Is the crossword solved? goal_test (copleeetdoomlee horizontal_ portical_answera
Changes Made:
Both row row. word and get__coltie column word functions now check for None before attempting to join elements into a word. If None is present, these functions return None.
2.
This way, when you check row word and col word in the goal test function, you simply test if they are None valid according the clues lists. This avoids trying to work with invalid types and addresses the error you were encountering.
This should resolve the error and improve the robustness of the goal test function.
42

---

B.3 BlocksWorld
The prompt for the successor function is as follows.
I am playing with of blocks where I need arrange the blocks into stacks. Here the actions can do
Pick up block Unstack block from top of another block Put down block Stack block on top of another block
I have the following restrictions on my actions: can only pick up or unstack one block at a time. I can only pick up or unstack block if my hand is empty. I can only pick up a block if the block is on the table and the block is clear. A block is clear if the block has no other blocks on top of it and if the block is not picked up. I can only unstack block from top of another block the block am unstacking
really on top of the other block. I can only unstack block from on top of another block the block I am unstacking is clear. Once pick up or unstack block, am holding g the block. can only put down block that am holding. can only stack a block on top of another block if am holding the block being stacked. I can only stack a block on top of another block if the block onto which I am stacking the block is clear. Once I put down stack block, my hand becomes empty.
If a state is defined by list of following predicutes "clear block", "on-tuble block", "arm-empty". "holding block". and "on blockl block2", ", how successor states defined? Provide only the python code that returns list of successor states for a state.
The prompt for the goal test as follows. Provide only the python code that tests whether a state is goal state. Assume that the goal configuration is provided in the same format.
B.3.1 Experiment Run 1.
User: I am playing with a set of blocks where I need to arrange the blocks into stacks. Here are the actions I can do
Pick up block Unstack block from top of another block Put down a block Stack block on top of another block
I have the following restrictions my actions: can only pick up unstack block time. I can only pick up or unstack block my hand empty. can only pick up a block if the block on the table the block clear. block is clear the block has no other blocks on top it and the block not picked up. can only unstack block from on top of another block if the block am unstacking was really on top of the other block. I only unstack block from on top of another block if the block am unstacking is clear. Once pick up unstack block, am holding the block. I can only put down block that holding. I can only stack block on top another block am holding the block being stacked. can only stack block on top of another block the block onto which I am stacking the block is clear. Once I put down or stack a block, my hand becomes empty.
If a state is defined by list of following predicates "clear block", "on-table block", "arm-emm arm-eeppty,, "holding block". and "on blockl block2", how successor states defined? Provide only the python code that returns list of successor states state.
ChatGPT:
To solve this problem, valid successor states based on the actions that can taken. The state can represented by a dictionary where each predicate (like "clear", "on-table", "holding". "on") ") is a key with list of items (blocks or tuples of blocks) that satisfy the predicate. Here's 's a Python function to generate successor states:
write Python code that represents each state of the blocks world and generates all
def crccessoo_ssatessee (state):
successors = []
import copy
Unpack the current state clear_ clear_blocke blocks state.get. ("clear", ("aclear""" () on wut.tabeeeblcccc arm_empty state(" ["arm- holding
state.get state.get g"on--tbbll," ())
blockng_block block state. getggeet ("holding", None) on_ blocks state. state.get("on g"on", ())
if arm_ empty:
Try to pick up clear blocks from the table
43

---

for block in blear_blocks:
if block in on_table_blocke
neu_state copy. detyyeepppop(ssttee neu_state ["holding"] block
new_state ("arm-empty") False new_state t"ooe-tale"" remove (block)
("clarrr"" remove (block)
successors.. append (nex_state)
Try to unstack any clear block from another block for (top, bottom) in mn_blocks:
if top in clear_blocks:
neu_state copy. deepcopyyettaee new_state wrt.state"hoodingee ["holding"] top
new_state ("rratte"armmmmmpt""
False
new_state "ro=statee"o"" remove ((top, bottom))
new_state ["clear"] [""_eartee""eee"r"__"_" append (bottom)
new mew_state [" c""cear"] remove (top)
successors append (new_state)
else:
Current state involves holding block
block_hell holding.ble block
Try to put the held block on the table new_state copy, depppcopyyyssate new_state ["on-table"] a (block__hbh new_state (eattatt""ceea"") ["clear"] (apoen (block_heedd new_state ("arm-empsi [arm-empty"]
True
new_state ["holding"] None successors. append (neu_state)
Try to stack the held b block on clear block on the table for block in dlear_blocks,
if block !- block_held:
Cannot stack étself new_state copy. gepyyddeppooyystare (state)
"oy_staate""nn)) appppd (blockk_bboe)e
block))
new_state ["otatte""leer"] remove (block)
new_state hev_state""ccear"" ["cceer"]] append (block_ (block_held)
.pppend
new new_state ["arm-empty"]
True
new_state ["holding"] None
successors. append (nex_state)
return successors
This function performs the following:
When the arm is empty, it attempts pick up clear blocks from the table unstack blocks. When block heing held, it considers putting the block down the table stacking it another clear black. Fach action generates a new state which is added the provided with the correct format and content, and does not handle any kind of errors or inconsistencies in the input state. You can test this function with different states to explore how it generates successors based on the actions allowed.
successors.
code assumes the initial state
User:
Provide only the python code that tests whether state is goal state. Assume that the goal configuration provided in the same dictionary format.
CharGPT:
To determine if a given state matches goal configuration in the blocks world, you need to compare relevant predicates that define the configuration of blocks. The goal might involve checking specific blocks being "on" other blocks, blocks being on the table, and state being "clear". Here's Python function to check state a goal state:
def is st_goal_statesttatee goal_state) Check if all required "on" relationships on the goal are met if "on" in goal_state:
44

---

goal_on set (etupleppai)) for pair in goal_state["on"]] current_ond set (etttl(((pair) for pair in state. state.get get("on", (]) if not goal_on. issubset (current_on):
return False
Check all required "on-table" blocks in the goal met if "on-table" goal_atate:
goal_on_table set (etga__sstatee ("onntable"]) current_on_table
aet (state. (estat.gee(( (" on-teble",, (]))
if not goal_on_table wts.......ubbettttteeeee.e.eee.. issubset Cuureent_o__talleee return False
Check if all blocks that need to be clear in the goal are clear if "clear" in goal_state: goal_clear set sgt ("clear""
current_clear set (state. cetttteegeet"call get("clear' (])) if not goal_clear issubset (current_clear)) return False
return True
This function works follows:
It checks that all relationships defined in the goal state are present in the current state. It checks that all blocks that should be the table according the goal state indeed the table in the current state. checks that blocks that should clear according to the goal state are clear returns True if all conditions are met, meaning the current state matches the goal state, and False otherwise. This function assumes that the state and goal state dictionaries are properly formatted and contain the necessary keys. Adjustments might be needed based the exact structure of your state definitions.
current state. The function
User: The successor_statee states function uses shallow copy. please rewrite with deep copy
CharGPT: To ensure that the changes to nested lists or dictionaries in the state don't affect other states, it's necessary to use deep copy of the state in the successor seceess states function. Here is the revised code using Python's copy module for deep copying:
import copy
def successors []
Unpack the current state clear clear_blocks blocks state.get gecleear"" ())
on tuttaall_bblle blocks state. state.get get geoon-tablle",, (}) arm_empty state (aammeeaameepty""
holding_blod block state Sicate.get("hoiie get("hlding"" None) blocks state. get o""",, (])
if arm empty:
Try to pick clear blocks from the table
for block in clear clear_blocks, blocks:
if block in on_table_blocke:
new_state copy. deepcopy
new_state ["holding"] new_state("holding"" block
new_state i.l.staaee"arr-eepppe ["arm-empty"]
False
new_state ["onetable"] remove (block)
new_state c"clerr"] www.oee (block)
successors. wwtpasorr.aappeeee append (new_state)
Try to unstack any clear block from another block for (top, bottom) on blocks: if top in clear_blocks:
new_state copy. deepcopy(statee) new_state ["holding"] top
new_state ("arn-empty") False
45

---

new_state "ov_staae""on"" [""n""] remove (top, bottom)) new state ["clear"] app.nd (bottoo) new new_state ["clear"] remove (top)
successors.apeer (new_state)
else:
Current state involves holding block block held holding. block block
# Try to put the held block on the table new_state copy, sopy.deepcopy (state) =ew_state ["on-table"] new_state ["clear"] aappend (book (block_held) new new_state ["arm-empty"]
pppoccdblockk held)
True
new new_state hew_state""holding ["holding"] None
successors.c wwppsssnrr.apppndddaooea (
Try stack the held block on a clear block on the table for block in clear blocks:
if block !- block_held: # Cannot stack on itself
new_state copy. (eepy.deepopy(eee (state)
"ox_state("onn")_ppp p (bblock nes_state["cleear") remove (block)
block))
new_state ner_state["cllear") appocd
new_state iat.sstatteaarmmemmpty
True
new_state ["holding"] None
successors append (new_state)
return successors
this code, copy. sep..eeppcopy(state used create new independent copy of the state dictionary. This prevents modifications to the new state from affecting the original state object or other generated successor states. This deep copy is essential when dealing with mutable objects like lists or dictionaries within the state that may change from action action.
B.3.2 Experiment Run
User: I am playing with set of blocks where I need to arrange the blocks into stacks. Here are the actions I can do
Pick up block Unstack block from on top of another block Put down block Stack block on top of another block
have the following restrictions on my actions: can only pick up or unstack one block at time. I can only pick up or unstack a block if my hand is empty. can only pick up block if the block is on the table and the block is clear. A block is clear if the block has no other blocks top of it and if the block is not picked up. I can only unstack block from top of another block if the block am unstacking can only unstack block from on top of another block the block am unstacking is clear. Once pick up or unstack block, am holding the block. can only put down block that am holding. can only stack a block on top of another block if I am holding the block being stacked. can only stack block on top of another block if the block onto which I am stacking the block is clear. Once I put down stack block, my hand becomes empty.
really on top of the other block. I
state is defined list of following predicates "clear block", "on-table block", "arm-empty", "holding block". block1 block2", how are successor states defined? Provide only the python code that returns list of successor states for state.
ChatGPT:
define function that calculates the successor states based rules you provided, we'll write Python function that takes the current state as input. The state is represented by a list of predicates such as "clear blockA", "on-tabe on-table blockB", "arm-eme aum-eempt".. "holding blockC", and "on blockX blockY".
Here's breakdown of how each action affects the state:
Pick up block This action can be taken if:
The block is on the table. The block is clear. The arm is empty. It results in the block being held by the arm and the arm longer being empty. Unstack block This action be taken if:
46

