KNOWLEDGE CROSSWORDS: Geometric Knowledge Reasoning with Large Language Models (2024)

Abstract: We propose KNOWLEDGE CROSSWORDS, a geometric knowledge reasoning benchmark consisting of incomplete knowledge networks bounded by structured factual constraints, where LLMs are tasked with inferring the missing facts to meet all constraints. The novel setting of geometric knowledge reasoning necessitates new LM abilities beyond existing atomic/linear multi-hop QA, such as backtracking, verifying facts and constraints, reasoning with uncertainty, and more. KNOWLEDGE CROSSWORDS contains 2,101 individual problems, covering diverse knowledge domains, and is further divided into three difficulty levels. We conduct extensive experiments to evaluate existing LLMs and approaches on KNOWLEDGE CROSSWORDS. Results demonstrate that baseline approaches struggle with larger knowledge networks and semantically-equivalent entity distractors. In light of their limitations, we propose two new approaches, STAGED PROMPTING and VERIFY-ALL, to augment LLMs’ abilities for error-aware backtracking and constraint verification. Our VERIFY-ALL significantly outperforms prior methods and is more robust towards problems in the hard subset. Further analysis shows that geometric knowledge reasoning poses new challenges to LLMs’ knowledge abilities, particularly in robustness towards varying option orders, complex structural constraints in knowledge networks, “none of the above” scenarios, and more.1

heading: 1 Introduction
text: Large language models (LLMs) encode wast amounts of world knowledge in model parameters (Petroni et al., 2019; Yu et al., 2023a). Existing tasks and datasets assess LLM knowledge abilities mostly by focusing on atomic (e.g., open-domain
∗ equal contribution 1Code and data are publicly available at
https://github.com/Wenwen-D/KnowledgeCrosswords.
QA) (Rajpurkar et al., 2016; Das et al., 2022; Joshi et al., 2017) or linear (e.g., multi-hop QA) (Press et al., 2023; Yang et al., 2018; Ho et al., 2020) settings, extracting one fact or a fixed concatenation of facts from LLMs. However, knowledge (and language) is naturally structured (Reagans and McEvily, 2003), going beyond linear arrangements, involving complex structural attributes, and forming an interweaving network that connects various entities and relations through multiple chains as illustrated in Figure 1.
Consequently, we ask: Can LLMs extend beyond linear compositionality and aggregate information from multiple chains along with various knowledge constraints? Specifically, can LLMs, with the help of their internal parametric knowledge and inherent reasoning patterns, infer missing facts in a network? We term such ability geometric knowledge reasoning. While compositional QA has been explored in the constrained setting of external knowledge bases (Zelle and Mooney, 1996; Cui et al., 2017; Ye et al., 2022; Neelam et al., 2022; Xie et al., 2022), we aim to investigate whether LLMs could reason with non-linear fact networks solely relying on their internal parametric knowledge. Formally, we define geometric knowledge reasoning as reasoning over a network by inferring missing entities based on the given contextual information, where such networks cannot be simply broken down into chains (like linear reasoning), risking the loss of structural information and constraints. Geometric knowledge reasoning with LLMs naturally necessitates new LLM abilities beyond those encountered in atomic or linear tasks, such as composing knowledge across multiple chains, reasoning with uncertainty, fact verification, error-aware backtracking, and more. Since state-of-the-art LLMs are trained on linear sequences of texts devoid of explicit structure, it is underexplored whether they could effectively apply their linearly acquired knowledge to solve geomet-
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ric reasoning tasks. LLMs with strong geometric knowledge reasoning abilities could serve as versatile relational databases, allowing controllable access to parametric knowledge through SQL-like conditioned prompting.
To this end, we propose KNOWLEDGE CROSSWORDS, a geometric knowledge reasoning dataset with 2,101 problems evaluating to what extent LLMs could achieve such desiderata. Each knowledge crossword consists of a list of constraints representing an incomplete fact network, and LLMs need to reconstruct the knowledge network while ensuring that all factual constraints are met. To solve knowledge crosswords, LLMs should ideally evaluate candidates for each blank, jointly consider factual constraints, verify intermediate solutions, and backtrack when encountering factual errors, until all constraints are met. For each incomplete fact network, we generate three sets of distractors as options that are progressively more plausible, resulting in easy, medium, and hard subsets for fine-grained evaluation. Each problem also comes with two settings: w/o knowledge, where LLMs solve knowledge crosswords solely with parametric knowledge; w/ knowledge, where a helpful (and noisy) paragraph is prepended to each problem.
We conduct extensive experiments to evaluate LLMs and approaches on KNOWLEDGE CROSSWORDS, ranging from simple zero-shot prompting to advanced ones such as self-consistency (SC) (Wang et al., 2022) and least-to-most prompting (LTM) (Zhou et al., 2022). Results demonstrate that baselines struggle with problems in the hard subset and have significant performance drops in the w/o knowledge setting. Advanced prompting methods such as SC and LTM barely improve LLMs due to their reliance on left-to-right reasoning patterns. To address these challenges, we propose two new instruction-based techniques, STAGED PROMPTING and VERIFY-ALL, aiming at
augmenting LLMs’ abilities for backtracking, constraint verification, and more. STAGED PROMPTING guides LLMs through an elaborate problemsolving process that progressively solves and simplifies the problem blank by blank, while VERIFYALL proposes candidates for all blanks and verifies them simultaneously. We find that VERIFY-ALL achieves top performance and is more robust with harder problems, while the success of STAGED PROMPTING is contingent on stronger base LLMs. Further analysis reveals geometric knowledge reasoning poses great challenges to LLM knowledge abilities, as they could be susceptible to a variety of factors such as option order, “none of the above” scenarios, number of distractors, special structural patterns, and more. We envision geometric knowledge reasoning as a challenging research question and KNOWLEDGE CROSSWORDS as a comprehensive testbed to evaluate LLM knowledge abilities in more complex and structured settings.

heading: 2 KNOWLEDGE CROSSWORDS
text: We propose KNOWLEDGE CROSSWORDS, a geometric knowledge reasoning benchmark to evaluate whether LLMs could reason with incomplete fact networks bounded by geometric constraints (Appendix B). An example knowledge crossword is presented in Figure 2.
Definition Each knowledge crossword consists of a question graph GQ = {(h, r, t)|h, t ∈ VQ, r ∈ R}, where VQ is the set of entities represented as nodes of GQ and R is the set of all possible relations between entities. Each (h, r, t) in GQ denotes a directed edge representing a factual association such as (Marvin Minsky, has won prize, Turing award). In the question graph GQ, certain nodes bi ∈ VQ are masked out as blanks B = [b1, b2, . . . , b|B|] for QA. The goal of each knowledge crossword is to find one combination
of answers for all blanks A = [a1, a2, . . . , a|B|] that satisfies all factual associations represented as geometric constraints in the question graph GQ.
Data Source We resort to encyclopedic knowledge graphs, specifically YAGO (Suchanek et al., 2023), as scaffolds of geometric knowledge reasoning to construct the KNOWLEDGE CROSSWORDS benchmark. Different from existing KBQA datasets (ComplexWebQuestions, Talmor and Berant (2018), GrailQA, Gu et al. (2021), inter alia) where LMs are required to reason with external KBs, LLMs solve knowledge crosswords with their internal parametric knowledge. We conduct preprocessing to remove certain relations in YAGO that are location-related, time-sensitive, or not selfevident. This is to ensure that the KNOWLEDGE CROSSWORDS is minimally affected by question ambiguity (Min et al., 2020; Cole et al., 2023), outdated knowledge (Yu and Ji, 2023; Hernandez et al., 2023), etc. We obtain the filtered knowledge graph as KG = {(h, r, t)|h ∈ H, r ∈ R, t ∈ T }, where H, R, and T are the sets of heads, relations, and tails respectively.
Question Graphs We first adopt two hyperparameters to control the property and difficulty of the generated question graphs (incomplete fact networks): question graph size sG, denoting the total number of nodes in a question graph, and blank size sB , representing the number of nodes masked
out as blanks that need to be filled. We start from a random center node c and retrieve the k-hop neighborhood of c as G(c)N . We then downsample G (c) N by randomly removing nodes with degrees higher than a dynamic threshold td in KG, until the largest weakly connected component in G(c)N has a size no greater than sG. This is motivated by the fact that entities with higher degrees are presumably less typical and more ambiguous, resulting in problem ambiguity (Shomer et al., 2023; Qian et al., 2023). We refer to the largest connected component in downsampled G(c)N as an answer graph GA.
We then randomly select sB nodes in GA with degrees larger than a threshold tb in GA and mask them out as blanks B to obtain a question graph. These high-degree blanks would be rich in geometric associations and provide more constraints to work with. The question graph is then linearized in triplet format, since converting interconnected triples into plain natural language and vice versa are noisy and prone to introduce errors and biases (Bai et al., 2023; Min et al., 2023). By employing hyperparameters and thresholds such as sG and sB , the dataset comes with built-in difficulty control measures to controllably generate diversified problems. As the question graph generation step does not guarantee answer uniqueness, we exhaustively search answers for each GQ in KG and only retain those with one valid answer combination.
Negative Sampling We mainly consider KNOWLEDGE CROSSWORDS in a multiple-choice setting, where several candidates are provided for each blank in GQ. This would require negative sampling for each blank to provide distractors in addition to the correct answer, while we identify a taxonomy of three progressive rules for distractors from loose to strict:
• Rule #1: Semantic Role: If a blank bi is the head (or tail) of an edge with relation ri, then the distractor for bi should be selected from the set of heads (or tails) of edges with the same ri.
• Rule #2: Network Proximity: The distractor for bi should occur in the neighborhood G(c)N around c, which further ensures that distractors are likely to be in a similar context as bi.
• Rule #3: Definite Constraint: If the other end of the edge that the blank bi is incident to is known, then we say such edge is a definite constraint for bi, and the distractor should satisfy at least one of such definite constraints for bi to fulfill Rule #3. Such distractors impose higher demands on LMs in the sense that LMs should jointly consider all constraints to exclude these distractors.
As a result, we obtain three negative sampling strategies with varying difficulty implications for knowledge crosswords: easy, where distractors only meet Rule #1; medium, where distractors meet Rule #1 and #2; hard, where distractors meet Rule #1, #2, and #3. We opt to separately assign multiple options for each blank in GQ, using either easy, medium, or hard strategies, resulting in three subsets of knowledge crosswords with increasing difficulty levels.
Relevant Knowledge Each knowledge crossword comes with two settings: W/O KNOWLEDGE, where LLMs solely rely on internal parametric knowledge to solve knowledge crosswords; W/ KNOWLEDGE where a helpful but noisy passage is prepended to the problem description. These knowledge passages contain both helpful information about the correct answers and irrelevant information generated by the three proposed negative sampling rules. Useful and irrelevant knowledge is then sampled to 1:3, shuffled, and presented before each knowledge crossword. LLMs would need to dynamically select relevant and useful information to facilitate geometric knowledge reasoning, which poses new challenges to LLMs (Shi et al., 2023a).
Subset #Qs Avg. #Nodes Avg. #Edges Avg. #Blanks
EASY 869 7.28 6.63 2.89 MEDIUM 873 7.28 6.64 2.89 HARD 359 7.09 6.41 2.62

heading: 3.2 Models and Settings

heading: 3.1 Baselines

heading: 3 Experiment Settings
text: (FEW-SHOT, COT, COT+SC, LTM), we present 5 in-context exemplars. The sampling temperature τ is set to 0.1 except for COT+SC; we sample 5 Chain-of-Thought responses with temperature τ = 0.7 for the CoT with Self-Consistency baseline.

heading: 4 Our Approach
text: We hypothesize that the left-to-right reasoning patterns in autoregressive language models (Yao et al., 2023) and prompting approaches (discussed in section 3.1) would fall short of solving knowledge crosswords, which require backtracking, maintaining problem states, verifying existing information, reasoning with structured constraints, and more. To this end, we introduce two instruction-based methods that promote these abilities, illustrated with a detailed example in Figure 2.

heading: 4.1 STAGED PROMPTING
text: The STAGED PROMPTING approach divides geometric knowledge reasoning into stages involving three steps: solve, update, and verify. At the beginning of each stage, LLMs maintain a current status of solved blanks and unresolved constraints (edges that involve unsolved blanks). In the solve step, LLMs propose a candidate for an unsolved blank based on internal knowledge; in the update step, LLMs update unsolved constraints using the newly proposed candidate for an associated blank; in the
verify step, LLMs reflect on the updated constraints in the update step and judge their validity. If an invalid factual association is identified as a result of the proposed candidate, LLMs backtrack to the problem status in the previous stage and propose another option; otherwise, LLMs proceed to tackle the remaining blanks until all blanks are filled and all constraints are met.

heading: 4.2 VERIFY-ALL
text: While STAGED PROMPTING presents an elaborate problem-solving process that tackles challenges such as backtracking and status updates, such complex reasoning might be hard to learn in context for LLMs. We additionally propose the VERIFY-ALL approach: candidates for each blank are simultaneously proposed, rather than in separate stages. A verification step is then employed to assess the validity of all filled constraints using these proposed candidates. If an error is detected, the LM should backtrack and propose an alternative set of candidates until no error is found.

heading: 5 Results
text: We evaluate approaches on KNOWLEDGE CROSSWORDS and present results in Table 2.
LLMs have preliminary abilities for geometric knowledge reasoning. Table 2 shows that all investigated approaches outperform the RANDOM
baseline, while LLMs could achieve 90+ FC scores on the easy subset and w/ knowledge setting. However, model performance (FC) drops by 29.7% on average on the hard subset compared to easy, even when the required knowledge stays the same, indicating that LLMs are far from robust on geometric knowledge reasoning.
Noisy knowledge does help LLMs solve knowledge crosswords. Despite the existence of irrelevant and confounding information, LLMs do benefit from the prepended noisy knowledge. On average, the W/ KNOWLEDGE settings exhibit a 34.3% FC gain compared to W/O KNOWLEDGE settings across all approaches. This indicates that LLMs possess preliminary abilities to selectively leverage knowledge and information.
Advanced prompting methods show little improvement. Specifically, COT, LTM and COT+SC do not greatly advance performance compared to ZERO-SHOT and FEW-SHOT prompting: the average PC and FC of COT, LTM and COT+SC is 5.6% and 10.4% less than those of ZERO-SHOT and FEW-SHOT. This suggests that the left-to-right reasoning patterns employed by these prompting techniques may not be applicable for knowledge crosswords, as these prompting methods fail to induce non-linear reasoning steps for verification and backtracking.
Promoting verification and backtracking improves geometric knowledge reasoning. With GPT-3.5-TURBO, VERIFY-ALL greatly outperforms all baselines with explicit self-verification.
Interestingly, after a closer look into the responses, we find that factual errors are rarely detected while the performance gain mainly comes from LLMs proposing more precise answers in a single attempt when specifically asking for fact verification. In addition, while GPT-3.5-TURBO struggles at learning complex reasoning steps required by STAGED PROMPTING, Table 2 shows that STAGED PROMPTING achieves impressive results with GPT-4 and generally outperforms all other methods including VERIFY-ALL in the W/ KNOWLEDGE setting. This indicates that the more elaborate instructions of STAGED PROMPTING work best with more advanced LLMs, as smaller models struggle to grasp these detailed reasoning steps.