---

The block is top of another block. The block is clear. The arm is empty. It results in the block being held, the block below it becoming clear, and the longer being emply. Put down block This action can be taken
The block is being held. results the block being the table, clear, and the arm becoming empty. Stack block - This action can be taken
block is being held. The target block is clear. results in the block being on top of the target block, the target block no longer clear, and the arm becoming empty. Let's write Python function that generates successor states for given state:
def httpreeeeeccccssssssttteeeetccie
successors O
clear_ble clear_blocks (pred.split() [1] for pred state if startsssssssstht((cceea"""
on table_blocke (pred.split()) [1] for pred in state if
holding_ block next (apred..pppicee [1] for pred in state if pred. startssiti (""ooiiinhhhhhh"""" None)
arm_empty "arm-empty" in state
on_top ((pred.split() [1], pred.split() [2]) for pred in state if pred. startswith("on
Pick up a block
if arm_empty:
for block in clear_blocks & tatttaallbllckke
new_state state[:]
new_state. reootee cf"leear (block)") wwr.atate.remov (f"on-table [block)")
new new_state eas_state.remove remove (aaoovee""ammmma"" ("arm-empty")) new_stater .tt.tt..e..pppaddle.e.... (f apppnddp (f"holding (block}")
successors. append app.ad (neu_statt)
Unstack block if arm_enpty:
for top, bottom top:
if top in clear_blockss
new_state = state[:]
pax_state.remove (f"clear {top}")
new new_state.remove of on"on (tep} (bottom)")
bex_statterreove remove ("arm-eett)))
new_state append (f"lodding (top}")
new_state (f" (bottom}")
successors. append (new_satt)stttee))
Put down block if holding. block:
new_state state[:]
new_state. remove hf"hilding hholding.blockk)
new_state. append of (holding. block)))
new_state append clear tholding.block!") new_state. append ("arm-empty") successors. (nev_state)
Stack block if Boddnng_bocck for block in clear_blocks. if block != holding_block
new_state statel:]
new sax_state.remon hfolddi hoolding.block!1
new new_state. remove (f"ovee""clear (block)")
new_state append (f (holding. block) (block)") new_state append ("arm-enpty") successors append (new_state)
return successors
This function computes the successor states by iterating through possible actions and checking the constraints of each action. It then returns list of new states based on the applicable actions.
47

---

User:
Provide only the python code that tests whether state is goal state. Assume that the goal configuration provided in the format.
ChatGPT: To check whether given state matches goal configuration, you can write a Python function that compares the states. Here's simple Python function that does this by ensuring every predicate in the goal state present the current state:
def is_
_e__oal__stte _ttte (current_ state, goal_state):
current_state_eet _oal_state_set = set agoal_ssstatt return goal_state_see
set (etureentt_stat)
current_state_ser
This function works converting both the current and the goal state into then checking if all elements goal state set are contained within the current state set. This approach handles cases where current state might have extra predicates not relevant to the goal, as long as all predicates describing the goal configuration are satisfied.
User: The generate successor states function the case stack block does not add clear for the block that was holding. Please rewrite.
You're re correct about the issue in the "Stack a block" section of the generate_suceeseee _ecc_usuuc_sstaaee function where it does not correctly handle the clearing status of the block being held after it is stacked. Here's 's the corrected version of the function that ensures the is correctly updated:
ChatGPT:
def generate_succeseeeeeeeeeeeeeeeee
(state):
successors
clear blocks =
clear_blocks
[1] for pred in state if sttrtatittt taall_ablcleeeeeeeeeeaabbaaataaa blocks ppreddsplit() [1] for pred in state if pred. startswith
holding block next next(pprd.sspitt [1] for pred state if ttrerssttaatwtthhhholohii None)
arm_empty "arm-empty" state
on_top = ((pred.split()) [1], pred.split() [2]) for pred in state if startswitttt
Pick up block
if arm_ empty:
for block clear clear_blocks blocks & on_table_blocks:
new_atate state[:]
hew_state.remove (f" (flcl (block)")
new_state of"ontabbe (block)")
new_state remove (erooeeearmmm ("arm-empty")
(f new_state (f (f"ohlding (block)")
successors append (nex_staaee
Unstack block if arm_ emmpempy::
for top, bottom in on_top: if top in clear blocks:
new_state state[:]
www.staaterrroooeeeeeee (cllea (top)")
(oroovee"rrmoo (top) (bottom)""
new www.atateeremove ("amm-erptt")
new_state append new_stateeappend (f {top}")
new new_state append cf"clear {bottom]"")
successors. www.:orosspppeeeee__ssssssss
Put down block if holding.bles block:
new_state state[:]
hew_state.remove remove hholding blockkng_block""
new_state append (f on-tablee (holding. "lock)")
new_state. append clerr tholding.block"1
new_state. (ireendd"armmeer" ("armeemtm)))) successors. append awwend (neu_state)
Stack block if holding.bloce
48

---

for block in
if block != holding- block:
nes_state state[:]
(1
booldingbloock]]
remove cl"ll (block)")
new_state apepsee.ppnnddccc of (holding budding_block block) (block)")
new_state.. app.nd (f"clear (holding.blockk")
new_ www.eatee.pppnddd appendd"arm-eertte ("arm-empty") successors. append (nev_atate)
return successors
Now, when stacking block, the function correctly adds the clear predicate for the block that was previously held, reflecting the that after being placed on another block, accessible again (unless it gets covered later). This is important for maintaining an accurate description of the state for future actions.
User. Also, when getting the on_top blocks from the state, we must differentiate the predicate "on x y" from "on-table x". They both start with "on". Please fix.
ChatGPT: You're right; the current way the function checks for blocks on top cach other can mistakenly include "on-table" predicates due to the substring match. To diffcrentiate properly between "on x and "on-table x". we can use more precise filtering method. Here's the updated version of the generate_ sneeratessuccessor states function with this correction:
def starrratessccesso__sseeee
successors clear blocks = clear_blocke (pred. splitt)) [1] for pred in state if pred. startssitch
[1] for pred in state if pred. startawith
on, hable_blocke (ppt((ppr..ppli [1] for pred state if teeesssttraawthhhhhdditit None)
holding_ block next
arm_empty "arm-empty" state
on_top = ((pred.split()) [1], pred.split() [2]) for pred in state if pred. startswihhtt ") and len
Pick up block
if arm_ empty:
for block clear_ clear_blocks blocks & on_table_blocks:
new_atate state[:]
new_state.remove (flc (block)")
new_state of--ntabbe (block)")
new_state remove ("arm-eeaammmmpttt))
(f new_state (f (f"ohlding (block)")
successors append (ppx_atteee
Unstack block if arm_ emmpempy::
for top, bottom in on_top: if top in clear blocks: new_state state[:]
cew.ttateerreoeeeeccceee clleea (top)")
new_state.remove re (top) (bottom)""
www.statee.emove ("am-eerpt"))
new_state append new_statte.apeen (f"hodding {top}")
new new_state append cflcl {bottom]"")
successors. wtw.:oossspppeeeee____s_so_.
Put down block if holding.bles block:
new_state state[:]
hew_state.remove remove hholding boockkng_block""
new_state append (f on-table (holding. "lock)")
new_state. append clerr tholding.blockk"
new_state. ("atondd"mmmmee"" (marm-eett))) successors. awwend (neu_state)
Stack block if holding.blocd
49

---

for block in
if block != holding block:
nes_state state[:]
vek_state.rrmove (foolding booldingbblockll
renoaee cllal (block}")
new_state arepsee.pppnddcccc of (holding buddnng_block block) (block)")
new_state.. append (f"clear (holding.blockk))
new_ www.aatee.pppdddddee apppnnd"arm-eertdeeeeeee ("arm-empty") successors. append (nev_atate)
return
This update that the function specifically checks for predicates that exactly match the format "on by splitting the string into exactly three parts, which helps accurately identify the "on" relationships between blocks, without confusing them with "on table" predicates.
B.3.3 Experiment Run
User: 1 am playing with set blocks where need to arrange the blocks into stacks. Here are actions can do
Pick up block Unstack block from on top another block Put down a block Stack block on top of another block
have the following restrictions on actions: can only pick up or unstack one block at time. I can only pick up unstack block if my hand is empty. can only pick up block if the block is the table and the block is clear. A block clear if the block has other blocks top it and if the block not picked up. can only unstack block from on top another block the block am unstacking was really on top of other block. I can only unstack block from on top of another block if the block am unstacking is clear. Once pick up or unstack block, am holding the block. I can only put down block that am holding. I can only stack block top of another block if am holding the block being stacked. can only stack block top of another block the block onto which I am stacking the block clear. Once put down or stack block, my hand becomes empty.
a state defined by list of following predicates "clear block", "on-table block", "arm-empty", "holding block", and "on block1 block2", how are successor states defined? Provide only the python code that returns a list of successor states for state.
ChatGPT: To define the successor states for a block stacking game based on the rules and actions you't 've described, we create Python function that checks the current state against possible actions (Pick up, Put down, Stack, Unstack) and returns all valid successor current situation, such as which blocks are clear, which are on table, whether arm is empty. which block being held, and the stacking relations between blocks.
Each state of predicates that describe the
Here's Python function to calculate successor states:
def wtw.suuccessor_sttteeeeeeee (state):
successor _statesettessseeeeeee = 0
arm_empty 'arm- emptyy in state
held_blos block None
on au__able__blen blocks set()
clear_blocke = set()
stack
0
Porsing the current state for predicate in state:
if
., block predicate.sspist
on, mn_table_blocke (bllckk
elif predicate.starteviteee clear")
block = predicate.split
clear_blocks add
elif www.ttttttttttolttttttttoo
block predicate.split
held_block block
elif predicate. sstattswithh'on
top, bottom predicate. Spldit
50

---

stack_reelationsttopl bottom
Possible actions if the arm is empty if arm_ cmm_enpt::
# Option to pick up a block from the table for block in klear_blocks tnt.rsttioooooottalleleeeee blocks)): new_state
new_state remove armm-empyy')
new_state. remove (f clear (block)')
new_state T...sateerrmooeeC. (f on-table (block)')
new_state (f holding (block)')
successor_states wtt.:ssttteeee.eedddeeeeee (nev_state)
Option to unstack one block from another for top, bottom in if top in clear_blocks:
new_state copy)
new www.state.rmmove ('armmerpty"
uew_state.remover clear (top)')
new_state.remove ' (top) (bottom)')
new_state add holding (top}')
new_state af clear {bottom}))
successor satissrrrstttteeeee wwwttstappeeenneeeeeee.ee. (ppaan(eesssstttte)
Possible actions if holding block elif held_block:
# Option to put down block on the table new_state state.copy() new_state add ('arm-empty')
new_state add (f' clear (held_ ()ock)
new_state add (f on-table [held_block]
new_state remoove (f holling (held_block)") successor sattasorssttteeessse appbend (new_staat))))
Option to stack for block in clear_ blocks:
held block on clear block clear_blocks:
if block != held_block: # Can 't stack on itself new_state = state.
new_state add(''am-emptyy"
new_state (fd (held_block) (block)')
new_state www.staaeeremoveedee (f' clear {block)'
new_state remove (f holdinn (held_blockk))
successor states.tts......e append (neu_state)
return Siccessor_states
This function performs following steps:
Parse current state to identify clear blocks, blocks on table, block being held, and relationships between stacked blocks. Determine possible actions based on whether the arm is empty or holding a block. Generate successor states for each valid action, taking care to update predicates accurately to reflect the new state of the world after the action. Return all successor states. You use this function providing it with set of predicates that describe current state, and it will return list of all possible successor states.
User: The generate_ successor states function in the case of stack block does not add clear for the block that it was holding. Please rewrite.
You're re right; when stacking block, the previously held block should still be clear after the action, nothing is on top in the new state. I'll update the code handle this correctly. Here's the revised version of the function:
CharGPT:
def get_successor_ set_ssccessor_stateess satprsssorssttteeeeeeee stw.tttttttsstt (state):
successor_ sottessssrssateess states []
arm_empty ''arm-em 'arm-empty' in state blld_block None
51