heading: 6 Analysis
text: Error Analysis Each knowledge crossword comes with W/ and W/O KNOWLEDGE settings. We conduct error analysis to investigate the impact of noisy passages on LLM problem solving and present in Figure 3. We label each problem based on whether it is answered (in)correctly in W/ KNOWLEDGE (K+(–)) and answered (in)correctly in W/O KNOWLEDGE (N+(–)).
Figure 3 reveals that for easy and medium problems, the main bottleneck is knowledge access since most N– questions are correctly answered under W/ KNOWLEDGE. However, for hard problems, the bottleneck is geometric knowledge reasoning abilities, given that the proportion of N–K– is consistently above 30%, showing that LLMs struggle to reason even when the required knowledge is provided. By including three subsets of varying difficulty in KNOWLEDGE CROSSWORDS, we successfully reveal the multitudes of LLM limitations in knowledge-intensive contexts, disentangling limitations of knowledge and reasoning.
None of the Above To study whether LLMs are subject to “none-of-the-above (NOTA) scenarios”, we add an instruction “Output ‘none of the above’ if none of the option combinations satisfy all the constraints.” and evaluate the performance of GPT-3.5-TURBO with the FEW-SHOT prompting. Specifically, we experiment with two settings: #1. The correct option is removed from candidates and LLMs should choose to report NOTA; #2. The correct option exists and LLMs should provide answers. Table 3 demonstrates that LLMs struggle
with the NOTA scenario, regardless of whether the knowledge crossword is indeed coming without correct answers.
Option Order As our proposed STAGED PROMPTING considers one candidate at one time, we expect that model performance may be worse for problems where the correct answer appears later in the prompt. Figure 4 demonstrates this negative correlation, which could be attributed to LLM hallucination (Ji et al., 2023) and falsely accepting earlier incorrect options. On the other hand, we see an opposite trend in the performance of COT. This indicates that COT does not consider the options sequentially and later options might influence the prediction more.
Structural Patterns We investigate whether special structural patterns of blanks in the question graphs might impact LLM performance. We identify three patterns: 1) A-B, where two blanks are connected by an edge; 2) A-B-C, where three blanks are on a chain; 3) cycle, where three more more blanks form a cycle. Figure 5 demonstrates a decrease in performance of A-B and A-B-C compared to the full dataset, showing a chain of blanks
would pose challenges to LLM knowledge reasoning. On the other hand, cycle exhibits performance gains: we hypothesize that it has a higher blank-toconstraint ratio (closer to 1:1) than other patterns, which gives LLMs more constraints to work with.
Number of In-Context Exemplars Despite the in-context learning ability demonstrated by LLMs (Brown et al., 2020), we find that more in-context exemplars fail to improve model performance on KNOWLEDGE CROSSWORDS. As presented in Figure 6, for questions with all three difficulty levels, the best performance is achieved at ZERO-SHOT except for the Full-Credit of hard problems. This indicates that left-to-right CoT reasoning could not be adequately learned in context for the problem of knowledge crosswords.
Difficulty of In-Context Exemplars We investigate the correlation between the difficulty of incontext exemplars and model performance by evaluating the performance with 5-shot COT using 4 different sets of in-context exemplars: easy, where all in-context examples come from the easy subset; similarly medium and hard; mixed, where a mixture of 2 easy, 2 medium, and 1 hard examples are
employed. Table 4 demonstrates that medium or mixed in-context examples work best, while solely employing easy or hard ones is marginally worse. As a result, we follow the mixed settings in the main experiments.
Fine-tuning and open-source LMs We additionally evaluate the geometric knowledge reasoning abilities of an open-source language model - LLAMA2-7B with 100 problems randomly selected across all difficulty subsets. Without fine-tuning, LLAMA2-7B demonstrates a performance close to random guess. After instruction-tuning with 1,471 knowledge crosswords randomly selected from all 2,101 questions, the Partial-Credit and Full-Credit become 17.7% and 12.0% higher than ZERO-SHOT prompting as reported in Table 5. This indicates that instruction tuning (Wei et al., 2021) could augment LLMs for solving knowledge crosswords, while to what extent they work with larger LLMs requires further research.
Number of Options As the number of options for each blank increases, the problem becomes harder due to the presence of more confounders. We expect to see a downward trend in model performance when there are more distractors per blank.
Unsurprisingly, the results in Figure 7 show that the performance is negatively correlated with the number of options per blank. We also observe that performance gap with random guessing is narrowing, suggesting that KNOWLEDGE CROSSWORDS might be difficult for LLMs in the W/O KNOWLEDGE generation setting. (Appendix A)

heading: 7 Related Work
text: Understanding LLM Knowledge LLMs could memorize and encode factual knowledge in model parameters (Petroni et al., 2019; Yu et al., 2023a). As a result, previous research focuses on investigating the extent to which LLMs retrieve and utilize factual knowledge (Yu et al., 2022; Chen et al., 2023a; Mallen et al., 2023). However, the knowledge abilities of LLMs also come with a wide range of limitations such as knowledge update (Hase et al., 2023), irrelevant information (Shi et al., 2023a), and more (Chen et al., 2022; Mruthyunjaya et al., 2023; Kandpal et al., 2023; Sun et al., 2023; Kandpal et al., 2023; Xie et al., 2023; Amayuelas et al., 2023; Wang et al., 2023e; Huang et al., 2023). As a result, various lines of research aim to expand the knowledge abilities of language models, such as better prompting (Press et al., 2023; Sun et al., 2022; Yu et al., 2022; Kojima et al., 2022; Ye and Durrett, 2022), retrieval augmentation (Shi et al., 2023b; Yu et al., 2023b; Asai et al., 2023; Borgeaud et al., 2022; Jiang et al., 2023b), search engine integration (Yu et al., 2022; Press et al., 2023; Kasai et al., 2022; Qin et al., 2023; Vu et al., 2023; Khalifa et al., 2023), and more. While these works primarily focus on evaluating and improving abilities to handle atomic (e.g., open-domain QA) or
linear (e.g., multi-hop QA) knowledge, we propose to assess whether LLMs could reason with fact networks bounded by geometric constraints that better align with the structural nature of knowledge.
Reasoning over Knowledge Graphs Simple and complex questions in KBQA have been extensively studied, covering varied tasks including temporal QA (Li et al., 2022; Shang et al., 2022; Chen et al., 2023b; Saxena et al., 2022; Xia et al., 2022; Mei et al., 2022; Ding et al., 2022; Kannen et al., 2022), conversational QA (Ke et al., 2022), general QA (Zhang et al., 2022a; Bai et al., 2022), and more. A myriad of methods have been proposed to tackle these problems, including enhancing reasoning over knowledge graphs (Cao et al., 2022b,a; Ye et al., 2022; Neelam et al., 2022; Xie et al., 2022; Patidar et al., 2023; Zhang et al., 2023; Gupta and Lewis, 2018; Zettlemoyer and Collins, 2009; Cui et al., 2017; Zhong et al., 2017; Shen et al., 2019) and incorporating the generating ability of language models (Liu et al., 2022; Shu et al., 2022; Tang et al., 2022; Hu et al., 2022; Zhang et al., 2022b; Jiang et al., 2023a; Wang et al., 2023b; Kim et al., 2023; Li et al., 2023; Guo et al., 2023; Aglionby and Teufel, 2022; Zhang et al., 2022c). However, it remains underexplored whether LLMs, with the help of its internal parametric knowledge, could perform geometric knowledge reasoning with elements similar to these works. We propose KNOWLEDGE CROSSWORDS to investigate LLMs’ ability to utilize their linearly acquired knowledge for structured knowledge reasoning scenarios.
Reasoning with Large Language Models LLMs have been evaluated on a myriad of reasoning tasks in an in-context learning setting, including math problems (Ling et al., 2017; Lewkowycz et al., 2022), logical reasoning (Srivastava et al., 2023; Huang et al., 2022), factual knowledge reasoning (Press et al., 2023; Feng et al., 2024), commonsense reasoning (Talmor et al., 2019; Fang et al., 2024; Wang et al., 2023c,d, 2024), and more. Leveraging the in-context learning behavior of LLMs, various prompting techniques (Wei et al., 2022; Zhou et al., 2022; Khot et al., 2022; Wang et al., 2022, 2023a; Schick et al., 2023; Gao et al., 2023) have been proposed to boost the reasoning ability. Specifically, Khot et al. (2022) and Yao et al. (2023) incorporate programs as guides to LLM generation. In
this work, we focus on the geometric knowledge reasoning ability of LLMs, which is different from existing left-to-right reasoning patterns, with minimal explicit program-based guidance, involving self-verification, backtracking, and more.