---

Now the held block is clear tt_table_blocke set () clear_blocks set()
stack_relations +
Parsing the current state
for predicate in state: if tttarrtttttattttttttoootoaeae
block appediiaate.pllt((
on adw.lee_block..addc@@oo..
elif
clear"") block predicate.splitt
Elear_blocks adblocck)
elif htw.:aatt..tartttttthhhhhhheet hholdiggg
- block = predicate.split()
held_block block
elif predicate predicate.startstet sttrrssittttttttthhe
top, bottom predicate.splite stack_relations [top] bottom
Possible actions of the is empty if arm_ empty:
Option to pick up a block from the table
for block in clear clear_blocks blocks........o...o interaectionnon. ton_taaeeeeneeeeeeeeeeaaaaaeaeee t new_state
siate..ooy()
new_state.rent remove arm-empty')
new_state. remove rewove (f clear (block)')
new_state. remove (f on-table (block)')
nev_state. add holding (block)')
muccessor_stated wetaaee.pppeeddaeeeeeeeeeaa (neusstatee))
Option to unstack one block from another for top, bottom in atack_relations stttettions.items iitemmm)) if top in clear_blocks
new_state state.cop copy()
lew_state.remov maamoveeaarmepptteeee cemm-empy')
new_state.remove clear (top)')
wwr_state.remove on (top) {bottom}')
new_state (f 'holding {top}')
new_state ad clear {bottom}'')
successor_statee wtt.tttee..pendeeeee.e. append (neu_state)))
Posssble actions if holding block elif beld_block:
Option to put down block on the table new_state state.copy()
new_state add (' ('arm ('arm-empty') new_state. add (f clear (held_blockk'')
new_state add (f on-table (held_ boold_block])
new_state remove 'olliin (held_block)] Siccessor_states append (new_state)
Option to stack the held block clear block for block clear_blele blocks:
if block != held_block:
Can't stack on stself new_state state.copy()
new_state. add add('armeenpyy))
new_state add (held_block) (block)')
new_state add clear (held_blockk'")
clear new_state remove clear (block)')
new_state rew.steeerrrmooe(C holding (held_block)
successor_state statesstteeeeeeeeeeeeeseeeeeeeet hatpesooor_aattes.pppannnneee.e. (neu_saae)t
return successor_statte
52

---

In this corrected version, when the held block is stacked on another block, the clear predicate for the held block is added the state indicate that nothing is on top of it. This maintains the integrity of the game's 's logic and ensures state reflects the actual conditions after each move.
User:
Provide only the python code that tests whether state is goal state. Assume that the goal configuration is provided the same format.
CharGPT: Here's Python function that checks whether given state matches specified goal state. This function assumes both the current state the goal state are provided as sets of predicates the same format. The function will return True all predicates in the goal state are present in the current state, indicating that the current state meets all the conditions of the goal state.
def is ___goal__aa_ Staal_stateeCee (current, atate, goal_ stale) nun
Check if the current state matches the goal state.
Args:
current_state (set): set of predicates describing the current state. goal_state (set): set predicates describing the goal state.
Returns:
bool: True if the current state matches the goal state, False otherwise.
n.n
return goal_ goal_state... issubaet (current_state
This function the issubset method of Python sets, which returns True if all elements of the goal_state present in the purrent_state, This approach ensures that the current contains least all the conditions defined by the goal, but may also contain additional predicates not specified in the goal, reflecting possibility of multiple configurations that satisfy the goal requirements.
B.3.4 Experiment Run 4.
User: I am playing with a set blocks where need to arrange the blocks into stacks. Here are the actions can do
Pick up block Unstack a block from on top of another block Put down a block Stack a block on top of another block
I have the following restrictions my actions: can only pick up or unstack one block at time. I can only pick up unstack block my hand empty. can only pick up block if the block is the table and the block clear. block clear the block has no other blocks on top unstack block from on top of another block the block am unstacking was really on top of the other block. I can only unstack block from on top of another block if the block I am unstacking is clear. Once pick up or unstack block, am holding the block. I can only put down block that am holding. I can only stack block top of another block if I holding the black being stacked. the block onto which am stacking the block clear. Once put down or stack a block, my hand becomes empty.
and the block not picked up. can only
only stack block top of another block
a state is defined by list of following predicates "clear block", "on-table block", "arm-empty", "holding block". and "on blockl block2". how are successor states defined? Provide only the python code that returns list of successor states for state.
ChalGPT: To define the successor states for given state in this blocks world, we can implement function in Python. This function will take the current as input and return list of all possible successor states based on the actions described (pick up. put down, stack, unstack). Each state can represented as of predicates that describe the conditions in the environment.
Here is the Python code that defines the function to generate successor states:
def srccessor_statese (state): auccessors 0
Helpers to check conditions and update states def is_ us_clear(bloc (block): return f"clear (block)" in state
53

---

def ta_on_table(bloctt)
return foon-table [block]" in state
def is (.booddingbblock::
return I 1"holding (block)" " in state
def is is_arm_empty()
return "arm-empty" in state
def bodd_lloo_o_o_ooplolloe for predicate state: if wredicate.startevithhccc ") and predicate. endswithh (block). return predicate.split())
return None
Actions
def pick_upbblock):
if as_on_table (block) and cllerr and is_arm_enpty()
new_state copy)
new_state remove (fn-ttabbe (block)")
new_state. new_staaee.reove re nev_atate www.stateeremoveeee remove (emmaetrremeeeammee ("arm-empty"))
[block}")
new_state add (f"oldinn holldng (block)") successors. (ap.uoresappeeddessssttttt. (nex_stae))
def put put_dowm(blockk)
if is ts__holdingbblo. (block):
new_state staye.copy(
new_state.. add of"on- (block)")
new_state add clerl {block)") new_state add ("arm-eempty"" new_state. remove (fohhiiig {block)") successors apenndnee
def unstack (block, below):
if is_ ss_cceerrbblock and f"on (block) (belov}" in state and is_arm_ _0_arr__mpppt exmm_)) new new_state state.copy()
new_ aemove(f"on (block) (below}")
new_state (f (below)")
new_state remove ("arm-emptt)))))
nsw_state. www.sttatereemoeeeee. (f" (block)")
hew_state.o ww__stateeadd(1"" afd(dino (block)") successors. Srceessors.appendnnnee append
def stack (block, target):
if is h..holddingbllock and is_cleer (target):
new_state state.copy()
new_state aew_staate.ddd("or [block} (target}")
new_state. (fmevee (target)")
new_state aatpaatteeddd"aaammmmmeeppp
new_state reu_state..remveed (f hholding (block)")
successors. append (new, statte
Try each action with each block
blocks = spriicate.slit(() [-1] for predicate in state if "block" in predicate. [-1]] for block in blocks:
pick_up(block: put put_doumbblock for other_ block in blocks:
if block !- other block: unstack (block, other other_block) block) stack (block, other_block
return successors
54

---

Example le usage:
surrent_state set (l"clear A", "on-table A", "clear B", "on-table B", "arm-empty"] successor_ siceessorssaatessccurrretsttsass
This code defines actions (pick up. put down, unstack, stack) checks necessary conditions before applying an action. Each action, when applicable, modifies the state accordingly and adds the new state to the list of successors. The function assumes that each state is uniquely represented by set of predicates and each block involved in the predicates is uniquely identifiable.
User: Provide only the python code that tests whether state is goal state. Assume that the goal configuration provided in the same format.
ChatGPT: To test whether given state is goal state in the blocks world, can implement Python function that compares the current state against specified goal configuration. The function will check if all the predicates the goal configuration are present in the current state. Here's the Python code this function:
def is ss_goal_sttae state (ccurent_stater goal_state)) au.
Checks if the current state meets all the conditions of the goal state.
Parameters:
current_state (set): set of predicates representing the current state. goal_state (set); set of predicates representing the goal configuration.
Returns:
bool: True if the current
goal state, False othervise.
return goal_state <= current_state
This function uses set containment check (<=), which checks if all elements of the goal state are present in the current state, thus cnsuring that all conditions required by the goal are satisfied in the current state.
User: The successo states function in the case of stack a block does not add clear for the block that it was holding, Please rewrite.
You're correct; the case of stacking a block on top of another, the code should ensure that the block being stacked becomes "clear" after the action, as there would be nothing on top of it. Let's revise the stack function within the successor states method correctly reflect this:
ChatGPT:
def successor_ states successors []
Helpers to check conditions and update states def is is_clear (block): return f"clear (block)" in state
def is_on_ ta_on_ttbblebllock)
return fonnttbble (block)" in atate
def is bbloingk
return f"holding (block)" state
def em
return "arm-ema emmtye in state
def hind_block_on (aacck_oo_top(block)
for predicate in state: if predicate.co starresithh ("on ") and predicate. th (block): return predicate.split() [1]
return None
Actions def :ick_u)bblock): (block):
55

---

if us_on_table (block) and is_clear(blochh and is_arm_enpty O nev_state state.copy()
new_state remove on-taall {block)")
we..tate.remov (f cflar (block)")
new_state penrateerrmove rww.oeeareeeeeeeeeaare (marmemmttt)))))
new_state add af (block)")
successors.co appced (new_
def put pat_doom(bloock/
if is ss_holding((ien) (block):
new_state state.copy()
new_state (f [block}")
new_state add clear {block)")
new_state add(marm-mmpty new law_state.o remove (fhhoding (block)") successors. append append (nex_state)
def unstack below):
if is_clear(block) and f"on (block) (below)" in state and ar__rmmemmptyllp new_state state.copy()
new_state remove(1"oz (block) (below)")
new_state add ae clear (below)")
nev_state rew_state.remove rewove ("arm-empty")
new_state weetraaterreoveeeee.eee new_state of hf (block)") successors.o cetesorrsaappennn append (nex_state)
(block)"")
def stack (block, target):
if hw.hooldingbblooe and is_ clacrrrrr (target):
new_state state.copy()
new_state on {block} (target)")
new_state reweeeee (target}")
new_state add (fs_statee.dddfcc1111 {block)")
new_state add("arnemmpty") new_state. eentateeremove. "holding (block)")
successors append (nex_state)
Try each action with each block
blocks {predicate.splite [-1] for predicate in state if "block" predicate. split [-1]] for block in blocks:
pick_up(block) (tt__ommbblock) other_block if block != other_block-
for other block in blocks:
unstack (ustackbblokk, other_blocks stack (block, ttherblock:
successors
Example le usage:
current_state set set( (["clear A", "on-table A", "clear B", "on-table B", "arm- farm-enpty")) successor_states (curreeettee
This updated version stacked, ensuring it's clear because there's 's nothing on top of after the action. This update reflects the logic needed to maintain a consistent state representation after stacking operations.
stack function now correctly adds the clear block predicate block being
User: The blocks are not correctly identified from this state representation. Please fix
ChaiGPT: You't re right; the approach identifying blocks from state representation previous code flawed because it inaccurately captures block names when parsing predicates. I'll correct this by implementing a more reliable method to extract block names directly from the predicates involved in the state.
Here's 's the revised version of the auccessor_state, function that includes an improved way to determine the set of blocks based the current state:
56

---

def saccessor_statess (state):
0
Helpers to check conditions and update states def us_clear(blorg (block):
return f"clear (block)" in state
def is_on is_on_table
return foonttbble (block)" in state
def is (blcoo)) return f'holding [block)" state
def is_ es_arm_eppty()
return "arm- cemm-empty in state
def find bww.bookkoontoppllcck...
for predicate in state: if predicate predicate.starrtswtth"" ") and predicate. endsvi th (block): return predicate.splie split
return None
Actions
def pick_up(block):
if is_ (a_on_table(block) and (slock)((block)) and is 1s_ayy)enpty()
new_state copy(
iil.staterreovv (f {block)")
new remove (f"clear (block)")
new_state remove ("arm-enpty"))
new_state add hooldig "block)")
successors.. appeed (p
def pat_dowm(block))
if is ts_holding (block):
new_state state.copy()
new_state add of (block)")
new_state (fd {block)")
new_state. www.sttttaddd"""...... ("aam-emmpyy") new_state. remove (f (foohlding (block}") successors. append append (nes_state)
def unstack(block, below):
if (blcck)rbblock) and f"on (block) (below)" in state and ae_arm_empty() new_state state.copy()
new_state rew_staaterreooee(""M (block) (below)")
new_state add nev_stateeadd(("clll (belou)")
wwt.state.reoove www.veeeerm-eeeeeeeaeeeeeee ("arm-empty")
nev_state remove (f"clear (block)")
new_state add hollddng "block)")
successors. append
def sbockk, target):
if is (bloolliingbblock) and is_ clear (targett:: new_state state.copy() new_state. on (block) (target]")
new_state (eu_staate.rmoov remove(f"clea) (target)")
new_state (f (block)") new_state add add emary-empty") new_state. (f (block)") successors. append (new_state)
Estract blocks from the state blocks set() for predicate in state:
parts predicate.splie split()
57

---

Add block names (assuming the last word in each predicate is block name or another block) if "table" in parts [-1]: avoid adding 'table' block
blocks upatte
Try each action with each block
for block in blocks:
if "block" in block: Filter ensure only block processed
if True:
pick_up(block)
put_down (block)
for other_blocd block blocks:
block != other_block
(blocck other, other_block) block)
stack other_block)
return successors
Ecample usage:
current_state set set cl"clear A", "on-table A", "clear B", "on-table B", "arm- cemmmempt")] successor_states (current_state)
In this revised version, have improved the block identification by iterating over all predicates and extracting block names from them directly, ensuring that only valid block names are considered actions. The revised code assumes that block names can be identified as they don't contain the word "table", thereby excluding predicates like "on-table". This approach provides more robust way to handle different naming conventions or additional predicates that might be added to the state.
User: There is no need for checking if the string "block" is part of the block name, it might not be. Please update the code.
CharGP:: You're re right; assuming that the term "block" will be part of the block's name in predicates is restrictive and may not apply to all scenarios. Instead, we'll directly gather the block names by processing the appropriate parts of each predicate more accurately. Here's an updated version of successor states function that extracts block names based on the structure of predicates without assuming specific naming conventions:
Sateeassorrtttteeeeeeeee def successor__ states (state):
successors = 0
Helpers check conditions and update states def (s_clear(block)
return f"clear "block)" in state
def is is_on_ tabbbcbbll)l))k): return toon-table [block)" in state
def is hhoding(bbocc))
return 1 1"holding (block)" " in state
def is ernpytt
return "arm-empty" in state
def find_block, baaabllooooo__to@tolee topbokkb))ckk
for predicate state: if seeiicate.starssvittte.o ") and predicate.endsvitthe block) return predicate.splitd
return None
Actions
def bick_upbllock:
if is_on_table (block) and is_clearblloct (block) and is_arm_enpty(:
nev_state copy()
new_state. remove of"on-table (block)")
nev_state. (f"clear (block)")
nev_state remove ""arm-empty"" new_state (f (f"holiig (block)")
58

---

nuccessors.appenddeee....
def (r__doummblock))
if is Ls_hoodingg(bll (block):
new new_state state.copy(
new_state add on- (block)")
new_state add cleea {block)")
new_state add ("arn-empty") new_atate. eenrateerrmoee (f (f"holding [block)")
successors append (nev_sstee
def unstack(block, below)
if is ts_clear(block, and f"on (block) [below}" state and _s_arm_empty" new_state state.copy)
new_state.o mew_stateeremove remove (block) (below)")
new_state new_state. removea ("arm-empty"))
{below}")
new_state remove remove cl"cl [block}")
new_state adu_ttate.add("" add (foldding (block)")
successors. aptpeessrrsapppeddddee (peu
def stack target):
if is 1s_holding(bloct) and is clear(ttrgee)) new_state state.copy()
new_state add (block) (target)")
new_state remove (f" (target)")
new_state (fd clear (block)")
add("arm) new new_state aadrateeadd(ce ("arm- (aarr---mpt") new_state remove (f "holding (block)") successors. append (new (nea_state)
Extract blocks from the state blocks = s) for predicate in state:
parts predicate.split/
if parts[0] in ["on", "clear", "holding", "on-table"]: # Assuming the block names are always the last element in predicates that involve block names blocks. (dd if parts
"on": also add the block beneath if predicate starts with "on" blocks. blocks.add (pd
Try each action with each block for block in blocks:
pick_up(block) put pat_dommbblock for other block in blocks:
if block != othck_block: unstack (block, other other_block) block) stack (block, other_block
return successors
This version extracts block names from predicates based their position in the string, which is typically the last part for predicates like "clear", "on-table", and "holding". For predicates starting with "on", both blocks involved are considered. This method ensures blocks are appropriately identified and handled without presuming the presence of specific substrings like "block" in the block names.
B.3.5 Experiment Run 5.
User: 1 am playing with a set of blocks where need to arrange the blocks into stacks. Here are the actions can do
Pick up block Unstack block from on top of another block Put down a block Stack block on top of another block
59

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I have the following restrictions up unstack block my hand empty. can only pick up block if the block is the table and the block clear. block clear the block has no other blocks on top unstack block from on top of another block the block am unstacking was really on top of the other block. I can only unstack block from on top of another block if the block am unstacking is clear. Once pick up or unstack block, am holding the block. I can only put down block that holding. I can only stack block top of another block if holding the black being stacked. can only stack block top of another block the block onto which am stacking the block clear. Once put down or stack block, my hand becomes empty.
actions: can only pick up or unstack one block at time. I can only pick
and the block not picked up. can only
a state is defined by list of following predicates "clear block", "on-table block". "arm-empty", "holding block". and "on blockl block2", how are successor states defined? Provide only the python code that returns list of successor states for state.
CharGPT: To generate successor states based the actions and restrictions you've described, you'l '1l want function that checks the current (represented by of predicates), applies possible actions based the rules, then produces the new states. Here Python function that accomplishes this:
def successor_stee satreassorrssttteeeeeeeee
Break down the state into predicates clear {b for b in state on on_table = {b for b in state if estartawith""on-ttalet ")} arm_empty "arm sarm-empty" in state
..startavith"clee ")}
holding [b[8:] for in state if b.startswith ")}
on [[b[3:], b2) for state for b2 in state on ") and b.nddwwithhb22 (b2)) successors
if arm arm_emmaa empty:
Pick up a block from the table
for in clear:
block (.split()
if f"on-table (block)" in state:
new_state state.copy()
ros_stateeremovee"mmmeee
new_state. remove remove cl"r {block}")
new (f" new_state add (ww.ssaae..ddd(fffffe afoldingg (block}")
(block}")
successors wwwpssorssapppeeooo append
Unstack a block from another for (block1, block2) in on: if f"clear [block1}" in state:
new state.copy() rew_staate.reovee "iaitatte_rroovee""aaaeeepp"'"" ("arm-empty")) new new_state.remove (f" (block1)")
new_state (Fr {block1} {block2) new_state add(f" (block2)"")
new_state adoli"n (block1)"" successors. append (new_state)
else:
Put down block
for block holding:
new_state state.copy()
new_state www_staaee.dddd"aaaa add("armeempty")
new_state ait_sttatedddff"" (fd (block)")
new_state add of (block)")
new_state. deu_state.remove rf "holding (block)") successors secceesors..apendnnnn
Stack block top of another clear block
for block holding:
for clear_ block in clear:
if clear biaar_blokkeae split [1) !- block:
new_state state.copy.
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---