heading: 8 Conclusion
text: We propose KNOWLEDGE CROSSWORDS to investigate LLMs for geometric knowledge reasoning, i.e. inferring missing information from an incomplete fact network bounded by geometric constraints. Extensive experiments demonstrate that while existing prompting approaches struggle to solve problems in the hard subset, our proposed STAGED PROMPTING and VERIFY-ALL strategies advance model performance while augmenting LLMs with abilities to verify facts, backtrack, and more. Further analysis reveals that LLMs are brittle to “none-of-the-above” scenarios, challenging structural patterns, spurious correlations such as option order, and more.
Limitations
Limited Data Source We construct KNOWLEDGE CROSSWORDS based on only the encyclopedic knowledge graph YAGO, which covers topics on general knowledge about people, cities, countries, movies, and organizations from Wikidata. Since we will make the code publicly available, we leave it to future work on evaluating the geometric reasoning ability of LLMs on different topics with various data sources, such as biomedical knowledge graphs (Chang et al., 2020) and networks in social sciences (Feng et al., 2022).
Answer Uniqueness Due to the incompleteness of knowledge graphs, it is possible that the answer to a problem in KNOWLEDGE CROSSWORDS is not unique. However, such likelihood is presumably low and does not hurt the general evaluation of the geometric reasoning ability of LLMs.

heading: Acknowledgements
text: This material is based upon work supported by the National Science Foundation under CAREER Grant No. IIS2142739 and NSF Grant No. IIS2203097. We also gratefully acknowledge support from Alfred P. Sloan Foundation Fellowship.

heading: A Discussion
text: Geometric Reasoning in the W/O KNOWLEDGE generation setting While we mainly focus on solving knowledge crosswords in a multiple-choice setting, it is interesting to evaluate the geometric reasoning ability in the W/O KNOWLEDGE generation setting. Specifically, the problems in KNOWLEDGE CROSSWORDS have unique answers, which should be useful when switching to the W/O KNOWLEDGE generation setting as answer uniqueness makes evaluation easier and makes the problem clearer. Our preliminary experiments show that solving knowledge crosswords in the W/O KNOWLEDGE generation setting is much harder. Considering the model performance in the multiplechoice setting, one method that might be promising is to prompt LLMs themselves to generate candidates for each blank and thereby transform the W/O KNOWLEDGE generation problem into a multiplechoice problem.
Performance gain of VERIFY-ALL While VERIFY-ALL helps LLMs obtain large performance gains in solving knowledge crosswords, it is quite intriguing when investigating where such gains come from. Specifically, in the W/ KNOWLEDGE setting, among all 359 hard problems, we find only 3 problems whose solution with VERIFYALL involves detecting errors in verification and repropose candidates. Among the 3 problems, 2 are answered correctly by both VERIFY-ALL and COT, and both methods fail the other problem. This leads to an interesting implication that the performance gain comes from LLMs proposing more precise answers in the first attempt, and that LLMs can jointly consider all constraints rather than consider one by one. We envision the study of such performance gain and the application of the insight as possible future directions.
Application of geometric knowledge reasoning Despite the difficulty of the task, LLMs do show preliminary geometric reasoning ability over incomplete fact network. While such ability still has a long way to achieve perfection, this finding opens up the possibility of using LLMs as flexible relational databases and accessing the parametric knowledge with prompts similar to SQL (structured query language).
Same prompting approach with different LLMs While GPT-3.5-TURBO does not benefit from
Hyperparameter Value
degreec 5, 7, 9 sG 6, 7, 8, 9, 10, 11 sB [ 1 4 · sG, 12 · sG] mr 1.1, 1.2, 1.3 mb 1, 1.1

heading: B KNOWLEDGE CROSSWORDS Details
text: B.1 Benchmark Construction Details
1. YAGO filtering
(a) We remove edges in YAGO with the following relations: (i) Location-related: isLocatedIn, livesIn, happenedIn, diedIn, wasBornIn; (ii) Time-sensitive: worksAt,
playsFor, isAffiliatedTo, isPoliticianOf, isLeaderOf; (iii) Not self-evident: influences, owns, isKnownFor, dealsWith, imports, exports, created, isInterestedIn, dealsWith, isConnectedTo.
(b) The remaining relations in YAGO are: graduatedFrom, hasMusicalRole, hasAcademicAdvisor, hasChild, wroteMusicFor, hasCapital, actedIn, hasOfficialLanguage, hasWonPrize, hasGender, hasCurrency, directed, isCitizenOf, participatedIn, isMarriedTo, hasNeighbor, edited.
2. Modified k-hop neighborhood
(a) A center node c is randomly selected from nodes with degree degreec in filtered YAGO.
(b) We retrieve a modified 5-hop neighborhood G(c)N : in each layer, we retain at most 8 nodes. This approach assists us in obtaining a subgraph with a relatively long diameter while avoiding excessive density.
3. Downsample to GA (a) We repeatedly remove nodes from G(c)N un-
til the number of nodes in the largest connected component in G(c)N is no more than question graph size sG.
i. Sort the nodes in G(c)N by degree in filtered YAGO in descending order as vsorted,YAGO.
ii. Denote reduce range multiplier as mr. Then reduce range rr is calculated as mr · (|V(c)N | − sG) where V (c) N is the set
of nodes in G(c)N . iii. Randomly pick a node in
vsorted,YAGO[(rr− 1)/2 : (rr− 1)] and remove this node from G(c)N .
(b) Following the abovementioned approach, we downsample G(c)N to GA by removing nodes with relatively high degree in filtered YAGO and introduce randomness in this process.
4. Generate blanks to get GQ (a) To mask sB nodes in GA as blanks, denote
blank range multiplier as mb and calculate blank range br as sB ·mb.
(b) Sort the nodes in G(c)N by degree in GQ in descending order as vsorted,GA .
(c) We then randomly select sB nodes from the first br nodes in vsorted,GA as blanks.
Specifically, the hyperparameters we used for benchmark construction are listed in Tabel 6.
B.2 Relevant Knowledge
In the W/ KNOWLEDGE setting, relevant knowledge is prepended to each problem. Specifically, for each triplet in the constraint, we present four pertinent triplets, with one of them reserved for the triplet containing the correct answer. And the other three are randomly sampled from YAGO following similar method as negative sampling, with priority given to the triplets that satisfy all the criteria (Rule #1, #2, and #3). By sampling relevant triplets in such a way, we provide necessary knowledge (filled with correct answers) as well as confounding knowledge that makes the solving process non-trivial. The sampling of confounding knowledge also simulates the possible information that one may consider when solving the question, with those satisfying all three criteria having the highest likelihood of being considered intuitively.
B.3 Experiment Details
Within 4k-context, in the w/o Relevant Knowledge setting, the numbers of finished responses for easy, medium, and hard questions are 755, 781, and 341 respectively; in the w/ Relevant Knowledge setting, the numbers of finished responses for easy, medium, hard questions are 759, 769 and 328 respectively. The credits are calculated based on these finished responses only.