new_state. add hwwpssttttteddaaaamaaaaaaaaaaaae ("arm-empty"))
nex_state remove cf*c [clear_block.ee blockk () [1]}")
nei_state remove (f"holding {block}")
nea_state add on (block) (clear_block. baatrbbooc..ppp... [1])")
new_state. add cl"re {block)")
successors nuccessors.apponnnuvu.tcoo
return successors
current_state set ( (["clear A", "on-table A", "clear B", "on-table B", "arm-enpty"))
print (successor_ states (current_state))
import prt.ppee_lloockkee blocks def
dict_to_listtetattee
('clear': ['b', 'on-table': ['d'], 'arm-empty's True, 'holding': None, 'on': [l'a', 'c), (bb, 'a') {: ['d'], 'on': [('a', 'c'), ('b', 'a'), ('c', 'd')}} new_state
if "arm-empty" in state and state "estee"armi emprm= new_state.appent ("arm-empty")
if 'holding' in state and state['holding is not None: new_state.apper append (p hf"lnldin (statel (state("holding))
for b in state ('cear' :
new_state append cflc {b}")
for in state -tatee'on-table
new_state append (f" (f"on-talle (b}")
for top, bottom in state['on']]
new_state ttt.sttee.apppddeee.. on (top) (bottom)""
return new_atate
def blocks_state_dicee
return set (bllccksssttteeeeeeeeeeee ticttdditttttt_ssssttttttt
init, goal parse_problem_ pttp://dw...bboooooprrrppdp....d
hateee/occcccsssstrrrretttttttt "ddd")
init htts__sttt__iicc_ttee (init)
goal =
(goal)
print("Init")
print(init)
print(goal)
print("---....>..
init. copy(
for in range(10):
print(curr)
succ successor_states (curr)
print(suce) print("................ if (succ)))) > 0:
import random
random. choice (aucc)
This function assumes the state parameter is set strings representing the predicates. The actions available depend on whether the robot's 's arm is empty or holding block, and whether certain blocks are clear and available on the table or can be unstacked from another block. This function then modifies the state according to the rules of each action and adds each resulting state to the list of successor states.
User: Provide only the python code that tests whether state is goal state. Assume that the goal configuration provided in the same format.
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ChatGPT: To test whether given state matches goal configuration, can use Python function that simply checks if all predicates specified in the goal present in the current state. Here's concise function perform this check:
def is as_goal_stta (sta_ttettttttee goal): return goal. issubset (state)
This function receives two sets: state and goal, where each is of strings representing the predicates. The function Truc predicates in the goal contained within the current state, signifying that the state meets the goal configuration, and False otherwise. This approach straightforward and leverages Python set operations for efficiency.
User: The successor_state function does not correctly parse the state into predicates. Please rewrite.
Certainly! seems was a misunderstanding how predicates "on" were being parsed. adjust the parsing logic in successor _sttttsssssss function to correctly handle the predicates relations between blocks. Here's the corrected version of the function:
ChaiGPT:
def seceessoo_stateseeeeee
Parse the current state
clear_ clear_blocke blocks (b. [1] for state if ststrttswith ""eartssichhcllea""" ("clear")) on ho_ttabl_blloce = (b.appit()) [i] for b in state if (ttarttavtth""ontaalleeee arm_empty "arm- "arm-empty" in state
holding_bloch blocks (b.split() [1] for in state if D.startswith(hholddnng en_top_of ((b.split() [1], b.split() [2]) for in state b.startswith("on ")}
successors 0
if arm_ empty:
Pick up a block from the table
for block in clear blocks intersactiionnonntaab.. blocks)
new_state copy)
rew_state.remove(" m"ammo emptyy")
pev_state.remove ccllea (block)")
new_state. rew_staate.remoee (f" of-tnt (block}")
new new_atate. af (block)")
Unstack a block from another for (top, bottom) in on_top_of: if f"clear (top}" in state:
new_state state..copy new new_state.remon ("armeeeptt"""
new_state. remove cl"re {top}")
new_state (e (top) {bottom}""
new_state cfd ees_state.add(("cleer {bottom}"")
new_state add hholddng (top)")
successors append (new_statee
else:
Put down block
for block holding.blocks.
new_state state.copy ()
new new_state add add("arm-empty")
new_state "f__staaee.ddd(f (fd(f"clear (block)")
new_state aad.sttte.ddd(f"........ (fd (block)") new_state remove hholding (block)") successors.apaee (ngendd(ee... (nex_state)
Stack block top of another clear block for block holding_block. blocks:
for target in clear_blocks
if block != target: # Cannot stack block on itself nev_state copy()
new_state add ("arm-emptyy
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new_state remove (f" cf*rl (target)")) nex_state remove hfoldngg "block)") new_state on [block] {target)") nea_state add clerr {block)") successors.o append
successors
In this revised function:
I ensure proper parsing of each predicate correctly identify which blocks clear, which the table, which one being held, and relationships between blocks (on top of). Operations like picking up block from table and unstacking are allowed only when arm is empty, and the corresponding block clear either on the table or on top of another block. The stacking and putting down operations are processed when holding block, respecting the restrictions clear targets and table placements. This code is more structured and should correctly handle the transition between states according to the provided rules and predicates.
User: When getting the top blocks from the state, must differentiate the predicate "on from "on-table x". They both start with "on". Please fix.
ChatGPT:
You'r re right; the initial parsing of the predicates didn't 't differentiate between "on-tahle" and "on" predi- correctly. I'll adjust the function parse these predicates appropriately. Here's the updated successor states function:
def swccessor_statteeeeeeeeeeessssse (statee)
Parse the current state
clear_ blocks = {b. s) [1] for in state if b. ..statttiich"cleeeat clear_blocks
on_ taattalle_____ blocks [b. (b.split( [1] for b in state if b.startssiithhoon-tabll arm_empty "arm-empty"
state
holding_blocke (b.split() [1] for state if Shartawith("hodinnn
on_top_of {(b.splittt..... [1], b.split() [2]) for b in state if b.startawith ") and len len(b.split()) = 3
successors
if arm arm_ empty:
Pick up a block from the table
for block in clear_blocks blocks.c....... bleer.blocce.inteereeeee. iaterreectionnoon__bblee... :lock))
new_state
copy((
new_state. remove ("arm-empty")
sev_state.remove (f"clear (block)")
new_state hrr.stateereeooeee (f"veeffooo (f"onttable (block)")
new_state hew_state.add(f (f (block)")
successors. srncessors.aapendnnnee
Unstack a block from another for (top, bottom) in on_top_of: if f'clear (top)" in state:
new_state state.copy
ret.staee.reoooe ("arm-erpty")
sew_state.remore (f" {top}")
hew_state.rrmoveeff remove on {top) [bottom}"")
new_state add(f"clear {bottom}""
new_state. afd(f""olding (top)")
successors. append (new_state)
else:
Put down block
for block in golding_blockss new_state state.copy() ()
new_state www.etaate.ddd"aamm..o addmm---eempty"
new_state add (f" (block}")
new_state add of (block)
waw.staaeerrmooo((""""""e. successors (wr.esorss.ppooodeesssstttteee
(block)
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---

Stack block top of another clear block for block in holding-bloce blocks:
for target in clear_blocker if block !- target: Cannot stack block on itself
nev_state state.
new_state. armmeettetttt
rew_state.remove cf (target)")) remove remove hfoohdnig "block)")
new_state. add {block} (target)") new_state new_state.add add cf {block)")
successors auccessors.appnndnevssstteed
return successors
In this revised function:
The top of (which corresponds ensures that only predicates starting with "on" and having exactly three components Y") are included. This ensures accurate recognition and handling the predicates representing blocks stacked on cach other while avoiding confusion with "on-table" predicates. The operations within the function continue to account for blocks being picked up. placed, or stacked according to the rules.
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---

B.4 PrOntoQA
The prompt for the successor function is follows.
Logical deduction performed by applying rules of the form X then to our current knowledge about a world, extending our knowledge with each rule application. rule if then Y applicable if we know If a state defined as our current knowledge of the world, how are successor states defined, given collection of rules? Provide only the Python implementation of successor states function
The prompt for the goal test is follows.
Provide only the python code that tests, given target fact, whether state goal
B.4.1 Experiment Run
User: Logical deduction is performed by applying rules of the form if X then Y to our current knowledge about world, extending our knowledge with each rule application. A rule if X then Y is applicable if we know X. If state defined as rules? Provide only Python implementation.
knowledge of the world, how
states defined, given collection of
ChatGPT: In the context of logical deduction using of rules, can be considered our current knowledge about world. Each element this a fact or a truth we know about the world. Successor states are defined by applying applicable rules to the current state to extend our knowledge.
representing
Here is a Python implementation to model this behavior. This implementation defines function that takes the current state and a collection of rules, and produces the set of all successor states by applying these rules:
def apply_rule (state, rule): "" AAplyy a rule the state if applicable. "". if rule[o] state:
return state (rule[1]}
return state
def generate_ seseraatesuccessor prccessor_statessstta.. rules):
"""Generate all successor states by applying rules the current state. **.
new_states = aet()
for rule in rules:
new_state roply_rulesstate,, rule)
if new_state !- state: Only add new states
new add (nev_state))
return nev_states
Ezample usage
current_ state = ('rain')
rules [('rain', 'wet'), ('sun', 'dry') ) ('wet', 'puddles')
Generate successor states
successor_stan generate_
rules)
for state successor_stater print(stated print (genneate, successon.statss(stats rules))
In this code:
apply_rule checks if the antecedent of rule (first element of the tuple) is in the current state. If it is. it adds the consequent (second element) to the state, generate_successor successor_ states iterates over all rules and applies them the current state, collecting new states that result from applying each rule. Each new state added only it different from the current state, avoiding duplicates. This model assumes that the rules are simple implication relationships and that the state can be represented as a set of facts. The generated successor states are those that include at least one new fact derived from applying the rules.
User: Provide only the python code that tests, given target fact, whether a state is goal state.
CharGPT: To test whether a given state goal state based on the presence of target fact, you can define a simple function in Python. This function will check if the target fact is in the current state. Here is how can implement this:
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---

def st_goal_stateestatee target_fact);
"""Check if the state contains the target fact. "** return target_fact in state
Ezample le usage state ('rain', 'wet', 'puddles') target_fact 'wet'
Test if the state is goal state (rr_gntss_goa_ssee stataall_steetttttee target_fact)) Output: True
This function is goal state takes two arguments: the state (a set of facts) and the target fact (a fact that you want to check for). It returns True if the target fact is present in the state, indicating that this state is a goal state with respect to the target fact.
B.4.2 Experiment Run
User: Logical deduction is performed by applying rules of the form if X then Y current knowledge about world, extending knowledge with each rule application. A rule if X then Y applicable if we know X. state defined as our current knowledge the world, are successor states defined, given a collection rules? Provide only the Python implementation of successor states function.
ChatGPT: implement function for generating successor states using logical deduction approach, you'l need a way to represent the current state of knowledge and the rules. Here's how you might do in Python:
Represent the state as a set of facts. Each fact can be a simple string or a more complex structure, depending your application. Rule Representation: Represent rules tuples (X, Y). where X is condition (or of conditions) and Y is the conclusion. X can be single fact current state. so, it adds
State Representation:
of facts, Successor
Function: The function checks each rule to see conditions are met nule's conclusion to the state, forming a new successor state. Below is the Python implementation for generating successor states:
def successor_ Saaccessor_tttttee states (current_state rules)
Generate all successor states from the current state by applying applicable rules.
Arga:
current, current_state state (set): A set of facts known in the current state.
rules (list of tuples): A list where each tuple represents a rule in the form (conditions, conclusion), where "conditions" can be single fact set of facts.
Returns:
list of sets: list of all possible successor states.
new_states 0
for conditions, conclusion in rules:
Check if the rule's conditions are met in the current state if isinstance (sinstaaceconniiiionn (conditions, set):
if conditions issubset (current_state))
Apply the rule: create new_state current_state, (tt.:at_ttttttttnnoooonoocoooooo ((oon((connl)s)on)) new_states. append www.staates.appeddnneee (neu_staeett
state with the conclusion added
else:
if conditions in current_state:
new_stata current_state www.uniiinnnoo ((noo((cconllusion))
new_states (ww_states.appenddnee... (nen_stat)))
return new_states
This function checks each rule to see the conditions are met within the current state. If they are, it creates a new state that includes the conclusion of the rule. This implementation also supports both single conditions and sets of conditions. Adjustments might he needed depending the specific requirements and the complexity of the conditions and conclusions.
User: Provide only the python code that tests, given target fact, whether state is goal state.
66