heading: C Additional Analysis
text: Error Analysis We provide error analysis conducted with ZERO-SHOT here for reference. Results in Figure 8 shows that all three methods share similar trends.
Number of Blanks We study the impact of the number of blanks in the problem on the model performance. Specifically, we randomly select 100 problems with three blanks (# of blanks = 3) from all three difficulty levels and construct two additional versions of these problems by filling in one (# of blanks = 2) or two (# of blanks = 1) answers to the blanks. We evaluate the performance of various methods (ZERO-SHOT, FEW-SHOT, COT, VERIFY-ALL) and two settings (W/ KNOWLEDGE or W/O KNOWLEDGE) on these three versions of the dataset and present the results in Table 7. We find that ChatGPT performs well on the simpler (# of blanks = 1) version, demonstrating a strong knowledge ability. However, its performance suffers when the required reasoning steps increase and the geometric structures involved become more complex.

heading: D Qualitative analysis
text: In this section, we provide examples of knowledge crosswords that GPT-3.5-TURBO answers correctly or wrongly using STAGED PROMPTING and VERIFY-ALL. In-context exemplars are omitted in this section to save space and can be found in Appendix E. Table 8 and Table 9 show results using STAGED PROMPTING; Table 10, Table 11 and Table 12 show results using VERIFY-ALL.