---

ChaiGPT: Here's Python function that tests whether given state contains larget fact, effectively checking if is a goal state:
def is_goal_state state (state, target_fact):
uun
Check if the state contains the target fact, indicating it is goal state.
Args:
state (set): The current state represented as set of facts.
target_fact (str): The fact that defines the goal state.
Returns:
bool: True the target fact
state, False othervise.
uun
return target_fact in state
This function simply checks if the target fact is member of the state set, returning True if the state meets the goal criteria (i.c., contains the target fact) and False otherwise.
B.4.3 Experiment Run 3.
User: Logical deduction is performed by applying rules of the form if X then Y current knowledge about a world, extending our knowledge with cach rule application. A rule if then Y applicable know X. state defined as our current knowledge the world, how are successor states defined, given a collection of rules? Provide only the Python implementation of successor states function.
CharGPT: To provide Python implcmentation for defining successor states based on given collection of rules and a current state of knowledge, we need a function that takes the current state (a set of known facts) and the rules (formulated "if X then Y") and produces all possible successor states by applying these rules.
Here's straightforward Python function to achieve this:
def successor_ setteessor_tatteeeeee statee *..
rulea)
Generates all possible successor states by applying rules to the current state.
of strings representing known facts. :param rules: list of tuples (X, Y) representing the rules 'if x then Y
:param current_state:
:return: A set of sets, each representing a possible successor state.
new_states = s) # To store unique successor states
Iterate through each rule and apply it if possible for x, y in rules:
# Check if the condition of the rule is satisfeed in the current state if in current_states Create
state by adding the consequence of the rule to the current state new_state current_state.com copy
new_state.
new_states (frozense (nev_state) Add as frozenset for immutabildity
return new_states
Ezample le usage:
current_stad ('I') rules [(',, 'Y"), ('Y', 'Z')]
print(successeeeee (wwccessorreeeeeooooo (tarret state, rules))
This function works by iterating through each rule and checking if the antecedent (condition) of the rule already known fact in the current state. If it is. it creates new state by adding the consequent (result) of the nule to copy of the current state. Each new state added to of successor states, ensuring that only unique stored. Fach state is stored frozenset
immutability and to allow sets of
User: Provide only the python code that tests, given target fact, whether state is goal state.
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---

ChatGPT: Here's Python function to check whether given state includes specific target fact, thus determining if it's goal state:
def is ___goal _e__oal_stateeeee_____s_s_ (state, target_fact):
Checks if the target fact in the given state.
:param state: set of strings representing the current state.
:param target_fact: A string representing the target fact to be achieved. :return: True if the target fact is in the state, False otherwise. aun
return target_ in state
Ezample usage: state ('I', 'Y', '2'}
target_fact
target.fact))
Output: True
User: In successor_statee I get AttributeError: 'frozenset' object has attribute "add'
The error arises because I incorrectly used frozenset for the new state, which is immutable and doesn't support the add method. We should first modify the mutable set before converting to frozenset when adding it new_states. Let's correct the implementation:
CharGPT:
def seceessor_stateer (current_statee rules)
Generates all possible successor states by applying rules to current state.
:param current_stated A set of strings representing known facts. :param rules: list of tuples (X, Y) representing the rules 'if X then Y'. :return: A of frozenseta, each representing possible successor state. aun
new_states = s) # To store unique successor states
Iterate through each rule and apply it if possible for x, y in rules:
Check if the condition of the rule satisfied in the current state
if x in current_state:
# Create new state by adding the consequence of the rule to the current state new_state set (current_state, new_state. ay) hew_states.adee (frozenst (nev_state)) Store as frozenset for immu tabelaty
Nake mutable copy of the current state the new fact
return new_states
Ezample usage: current_state ('I') rules = [('I' (Y'), ('Y, 'Z')]
rules))
This revised function ensures that the current_state is copied into mutable set before adding new elements. After the modifications done, it is then converted into frozenset for storage in the sow_states set. This approach maintains the immutability of states within new states and avoids errors related modifying frozenset.
B.4.4 Experiment Run
User: Logical deduction is performed by applying rules of the form if X then Y current knowledge about a world, extending our knowledge state defined as our current knowledge the world, how are successor states defined, given a collection of rules? Provide only the Python implementation of successor states function
each rule application. A rule if
applicable know
CharGPT: To implement Python function for generating successor states from a given state using a set of logical deduction rules, we'll assume that the state is represented as set of facts (truths known about the world), and the rules represented tuples (X, Y). meaning "if X then Y". rule applicable if all elements of X
68

---

are present in the current state. When rule is applied. Y is added to the state to create a successor state. Here's how this can be implemented:
successor_ (tw.etttttterrrrtttssstttte rules)
*.. Generate all successor states from the current state using the given rules.
:param current_statee
facts known about the world.
:param rules: list of tuples (X, Y), where each is a tuple representing conditions and singl
:return: A aet of new states, each differing from the current by the addition of new fact derived fro
new_states set()
Check each rule to see if ét can be applied
for conditions, result in rules:
if all (condition in current_state for condition in conditions):
If all conditions are met, create new state by adding result the current state new_state current_state. sopreett new new_state. add (dd(uul))))
new_states add (frozenset (new_state) Use frozenset to ensure the state can be added to set
return new_states
Ezample le usage
current_stan state ('rainy', 'umbrella, open')
rules [(({ 'rainy' }, "wet_groundd), (f 'umbrella, oamrellla_open,,, 'rainy' }, 'dry. unbrella
Convert rule format to match the function expectation
formatted rules = [((condition.)) result) for condition, result in rules]
new_states successor_ states (current, state, formatted, rules) print ("New States:", [set(state) for state in new, new_ states))
In this example:
surrent_state contains facts 'rainy' and 'umbrella_opoor rules specify that if it 'rainy'. then the 'wet_ground should be true, and if both uumbeella open' and 'rainy' are true, then 'dry _nner _under_umbrella" uumreellaa should be true. The function successor states checks each rule, determines if the conditions met, and generates new by adding the results applicable rules. new state a superset
current state one additional fact.
User: Provide only the python code that tests, given target fact, whether state is goal state.
ChatGPT; Here's the Python code that tests whether given state contains target fact, thereby determining if goal state:
i ie_goal ga__goll_satteeelee sttatestate,, target_fact):
Determine if the given state is goal state based the presence of the target fact.
:param state: set facts known about the world.
:param target_fact: The fact that if present in the state, identifies it as a goal state. :return: True if the target fact is in the state, False otherwise.
return target_fact in state
Ezample usage
state ('rainy', 'wet_ ground,, cumbrella_open '} factet_fact 'wet_ground' result is_goal_stode state(saae, target_fact) print goal state:", result)
This function simply checks if the target fact is part of the current state. If it is, the function returns True, indicating that the state is goal state; otherwise, it returns False.
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User: In successor states function, assume that the rules are just pairs of facts, a single fact condition. Also, you cannot add to frozenset
Given the revised assumptions where each rule is just pair of facts and each condition is represented by single fact, and considering that cannot add frozenset, can refactor the successor states function. The function will now operate on copy the current state converted a mutable set, apply the changes, and then convert back to frozenset for immutability and use in a set stales. Here's the updated implementation:
ChatGPT:
def sectessor__tattee stwretttttttttrtttttttttaae
rulea):
Generate all successor states from the current using the given rulea, where each rule is pair meaning X then Y. Each single fact condition.
:param current_state: A frozenset representing the current state of known facts.
sparam rules: list of tuples (x, Y), where is fact and Y is fact that follows if is true. :return: A of new states, each differing from the current by addition new fact derived fro
new new_states = set()
mutable_current_stee = set aeturrent_stacte
Convert frozenset to mutable set
Check each rule to see if at can applied
for condition, result in rules:
if condition in mutable_ cectll_cureet__tee
state by adding the result to the current state new_state mutable, caaibbe__urrenttttt sutreent,taae.cooyy.
condstion is met, create
new_state add (result)
new_states add (frozenset (nev_state)) Convert dack to frozenset for inmutabilis
return new_states
Ezample le usage
current_staie state frozenset (('ainyy' 'umbrella, openn )))
rules [('rainy', ground)) ('umbrella, open', 'dry, under_nde_utee umbrella
new_states wwccessor_staeeeeeeeeee states (current_state rules)
prênt ("New States:", (set(state) for state in new new_states])
This code:
Converts the immutable frozenset of the current state to mutable set to allow modifications. Iterates over each rule, checking if the condition is met in the current state. If the condition met, creates new state by copying the current state, adding the result of the rule, and then converting back to frozenset for use a set (to ensure immutability and unique state management). Returns set of all new states generated this way.
B.4.5 Experiment Run
User: Logical deduction is performed by applying rules of the form if X then Y current knowledge about a world, extending our knowledge with each rule application. A rule then applicable we know a state defined as our current knowledge of the world, how are successor states defined, given a collection of rules? Provide only the Python implementation of successor states function
In a logical deduction system, we can model successor states using function that takes the current state of knowledge (a set of facts) and collection of rules. Fach rule can be represented as tuple (X, Y) where if X (a precondition) is satisfied by the current state, then Y (a consequence)
CharGPT:
be added the state.
Here's Python function that computes the successor states by applying all applicable rules the current state:
def
rules)
Apply rules to the current state to generate successor states.
representing the current state (known facts) :param rules: list of tuples, each tuple
:param
rule (X, where if X is in the state, Y is added to the state :return: A new state formed by applying all applicable rules
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---