heading: E Prompts
text: We list the prompts for all experiments of Tables 2 and 5 in Tables 13, 14, 15, 16, 17, 18, 19.
Table 13: Prompts used in main experiments with exemplars and an example question. “Knowledge” for each problem is only applicable in the “W/ KNOWLEDGE” setting.
An example of knowledge crossword
Instruction: Pick the correct answer for each blank that satisfies all the given constraints. Desired format: blank i: Z ... Constraints: (Paz Vega, acted in, blank 1); (Jada Pinkett Smith, directed, blank 1); (blank 2, acted in, Prom Night (2008 film)); (blank 2, acted in, blank 1). Options: blank 1: A. The Human Contract, B. The Spirit (film), C. The Six Wives of Henry Lefay blank 2: A. Johnathon Schaech, B. Idris Elba, C. Brittany Snow Answer: blank 1: A, blank 2: B
Prompt Response
UPPERBOUND Instruction: Pick the correct answer for each blank that satisfies all the given constraints. Desired format: blank i: Z ... Knowledge: (Paz Vega, acted in, The Human Contract); (Jada Pinkett Smith, directed, The Human Contract); (Idris Elba, acted in, Prom Night (2008 film)); (Idris Elba, acted in, The Human Contract). Constraints: (Paz Vega, acted in, blank 1); (Jada Pinkett Smith, directed, blank 1); (blank 2, acted in, Prom Night (2008 film)); (blank 2, acted in, blank 1). Options: blank 1: A. The Human Contract, B. The Spirit (film), C. The Six Wives of Henry Lefay blank 2: A. Johnathon Schaech, B. Idris Elba, C. Brittany Snow Answer:
ZERO-SHOT
Instruction: Pick the correct answer for each blank that satisfies all the given constraints. Knowledge: (Paz Vega, acted in, The Human Contract); (Joe Roberts, acted in, Our Hospitality); (Paz Vega, acted in, The Six Wives of Henry Lefay); (Paz Vega, acted in, The Spirit (film)); (Charlie Chaplin, directed, A Woman of Paris); (Charlie Chaplin, directed, The Champion (1915 film)); (Charlie Chaplin, directed, Limelight (1952 film)); (Jada Pinkett Smith, directed, The Human Contract); (Johnathon Schaech, acted in, Prom Night (2008 film)); (Joe Roberts, acted in, Our Hospitality); (Brittany Snow, acted in, Prom Night (2008 film)); (Idris Elba, acted in, Prom Night (2008 film)); (Joe Roberts, acted in, The Haunted House (1921 film)); (Idris Elba, acted in, The Human Contract); (Joe Roberts, acted in, Our Hospitality); (William Hopper, acted in, The Bad Seed (1956 film)). (Optional) Desired format: blank i: Z Constraints: (Paz Vega, acted in, blank 1); (Jada Pinkett Smith, directed, blank 1); (blank 2, acted in, Prom Night (2008 film)); (blank 2, acted in, blank 1). Options: blank 1: A. The Human Contract, B. The Spirit (film), C. The Six Wives of Henry Lefay blank 2: A. Johnathon Schaech, B. Idris Elba, C. Brittany Snow Answer:
Table 14: Prompts used in main experiments with exemplars and an example question. “Knowledge” for each problem is only applicable in the “W/ KNOWLEDGE” setting.
FEWSHOT
Instruction: Pick the correct answer for each blank that satisfies all the given constraints. Knowledge: (Charlton Heston, acted in, True Lies); (Eliza Dushku, acted in, True Lies); (Tom Arnold (actor), acted in, True Lies); (Bill Paxton, acted in, True Lies); (Charlton Heston, acted in, Chiefs (miniseries)); (Stephen Collins, acted in, Chiefs (miniseries)); (Paul Sorvino, acted in, Chiefs (miniseries)); (Danny Glover, acted in, Chiefs (miniseries)). (Optional) Constraints: (blank 1, acted in, True Lies); (blank 1, acted in, Chiefs (miniseries)). Options: blank 1: A. Bill Paxton, B. Charlton Heston, C. Paul Sorvino Answer: blank 1: B
Instruction: Pick the correct answer for each blank that satisfies all the given constraints. Knowledge: (Joe Roberts, acted in, Our Hospitality); (Joe Roberts, acted in, Neighbors (1920 film)); (Taye Diggs, acted in, Rent (film)); (Joe Roberts, acted in, Three Ages); (Bradley Cooper, is married to, Jennifer Esposito); (Taye Diggs, is married to, Idina Menzel); (Charlie Chaplin, is married to, Mildred Harris); (Mary, Queen of Hungary, is married to, Jobst of Moravia); (Idina Menzel, acted in, Enchanted (film)); (Idina Menzel, acted in, Rent (film)); (Joe Roberts, acted in, Neighbors (1920 film)); (Joe Roberts, acted in, Three Ages); (Idina Menzel, is married to, Taye Diggs); (Bradley Cooper, is married to, Jennifer Esposito); (Mary, Queen of Hungary, is married to, Jobst of Moravia); (Charlie Chaplin, is married to, Mildred Harris). (Optional) Constraints: (blank 1, acted in, blank 2); (blank 1, is married to, Idina Menzel); (Idina Menzel, acted in, blank 2); (Idina Menzel, is married to, blank 1). Options: blank 1: A. Kelly LeBrock, B. Napoleon, C. Taye Diggs blank 2: A. Halloweentown High, B. Magnolia (film), C. Rent (film) Answer: blank 1: C, blank 2: C
Instruction: Pick the correct answer for each blank that satisfies all the given constraints. Knowledge: (Andy García, acted in, Smokin’ Aces); (Andy García, acted in, Ocean’s Thirteen); (Andy García, acted in, The Untouchables (film)); (Andy García, acted in, The Pink Panther 2); (Jeremy Piven, acted in, Smokin’ Aces); (Joe Roberts, acted in, Our Hospitality); (Joe Roberts, acted in, Three Ages); (Joe Roberts, acted in, Neighbors (1920 film)); (Virginia Madsen, acted in, Scooby-Doo! in Where’s My Mummy?); (Jeremy Piven, acted in, Scooby-Doo! in Where’s My Mummy?); (Mindy Cohn, acted in, Scooby-Doo! in Where’s My Mummy?); (Grey DeLisle, acted in, Scooby-Doo! in Where’s My Mummy?). (Optional) Constraints: (Andy García, acted in, blank 1); (blank 2, acted in, blank 1); (blank 2, acted in, Scooby-Doo! in Where’s My Mummy?). Options: blank 1: A. Things to Do in Denver When You’re Dead, B. Smokin’ Aces, C. Beverly Hills Chihuahua blank 2: A. Ron Perlman, B. Casey Kasem, C. Jeremy Piven Answer: blank 1: B, blank 2: C
Instruction: Pick the correct answer for each blank that satisfies all the given constraints. Knowledge: (Miriam Margolyes, acted in, Harry Potter (film series)); (David Thewlis, acted in, Harry Potter (film series)); (John Cleese, acted in, Harry Potter (film series)); (Richard Harris, acted in, Harry Potter (film series)); (Joe Roberts, acted in, Three Ages); (Maureen Lipman, acted in, A Little Princess (1986 TV serial)); (Nigel Havers, acted in, A Little Princess (1986 TV serial)); (Miriam Margolyes, acted in, A Little Princess (1986 TV serial)). (Optional) Constraints: (blank 1, acted in, Harry Potter (film series)); (blank 1, acted in, A Little Princess (1986 TV serial)). Options: blank 1: A. Miriam Margolyes, B. Maggie Smith, C. Emma Watson Answer: blank 1: A
Instruction: Pick the correct answer for each blank that satisfies all the given constraints. Knowledge: (Joe Roberts, acted in, Neighbors (1920 film)); (Dinah Sheridan, acted in, The Railway Children (film)); (Joe Roberts, acted in, Three Ages); (Dinah Sheridan, acted in, 29 Acacia Avenue); (Dinah Sheridan, is married to, John Merivale); (Charlie Chaplin, is married to, Mildred Harris); (Dinah Sheridan, is married to, Jimmy Hanley); (Bradley Cooper, is married to, Jennifer Esposito); (Joe Roberts, acted in, Neighbors (1920 film)); (Jimmy Hanley, acted in, 29 Acacia Avenue); (Joe Roberts, acted in, Three Ages); (Joe Roberts, acted in, Our Hospitality); (Charlie Chaplin, is married to, Mildred Harris); (Dinah Sheridan, is married to, Jimmy Hanley); (Charlie Chaplin, is married to, Oona O’Neill); (Dinah Sheridan, is married to, John Merivale); (Charlie Chaplin, is married to, Mildred Harris); (Charlie Chaplin, is married to, Oona O’Neill);(John Merivale, is married to, Jan Sterling); (Paul Douglas (actor), is married to, Jan Sterling). (Optional) Constraints: (Dinah Sheridan, acted in, blank 2); (Dinah Sheridan, is married to, blank 1); (blank 1, acted in, blank 2); (Dinah Sheridan, is married to, blank 3); (blank 3, is married to, Jan Sterling). Options: blank 1: A. Liam Neeson, B. Jimmy Hanley, C. Nancy Wilson (rock musician) blank 2: A. Courage of Lassie, B. 29 Acacia Avenue, C. Listen to Me (film) blank 3: A. José María Aznar, B. John Merivale, C. Prince Harald of Denmark Answer: blank 1: B, blank 2: B, blank 3: B
[ZERO-SHOT prompt]
Table 15: Prompts used in main experiments with exemplars and an example question. “Knowledge” for each problem is only applicable in the “W/ KNOWLEDGE” setting.
COT
Instruction: Pick the correct answer for each blank that satisfies all the given constraints. Knowledge: (Charlton Heston, acted in, True Lies); (Eliza Dushku, acted in, True Lies); (Tom Arnold (actor), acted in, True Lies); (Bill Paxton, acted in, True Lies); (Charlton Heston, acted in, Chiefs (miniseries)); (Stephen Collins, acted in, Chiefs (miniseries)); (Paul Sorvino, acted in, Chiefs (miniseries)); (Danny Glover, acted in, Chiefs (miniseries)). (Optional) Constraints: (blank 1, acted in, True Lies); (blank 1, acted in, Chiefs (miniseries)). Options: blank 1: A. Bill Paxton, B. Charlton Heston, C. Paul Sorvino Answer: (Charlton Heston, acted in, True Lies); (Charlton Heston, acted in, Chiefs (miniseries)). Therefore, blank 1: B
Instruction: Pick the correct answer for each blank that satisfies all the given constraints. Knowledge: (Joe Roberts, acted in, Our Hospitality); (Joe Roberts, acted in, Neighbors (1920 film)); (Taye Diggs, acted in, Rent (film)); (Joe Roberts, acted in, Three Ages); (Bradley Cooper, is married to, Jennifer Esposito); (Taye Diggs, is married to, Idina Menzel); (Charlie Chaplin, is married to, Mildred Harris); (Mary, Queen of Hungary, is married to, Jobst of Moravia); (Idina Menzel, acted in, Enchanted (film)); (Idina Menzel, acted in, Rent (film)); (Joe Roberts, acted in, Neighbors (1920 film)); (Joe Roberts, acted in, Three Ages); (Idina Menzel, is married to, Taye Diggs); (Bradley Cooper, is married to, Jennifer Esposito); (Mary, Queen of Hungary, is married to, Jobst of Moravia); (Charlie Chaplin, is married to, Mildred Harris). (Optional) Constraints: (blank 1, acted in, blank 2); (blank 1, is married to, Idina Menzel); (Idina Menzel, acted in, blank 2); (Idina Menzel, is married to, blank 1). Options: blank 1: A. Kelly LeBrock, B. Napoleon, C. Taye Diggs blank 2: A. Halloweentown High, B. Magnolia (film), C. Rent (film) Answer: (Taye Diggs, acted in, Rent (film)); (Taye Diggs, is married to, Idina Menzel); (Idina Menzel, acted in, Rent (film)). Therefore, blank 1: C, blank 2: C
Instruction: Pick the correct answer for each blank that satisfies all the given constraints. Knowledge: (Andy García, acted in, Smokin’ Aces); (Andy García, acted in, Ocean’s Thirteen); (Andy García, acted in, The Untouchables (film)); (Andy García, acted in, The Pink Panther 2); (Jeremy Piven, acted in, Smokin’ Aces); (Joe Roberts, acted in, Our Hospitality); (Joe Roberts, acted in, Three Ages); (Joe Roberts, acted in, Neighbors (1920 film)); (Virginia Madsen, acted in, Scooby-Doo! in Where’s My Mummy?); (Jeremy Piven, acted in, Scooby-Doo! in Where’s My Mummy?); (Mindy Cohn, acted in, Scooby-Doo! in Where’s My Mummy?); (Grey DeLisle, acted in, Scooby-Doo! in Where’s My Mummy?). (Optional) Constraints: (Andy García, acted in, blank 1); (blank 2, acted in, blank 1); (blank 2, acted in, Scooby-Doo! in Where’s My Mummy?). Options: blank 1: A. Things to Do in Denver When You’re Dead, B. Smokin’ Aces, C. Beverly Hills Chihuahua blank 2: A. Ron Perlman, B. Casey Kasem, C. Jeremy Piven Answer: (Andy García, acted in, Smokin’ Aces); (Jeremy Piven, acted in, Smokin’ Aces); (Jeremy Piven, acted in, Scooby-Doo! in Where’s My Mummy?). Therefore, blank 1: B, blank 2: C
Instruction: Pick the correct answer for each blank that satisfies all the given constraints. Knowledge: (Miriam Margolyes, acted in, Harry Potter (film series)); (David Thewlis, acted in, Harry Potter (film series)); (John Cleese, acted in, Harry Potter (film series)); (Richard Harris, acted in, Harry Potter (film series)); (Joe Roberts, acted in, Three Ages); (Maureen Lipman, acted in, A Little Princess (1986 TV serial)); (Nigel Havers, acted in, A Little Princess (1986 TV serial)); (Miriam Margolyes, acted in, A Little Princess (1986 TV serial)). (Optional) Constraints: (blank 1, acted in, Harry Potter (film series)); (blank 1, acted in, A Little Princess (1986 TV serial)). Options: blank 1: A. Miriam Margolyes, B. Maggie Smith, C. Emma Watson Answer: (Miriam Margolyes, acted in, Harry Potter (film series)); (Miriam Margolyes, acted in, A Little Princess (1986 TV serial)). Therefore, blank 1: A
Instruction: Pick the correct answer for each blank that satisfies all the given constraints. Knowledge: (Joe Roberts, acted in, Neighbors (1920 film)); (Dinah Sheridan, acted in, The Railway Children (film)); (Joe Roberts, acted in, Three Ages); (Dinah Sheridan, acted in, 29 Acacia Avenue); (Dinah Sheridan, is married to, John Merivale); (Charlie Chaplin, is married to, Mildred Harris); (Dinah Sheridan, is married to, Jimmy Hanley); (Bradley Cooper, is married to, Jennifer Esposito); (Joe Roberts, acted in, Neighbors (1920 film)); (Jimmy Hanley, acted in, 29 Acacia Avenue); (Joe Roberts, acted in, Three Ages); (Joe Roberts, acted in, Our Hospitality); (Charlie Chaplin, is married to, Mildred Harris); (Dinah Sheridan, is married to, Jimmy Hanley); (Charlie Chaplin, is married to, Oona O’Neill); (Dinah Sheridan, is married to, John Merivale); (Charlie Chaplin, is married to, Mildred Harris); (Charlie Chaplin, is married to, Oona O’Neill);(John Merivale, is married to, Jan Sterling); (Paul Douglas (actor), is married to, Jan Sterling). (Optional) Constraints: (Dinah Sheridan, acted in, blank 2); (Dinah Sheridan, is married to, blank 1); (blank 1, acted in, blank 2); (Dinah Sheridan, is married to, blank 3); (blank 3, is married to, Jan Sterling). Options: blank 1: A. Liam Neeson, B. Jimmy Hanley, C. Nancy Wilson (rock musician) blank 2: A. Courage of Lassie, B. 29 Acacia Avenue, C. Listen to Me (film) blank 3: A. José María Aznar, B. John Merivale, C. Prince Harald of Denmark Answer: (Dinah Sheridan, acted in, 29 Acacia Avenue); (Dinah Sheridan, is married to, Jimmy Hanley); (Jimmy Hanley, acted in, 29 Acacia Avenue); (Dinah Sheridan, is married to, John Merivale); (John Merivale, is married to, Jan Sterling). Therefore, blank 1: B, blank 2: B, blank 3: B
[ZERO-SHOT prompt]
Table 16: Prompts used in main experiments with exemplars and an example question. “Knowledge” for each problem is only applicable in the “W/ KNOWLEDGE” setting.
LTM
Instruction: Pick the correct answer for each blank that satisfies all the given constraints. Knowledge: (Charlton Heston, acted in, True Lies); (Eliza Dushku, acted in, True Lies); (Tom Arnold (actor), acted in, True Lies); (Bill Paxton, acted in, True Lies); (Charlton Heston, acted in, Chiefs (miniseries)); (Stephen Collins, acted in, Chiefs (miniseries)); (Paul Sorvino, acted in, Chiefs (miniseries)); (Danny Glover, acted in, Chiefs (miniseries)). (Optional) Constraints: (blank 1, acted in, True Lies); (blank 1, acted in, Chiefs (miniseries)). Options: blank 1: A. Bill Paxton, B. Charlton Heston, C. Paul Sorvino Answer: Considering (blank 1, acted in, True Lies), maybe blank 1: A, or blank 1: B; considering (blank 1, acted in, True Lies), (blank 1, acted in, Chiefs (miniseries)), maybe blank 1: B. Therefore, blank 1: B
Instruction: Pick the correct answer for each blank that satisfies all the given constraints. Knowledge: (Joe Roberts, acted in, Our Hospitality); (Joe Roberts, acted in, Neighbors (1920 film)); (Taye Diggs, acted in, Rent (film)); (Joe Roberts, acted in, Three Ages); (Bradley Cooper, is married to, Jennifer Esposito); (Taye Diggs, is married to, Idina Menzel); (Charlie Chaplin, is married to, Mildred Harris); (Mary, Queen of Hungary, is married to, Jobst of Moravia); (Idina Menzel, acted in, Enchanted (film)); (Idina Menzel, acted in, Rent (film)); (Joe Roberts, acted in, Neighbors (1920 film)); (Joe Roberts, acted in, Three Ages); (Idina Menzel, is married to, Taye Diggs); (Bradley Cooper, is married to, Jennifer Esposito); (Mary, Queen of Hungary, is married to, Jobst of Moravia); (Charlie Chaplin, is married to, Mildred Harris). (Optional) Constraints: (blank 1, acted in, blank 2); (blank 1, is married to, Idina Menzel); (Idina Menzel, acted in, blank 2); (Idina Menzel, is married to, blank 1). Options: blank 1: A. Kelly LeBrock, B. Napoleon, C. Taye Diggs blank 2: A. Halloweentown High, B. Magnolia (film), C. Rent (film) Answer: Considering (blank 1, acted in, blank 2), maybe blank 1: C and blank 2: C; considering (blank 1, acted in, blank 2), (blank 1, is married to, Idina Menzel), maybe blank 1: C and blank 2: C; considering (blank 1, acted in, blank 2), (blank 1, is married to, Idina Menzel), (Idina Menzel, acted in, blank 2), maybe blank 1: C and blank 2: C; considering (blank 1, acted in, blank 2), (blank 1, is married to, Idina Menzel), (Idina Menzel, acted in, blank 2), (Idina Menzel, is married to, blank 1), maybe blank 1: C and blank 2: C; Therefore, blank 1: C, blank 2: C