new_state set
$ Start with copy of the current state
changes True
$ To check if any fact added
while changes:
changes False
for precondition, consequence in rules:
if precondition new_state and consequence in new sew_state: mex_stateeaddccomsqquu cconceeuuee)
changes = True # A new fact was added, continue the loop
return new_state
Ezample usage current_ state ('X')
rules [('X', 'r"), ('Y, '21)]
new_state (ell.rreeescuurrnt staate, state, rules)
(reussttee Output should be ('X', 'Y 12'
This function works follows:
It initializes the new state with the current state. It iterates over the rules and checks if the precondition of any rule is met the current state. If precondition met and the consequence is not already in the state, the consequence is added. This process repeats until no new can be added, ensuring applicable rules have been exhausted. The function ensures that direct and indirect consequences of initial facts are considered, making it useful for computing successor states in logical deduction systems.
User: Provide only the python code that tests, given a target fact, whether a state is goal state.
CharGPT: Certainly! Below Python function that whether given state contains target fact, which would indicate it's a goal state:
def is es_goal_staan state (state, target_fact): a.n
Check if the state is goal state by determining if contains the target fact.
:param state: set representing the current state (known facts)
:param target_fact: The fact that defines the goal state :return: True the target_fact is in the state, False otherwise uun
return target_fact in state
Ezample usage
state ('I', 'Y', '2') target_fact 'p' result = goal__saae(state, target_fact) print(result) Output should be True
target_fact 'A'
result = is_ llagggllsattsstatteeee farget_fact) print Output should be False
This function takes set state containing known facts checks the target_ fact is among them, thus determining if the state is a goal state.
The implementation of apply_rules applies all possible rules iteratively. That is not what I asked for, asked for successor states function, returning all immediate successor states
ChaiGP::
Thank you for the clarification. To implement a function that generates all immediate successor states, we need a different approach. Instead of applying all rules iteratively until no more can be applied, we'll we'll generate new state for each single rule application, thereby obtaining all the states that rule application away from current state.
Here's Python implementation the function that generates immediate successor states:
def successor_stateee (state, rules):
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Generate all immediate successor states by applying each applicable rule once.
:param state: set representing the current state (known facts)
:param rules: list of tuples, each tuple is a rule (X, Y) where
if X is in the state and Y is not, Y is added to the state
:return: A list of states, each created by applying rule to the current state
successors
for precondition, consequence in rules: if precondition in state and consequence not in state: new_state set(state) new new_atate.o add (consequence) successors successors.appeun (ppend
return
Ezample usage
current_ strrett_statteee state ('X')
rules [('X' 'Y'), ('Y, '2"), ('X', 'A')]
alt_successors successor_stater (ww.sstattesrrettttttttt.. stareet__stte,, rules)
for idz, state in enumerate (all_ successors):
print "Sucessr fidm 1}: (state)")
This function iterates through each rule and checks precondition is met and consequence is not already in the state. If so, it creates new state with the consequence added and adds this new state to the list of successors. Each state in the list represents distinet state reached by applying rule from the initial state.
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Checklist
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Question: the main claims made the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
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is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper.
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guidance or search pruning techniques. But more importantly, we would human feedback in coming up with valid implementations of the search components, This could be done in ways similar to the way policy is obtained in generalized planning.
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78


Reviews:
[{"review_id": "BTKT2j4Uly", "reviewer": "Reviewer_uMM4", "summary": "The authors propose a Thought of Search: thinking before searching strategy to solve Automated Planning problems using LLMs. They use the GPT-4 LLM to generate a Python code for generating successor states and goal test functions which are the crucial parts of any search. They argue that this method is sound and complete and requires the least calls to the LLM before successfully solving the problem when compared to the relevant literature.", "questions": "1. Why did the authors use only GPT-4 for their experimentation? No reasons are presented in the paper. Presenting a comparison of the performance with different LLMs would have been interesting to see.\n2. What is the need to use LLMs to generate successor and goal test functions and solve the planning problems with a naive blind search? Is the objective to evaluate GPT-4's capability to generate Python code, or to assess its reasoning ability in solving planning problems?\n3. Comparing the number of calls to the LLM with other approaches may not be appropriate. Your methodology necessitates generating the search components once for each domain, whereas other approaches involve the LLM in solving every problem within the domain. A more relevant comparison would be the average time required to solve these problems and the success rate across different methodologies.", "limitations": "1. Showcasing results with only GPT-4 model.\n2. Incomplete comparisons with the other existing approaches as mentioned above.", "rating": 3, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 1, "strengths": "1. Analysis of Complexities for the existing methodologies of using LLM for Planning.\n2. Innovative use of LLM to reduce the number of calls made to solve the problems.", "weaknesses": "However, there are several concerns regarding the proposed work:\n1. If the Large Language Model (LLM) is being used to generate successor and goal test functions, which are already intrinsic components of automated planners, it is unclear what improvement is being made. The inherent memory and time complexities associated with solving these planning problems are not addressed or mitigated by the proposed approach. It appears that the LLM is simply re-writing (a few components of) a more basic planner, which does not seem sufficiently innovative for a NeurIPS-level conference.\n2. The process is not fully automated, as it requires human intervention to re-prompt the LLM until the correct code is generated. \n3. The methodology would have been more compelling if it had included an exploration of generating useful heuristic functions.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their feedback.  \n\n\n## Answers to the Questions:\n\n1. Our purpose was to show the feasibility of the approach rather than comparing which models are better at generating successor functions / goal test for the four search problems. For that purpose one language model is sufficient. Further, all papers we compare to (described in section 2) use GPT(3, 3.5, or 4) in their experiments and therefore, by using GPT model we could compare to their results without redoing their experiments, which we show in this paper to be so unnecessarily wasteful. \n\n2. The objective of this work is neither to evaluate GPT-4's capability to generate Python code, nor to assess its reasoning ability in solving planning problems. Our primary objective is to fill the gap in the current literature on planning with LLMs and  point out the inefficiency and pitfalls of the current trends. Please see the response to all reviewers for the discussion on this.   \nIn order to solve a search/planning problem, one needs to be able to capture the dynamics of that problem. Of course, for planning problems that are already expressed in a formal language, such as PDDL, one can simply use an existing PDDL planner, which internally performs a search, defining the successor and goal functions based on the PDDL. If the problem does not have a PDDL yet, but it can be easily captured in PDDL, one may prefer to do so (with or without the help of LLMs), as it is done in e.g., the cited work (Guan et al, NeurIPS 2023, Oswald et al, ICAPS 2024). Many search problems, however, are not easy to capture in PDDL, and therefore an alternative approach is needed. This is the case for many of the planning domains mentioned in the recent literature and we used some of them in our work. Probably the best example is 24 game, which has numeric features not easily captured by classical planning.  Please see the discussion in Section 3.  \n\n3. The number of calls to the language model is precisely how we measure complexity in this work. Additionally, we provide the overall time and accuracy comparison in the paper (more on this below).  \n- One of the major advantages of our approach is that we only need to call the LLM a constant number of times *per domain*, regardless of the number of problems in the domain. However, even if you want to take away this advantage and say that each domain has only a small number of problems, our approach needs less LLM calls per domain than the other approaches per problem in the domain. Further, after the successor function and the goal test were obtained, solving all the problems in a domain by search on a single core of a personal computer CPU typically takes as much time as a single call to the LLM. \n\n- Both the total search time (the average time is easy to derive since the number of instances is provided) and the accuracy/success rate results are provided in the paper. The success rate of our approach is 100% in all the tested domains. The success rates reported in the literature for 24 Game and for PrOntoQA are presented in lines 198 and 268, respectively. The success rate of ToT on mini crosswords is 20% and the success rate of RaP on BlocksWorld is 100%, 88%, and 42% for problems with solution length of 2, 4, and 6, respectively. We forgot to mention it in the paper, will add. The total search time results for our approach are provided in the text, lines 210, 226, 247, 278. To exemplify, solving all the 1362 instances of 24 game takes 2 seconds with the “fastest” successor generator and 7 seconds with the “slowest” among the 5 times we conducted the experiment. As mentioned before, our accuracy is 100% - we solve all 1361 solvable games and report unsolvable for the unsolvable one. In comparison, the ToT approach (Yao et al, NeurIPS 2023) restrict their experiments to 100 out of 1362 tasks and performs ~100 calls to an LLM per task. Assuming the same average number of calls and 7 seconds per call, the ToT approach would take around 10^6 seconds or 11 days, achieving the reported success rate of 75%. Even if the LLMs become significantly more efficient and the time of a single call would be cut down to 1s, it would still take ToT more than 1.5 days. \n\n\nPlease also see the response to Reviewer PhwG regarding automating the feedback.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "lv26vXGIeR", "reviewer": "Reviewer_PhwG", "summary": "Existing LLM-based planning approaches usually involve searching and multiple passes of the model, which leads to significant inefficiency and cost while failing to guarantee the correctness of the generated plans. Motivated by this issue, this paper first analyzes the soundness, completeness, and complexity of a series of existing planning methods, arguing that they do not meet the standard of these properties. A new algorithm, Thought of Search (ToS), is proposed to alleviate the heavy computing demand of the searching operation for planning. It queries LLMs for generating the code implementation of successor functions and goal tests. The proposed method achieves 100% success rate for four representative search problems, while the total time spent (on CPU) is shorter than or comparable with a single LLM evaluation time. Further discussion describes that, compared with other approaches, ToS is not only sound and complete but also more cost-effective and able to explore a much larger portion with only O(1) complexity.", "questions": "See weaknesses", "limitations": "The limitations have been discussed.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1. Impactful motivation. Recently, there have been increasing efforts to improve the planning ability of LLMs. However, since this line of research is still at its early stage, the efficiency of the proposed methods is easy to overlook, especially when the system contains multiple agents. Therefore, I strongly agree with the motivation of this paper (i.e.,  the need for sound and complete LLM-based approaches that uphold efficiency) and believe in its importance for future research.\n2. Comprehensive analysis. The paper systematically and comprehensively studies the properties of twelve related works that are commonly used or recently proposed, providing convincing support for the authors’ claim and, more importantly, valuable information to the community.\n3. Solid results. Across all four evaluated tasks, ToS consistently generates valid final solutions only and reaches 100% accuracy in a relatively short time. Notably, the searching operation is run on the CPU. These results are sufficient to demonstrate the effectiveness of the proposed method.", "weaknesses": "1. Current works on code generation using LLMs show that the correctness of the generated code is not guaranteed. In this paper, the obtained implementations in the experiments are also not always valid at the first trial and require human feedback. Thus, I am slightly concerned about whether ToS can generalize to more difficult tasks while maintaining its nice properties. Nonetheless, this is likely to be alleviated by combining with some automated optimization techniques as discussed by the authors. Therefore, I am still relatively optimistic about this approach at this point. Further experiments (if any) to address this concern are welcome.\n\n2. While the paper's contents are generally well organized, which I appreciate, there are quite a few typos and wrong words/phrases. For exemplification, a non-exhaustive list is written below. I highly recommend a careful and thorough check of the whole paper to fix all the mistakes.\n  - Line 22: ‘The purpose of our work is precisely that.’\n  - Line 99: ‘Reflection’\n  - Line 323: ‘a lower than reported by the approaches accuracy’", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their feedback and hope that we can somewhat alleviate their concerns and get their strong support for this work. \n\nExisting literature on code generation e.g., [1,2], as well as the literature on generalized planning with LLMs e.g., [3] shows evidence that automated feedback improves LLM performance. Our preliminary investigation of feedback automation for ToS supports this as well. \n\nWe will proofread the paper and fix the typos - thank you for pointing these out!\n\n[1] Chen, Xinyun, et al. \"Teaching large language models to self-debug.\" ICLR 2024\n[2] Zhang, Kechi, et al. \"Self-edit: Fault-aware code editor for code generation.\" ACL 2023\n[3] Silver, Tom, et al. \"Generalized planning in pddl domains with pretrained large language models.\" AAAI 2024.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "3HAu2K5qN8", "reviewer": "Reviewer_BYf6", "summary": "The authors propose a position paper that argues current works for LLMs for planning waste significant compute, on top of having poor algorithmic and empirical performance. The authors also propose ideas on how to use LLMs more efficiently and effectively by using them to preprocess search algorithms instead of using them directly during search. More specifically, the proposed method consists of using the LLM to generate the successor generator and goal state checker, alongside user feedback.", "questions": "No questions or clarifications that could change my opinion.", "limitations": "The authors adequately addresses limitations of their work, and also the checklist with appropriate justification.", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 4, "strengths": "The paper has several strengths as a position paper. It provides a simple yet original idea, namely that researchers should strive for responsible usages of LLMs for planning in terms of computing efficiency and also provides arguments that this would actually improve the performance of LLMs for planning as well.\n\nThe idea is complemented by an extensive survey of LLMs for planning methods with a summary of worst case complexities, and whether the algorithms are sound or complete. Furthermore, it is complemented by a novel methodology for using LLMs for planning which adheres to the authors' proposition: more efficient and effective LLMs for planning research. \n\nThe experiments are extensive with a wide variety of planning benchmarks, implementation details, and results. The results are quite positive as summarised in Table 1, with minimal calls to LLMs in comparison to existing approaches. The authors are also transparent about the limitations of their approach in its current state, being that it requires user interaction. Nevertheless, this does not undermine the proposition of the position paper.", "weaknesses": "With regards to the idea of the paper, there are no major weaknesses. Nevertheless, the paper could benefit from some additional details regarding experiments, and improved clarity in certain areas.\n\n- It may not be clear to some readers how %States could go over 100%. By its definition in line 344-345, it is not clear whether visiting the same state twice double dips into the percentage or not, but from the next 2 sentences, it does seem to be the case.\n- Minor formatting issue: the benchmark domains are not consistently capitalised, e.g. 24 game and crossword in line 357 but elsewhere they are capitalised such as in Table 1 and page 8.\n- Although the focus of the paper is in LLMs for planning, the paper misses more general related work regarding learning for planning/generalised planning. Such methods are magnitudes of orders more efficient than LLMs in evaluating learned heuristics (ToT or GoT in LLM terminology) or policies and solve problems magnitude of orders larger than problems solved by LLM research. Example works include learned heuristics [1] or generalised policies [2], as well as foundation models for planning [3].\n\n[1] Simon Ståhlberg, Blai Bonet, Hector Geffner: Learning General Optimal Policies with Graph Neural Networks: Expressive Power, Transparency, and Limits. ICAPS 2022: 629-637\n\n[2] Dominik Drexler, Jendrik Seipp, Hector Geffner: Expressing and Exploiting Subgoal Structure in Classical Planning Using Sketches. J. Artif. Intell. Res. 80 (2024)\n\n[3] Dillon Ze Chen, Sylvie Thiébaux, Felipe W. Trevizan: Learning Domain-Independent Heuristics for Grounded and Lifted Planning. AAAI 2024: 20078-20086", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their feedback and support. We hope that the reviewer could advocate for the paper.\n\nYour understanding regarding the %States is correct, we will clarify the text in the final version.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "GgnMFxsaKw", "reviewer": "Reviewer_4XWb", "summary": "The paper analyst the use of LLM in planning, and propose to write the successor function and goal test in code instead of directly solving the problem. The paper showed experiments that using the code can get a higher accuracy and lower number of calls to the LLM compared to LLM-based solutions.", "questions": "I am concerned about the current need of human feedback to successfully generate the code. While I can see from the appendix that a large portion of them might need such feedback, can you provide some statistics about this?", "limitations": "Yes.", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 2, "strengths": "The paper is very well-written and easy to follow. The discussion is very detailed and provide extra insights on the topic.", "weaknesses": "- First of all, I think the conclusion is not surprising or bring any insights at all. The idea of letting LLM to write code instead of directly solve problems has extensively appear in LLM for reasoning [1], and more specifically on LLM for planning [2][3]. \n- Because the conclusion does not provide any big insights, I am expecting to see a strong experiment, which is not the case in this paper. The benchmark selected, are quite classic from different manners, and similar code has appeared on GitHub for at least a year. It is hard to justify whether the success of generating code is coming from LLMs remembering similar context from Github or they do have the ability to correctly generate code. And even in this case, the authors mentioned “The mistakes GPT-4 makes when producing the code repeat from one experiment to another”, which is not a good sign if one wants to deploy similar methods to more general applications. \n- One minor drawback is that I feel the authors failed to cover some related works that are quite close, for example the ones I mentioned in the first weakness. A more thorough literature review are recommended for a more convincing paper.\n\n[1]. Zhou, Aojun, et al. \"Solving challenging math word problems using gpt-4 code interpreter with code-based self-verification.\" arXiv preprint arXiv:2308.07921 (2023).\n\n[2]. Liu, Bo, et al. \"Llm+ p: Empowering large language models with optimal planning proficiency.\" arXiv preprint arXiv:2304.11477 (2023).\n\n[3]. Guan, Lin, et al. \"Leveraging pre-trained large language models to construct and utilize world models for model-based task planning.\" Advances in Neural Information Processing Systems 36 (2023): 79081-79094.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for their feedback. We hope that our clarification regarding the statistics on when feedback was needed alleviate the reviewer’s concerns and can allow to raise their rating. \n\n## Answer to the Question:\n\nWe have provided the average number of interactions with the model (required to produce the correct code) in the paper. The average is computed over 5 separate runs. The number of times a feedback was needed is one less than the number of interactions - the first interaction asking to produce the code is also counted. For 24 game (line 204): 1.2 interactions on average for the successor function and 1 interaction for the goal test. That means that the goal test did not need any feedback and the successor function needed a single feedback once out of 5 runs. For Mini crosswords (lines 219-220)  the model required 2.4 interactions on average to produce a valid successor function and 1.4 interactions on average to produce the goal test. For BlocksWorld (line 239), 2.8 and 1 iterations on average for successor function and goal test, respectively. For PrOntoQA (line 275), 1.6 and 1 iterations on average for successor function and goal test, respectively. The sum of the two averages (successor function + goal test) is shown in the “Calls” column of Table 1, last row. \n\n## On weaknesses:\n* While we agree that the idea to ask the language models to produce code is not revolutionary, we are not aware of other work that uses language models to produce search components, such as successor function and goal test. Further, we consider this to be only one of the contributions of our work.  Please see the response to all reviewers for the discussion on the contribution of our work.\n\n\n* On related work: we will add the mentioned papers to the related work.\n    * Guan et al, NeurIPS 2023 is already cited in our work. The paper proposes generating a classical planning model (PDDL), under particular assumptions. The direction is complementary to our work and is probably the more efficient method, when applicable, as it allows using existing planners. Unfortunately, not all planning problems are easily captured by a classical planning model and in such cases our method can still help. One example of such case is the 24 game we experiment with in our work. \n    * Liu et al, Arxiv 2023 assumes that the PDDL domain (the major part of the PDDL model) already exists and proposes a way to use LLMs to produce PDDL problem instances (objects, initial state, goal). We do not make such assumption; rather we are asking LLM to convert natural language domain information to python code. \n    * While our focus is on using LLMs for generating code for successor and goal functions for search problems, Zhou et al. ICLR 2024 focuses on math reasoning problems. They illustrate that LLMs can be used to generate, execute (verify), and self-refine the python code for math problems. Their study corroborate our findings that LLMs can indeed be used for generating verifiable code with some feedback.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
The main objective of our work is to fill the gap in the current literature on planning with LLMs with regard to the computational complexity and properties of the proposed algorithms. Our main contribution is precisely this investigation. We show that the current literature proposes inefficient methods for producing unsound results. We present complexity analysis of the algorithms proposed and establish the lack of soundness and completeness of these algorithms.  We not only show just how inefficient the results are, but also show that there is another way by proposing a simple alternative.

We would like to highlight the importance of soundness, which is mostly overlooked by the existing literature body. With soundness, the solutions produced by search are guaranteed to be correct and do not require external validation. Without soundness, the produced solutions have a large potential to be invalid, and an automated validator would be needed. It is not clear from the literature on the algorithms for planning with LLMs studied in this work whether such validators were created and used to verify the claimed success rates. To clarify, let us use the 24 Game as an example, let's assume the initial state is [3 3 8 8] (one of the instances in the existing benchmark). An LLM could produce [24] as a successor  during search. If you do not validate each transition from [3 3 8 8] to [24], you would not be able to validate the answer. Note that there is no way to reach [24] from [3 3 8 8]. Producing a goal state does not mean solving the problem and a sound validator is required.

We feel that it is crucial to expose the scientific community to these results, giving our position a stage at this point in time, in an attempt to reduce the amount of work that continues the same trend in the recent literature.

Meta-review: I prepare a full review. Then, I present my meta-review

## Review

The paper reviews the literature on solving planning problems using LLMs, emphasizing the calls in terms of the number of LLM calls. The paper remark on the fragility of LLMs for planning, including GPT-4, and focus instead on code generation for the successor function and goal test function. In principle , the method is applicable in contexts where both functions could be obtained by code generation.  It mentions how samples of plan can be used to verify the planning model. While manual intervention is used to produce the code, other lines of work might be relevant.
The idea is quite simple, but it fills a gap in the literature.

If there is no data to evaluate the code generated, trust falls back on errors of other LLM-based approaches. On the other hand, perfect testing might require an unbounded amount of data or a simulator that could be used from a terminal.

Soundness: 3: good
Presentation: 3: good
Contribution: 1: poor
Rating: 7: accept
Confidence: 5: 


## Meta-review

The reviewers coincide on the value of the cost analysis for other prompting algorithms. The diversity of domains seems to be enough as the domains are different from each other. Regarding some of the weaknesses reported by reviewers :

- uMM4
	- “The generator function and goal detection are part of the planing algorithm. are used by search or planning algorithms. So, they are not the planner.
	- “The process is not fully automated. 
- PhwG
	- The code is not guaranteed,  as it happenes with popular approaches approaches like CoT.

I’m recommending the acceptance of the paper.
Decision: Accept (poster)