Instruction: Pick the correct answer for each blank that satisfies all the given constraints. Knowledge: (Andy García, acted in, Smokin’ Aces); (Andy García, acted in, Ocean’s Thirteen); (Andy García, acted in, The Untouchables (film)); (Andy García, acted in, The Pink Panther 2); (Jeremy Piven, acted in, Smokin’ Aces); (Joe Roberts, acted in, Our Hospitality); (Joe Roberts, acted in, Three Ages); (Joe Roberts, acted in, Neighbors (1920 film)); (Virginia Madsen, acted in, Scooby-Doo! in Where’s My Mummy?); (Jeremy Piven, acted in, Scooby-Doo! in Where’s My Mummy?); (Mindy Cohn, acted in, Scooby-Doo! in Where’s My Mummy?); (Grey DeLisle, acted in, Scooby-Doo! in Where’s My Mummy?). (Optional) Constraints: (Andy García, acted in, blank 1); (blank 2, acted in, blank 1); (blank 2, acted in, Scooby-Doo! in Where’s My Mummy?). Options: blank 1: A. Things to Do in Denver When You’re Dead, B. Smokin’ Aces, C. Beverly Hills Chihuahua blank 2: A. Ron Perlman, B. Casey Kasem, C. Jeremy Piven Answer: Considering (Andy García, acted in, blank 1), maybe blank 1: A, or blank 1: B, or blank 1: C; considering (Andy García, acted in, blank 1), (blank 2, acted in, blank 1), maybe blank 1: B and blank 2: C; considering (Andy García, acted in, blank 1), (blank 2, acted in, blank 1), (blank 2, acted in, Scooby-Doo! in Where’s My Mummy?), maybe blank 1: B and blank 2: C. Therefore, blank 1: B, blank 2: C
Instruction: Pick the correct answer for each blank that satisfies all the given constraints. Knowledge: (Miriam Margolyes, acted in, Harry Potter (film series)); (David Thewlis, acted in, Harry Potter (film series)); (John Cleese, acted in, Harry Potter (film series)); (Richard Harris, acted in, Harry Potter (film series)); (Joe Roberts, acted in, Three Ages); (Maureen Lipman, acted in, A Little Princess (1986 TV serial)); (Nigel Havers, acted in, A Little Princess (1986 TV serial)); (Miriam Margolyes, acted in, A Little Princess (1986 TV serial)). (Optional) Constraints: (blank 1, acted in, Harry Potter (film series)); (blank 1, acted in, A Little Princess (1986 TV serial)). Options: blank 1: A. Miriam Margolyes, B. Maggie Smith, C. Emma Watson Answer: Considering (blank 1, acted in, Harry Potter (film series)), maybe blank 1: A, or blank 1: B, or blank 1: C; considering (blank 1, acted in, Harry Potter (film series)), (blank 1, acted in, A Little Princess (1986 TV serial)), maybe blank 1: A. Therefore, blank 1: A
Instruction: Pick the correct answer for each blank that satisfies all the given constraints. Knowledge: (Joe Roberts, acted in, Neighbors (1920 film)); (Dinah Sheridan, acted in, The Railway Children (film)); (Joe Roberts, acted in, Three Ages); (Dinah Sheridan, acted in, 29 Acacia Avenue); (Dinah Sheridan, is married to, John Merivale); (Charlie Chaplin, is married to, Mildred Harris); (Dinah Sheridan, is married to, Jimmy Hanley); (Bradley Cooper, is married to, Jennifer Esposito); (Joe Roberts, acted in, Neighbors (1920 film)); (Jimmy Hanley, acted in, 29 Acacia Avenue); (Joe Roberts, acted in, Three Ages); (Joe Roberts, acted in, Our Hospitality); (Charlie Chaplin, is married to, Mildred Harris); (Dinah Sheridan, is married to, Jimmy Hanley); (Charlie Chaplin, is married to, Oona O’Neill); (Dinah Sheridan, is married to, John Merivale); (Charlie Chaplin, is married to, Mildred Harris); (Charlie Chaplin, is married to, Oona O’Neill);(John Merivale, is married to, Jan Sterling); (Paul Douglas (actor), is married to, Jan Sterling). (Optional) Constraints: (Dinah Sheridan, acted in, blank 2); (Dinah Sheridan, is married to, blank 1); (blank 1, acted in, blank 2); (Dinah Sheridan, is married to, blank 3); (blank 3, is married to, Jan Sterling). Options: blank 1: A. Liam Neeson, B. Jimmy Hanley, C. Nancy Wilson (rock musician) blank 2: A. Courage of Lassie, B. 29 Acacia Avenue, C. Listen to Me (film) blank 3: A. José María Aznar, B. John Merivale, C. Prince Harald of Denmark Answer: Considering (Dinah Sheridan, acted in, blank 2), maybe blank 2: B; considering (Dinah Sheridan, acted in, blank 2), (Dinah Sheridan, is married to, blank 1), maybe blank 1: B and blank 2: B; considering (Dinah Sheridan, acted in, blank 2), (Dinah Sheridan, is married to, blank 1), (blank 1, acted in, blank 2), maybe blank 1: B and blank 2: B; considering (Dinah Sheridan, acted in, blank 2), (Dinah Sheridan, is married to, blank 1), (blank 1, acted in, blank 2), (Dinah Sheridan, is married to, blank 3), maybe blank 1: B and blank 2: B and blank 3: B; considering (Dinah Sheridan, acted in, blank 2), (Dinah Sheridan, is married to, blank 1), (blank 1, acted in, blank 2), (Dinah Sheridan, is married to, blank 3), (blank 3, is married to, Jan Sterling), maybe blank 1: B and blank 2: B and blank 3: B. Therefore, blank 1: B, blank 2: B, blank 3: B
[ZERO-SHOT prompt]
Table 17: Prompts used in main experiments with exemplars and an example question. “Knowledge” for each problem is only applicable in the “W/ KNOWLEDGE” setting.
STAGED PROMPTING
Instruction: Pick the correct answer for each blank that satisfies all the given constraints. Knowledge: (Charlton Heston, acted in, True Lies); (Eliza Dushku, acted in, True Lies); (Tom Arnold (actor), acted in, True Lies); (Bill Paxton, acted in, True Lies); (Charlton Heston, acted in, Chiefs (miniseries)); (Stephen Collins, acted in, Chiefs (miniseries)); (Paul Sorvino, acted in, Chiefs (miniseries)); (Danny Glover, acted in, Chiefs (miniseries)). (Optional) Constraints: (blank 1, acted in, Harry Potter (film series)); (blank 1, acted in, A Little Princess (1986 TV serial)). Options: blank 1: A. Miriam Margolyes, B. Maggie Smith, C. Emma Watson Answer: Stage 1 - status: Current answer: None. Remaining constraints containing blanks: (blank 1, acted in, Harry Potter (film series)); (blank 1, acted in, A Little Princess (1986 TV serial)). Stage 1 - solve: From (blank 1, acted in, Harry Potter (film series)), candidate for blank 1: A. Miriam Margolyes. Stage 1 - status update: Current answer: blank 1: A. Miriam Margolyes. Filled remaining constraints with current answer: (Miriam Margolyes, acted in, Harry Potter (film series)); (Miriam Margolyes, acted in, A Little Princess (1986 TV serial)). Updated remaining constraints containing blanks: None. Stage 1 - verify filled constraints: Does any error occur in filled remaining constraints? No. Go to next stage. Stage 2 - status: Current answer: blank 1: A. Miriam Margolyes. Remaining constraints containing blanks: None. Stage 2 - solve: No more remaining constraints with blank. Final answer: blank 1: A
Instruction: Pick the correct answer for each blank that satisfies all the given constraints. Knowledge: (Miriam Margolyes, acted in, Harry Potter (film series)); (David Thewlis, acted in, Harry Potter (film series)); (John Cleese, acted in, Harry Potter (film series)); (Richard Harris, acted in, Harry Potter (film series)); (Joe Roberts, acted in, Three Ages); (Maureen Lipman, acted in, A Little Princess (1986 TV serial)); (Nigel Havers, acted in, A Little Princess (1986 TV serial)); (Miriam Margolyes, acted in, A Little Princess (1986 TV serial)). (Optional) Constraints: (blank 1, acted in, True Lies); (blank 1, acted in, Chiefs (miniseries)). Options: blank 1: A. Bill Paxton, B. Charlton Heston, C. Paul Sorvino Answer: Stage 1 - status: Current answer: None. Remaining constraints containing blanks: (blank 1, acted in, True Lies); (blank 1, acted in, Chiefs (miniseries)). Stage 1 - solve: From (blank 1, acted in, True Lies), candidate for blank 1: A. Bill Paxton. Stage 1 - status update: Current answer: blank 1: A. Bill Paxton. Filled remaining constraints with current answer: (Bill Paxton, acted in, True Lies); (Bill Paxton, acted in, Chiefs (miniseries)). Updated remaining constraints containing blanks: None. Stage 1 - verify filled constraints: Does any error occur in filled remaining constraints? (Bill Paxton, acted in, Chiefs (miniseries)) is incorrect. Redo stage 1. Stage 1 - status: Current answer: None. Remaining constraints containing blanks: (blank 1, acted in, True Lies); (blank 1, acted in, Chiefs (miniseries)). Stage 1 - solve: From (blank 1, acted in, True Lies), candidate for blank 1: B. Charlton Heston. Stage 1 - status update: Current answer: blank 1: B. Charlton Heston. Filled remaining constraints with current answer: (Charlton Heston, acted in, True Lies); (Charlton Heston, acted in, Chiefs (miniseries)). Updated remaining constraints containing blanks: None. Stage 1 - verify filled constraints: Does any error occur in filled remaining constraints? No. Go to next stage. Stage 2 - status: Current answer: blank 1: B. Charlton Heston. Remaining constraints with blanks: None. Stage 2 - solve: No more remaining constraints with blank. Final answer: blank 1: B
[ZERO-SHOT prompt]
Table 19: Prompts used in main experiments with exemplars and an example question. “Knowledge” for each problem is only applicable in the “W/ KNOWLEDGE” setting.
VERIFYALL
Instruction: Pick the correct answer for each blank that satisfies all the given constraints. Knowledge: (Charlton Heston, acted in, True Lies); (Eliza Dushku, acted in, True Lies); (Tom Arnold (actor), acted in, True Lies); (Bill Paxton, acted in, True Lies); (Charlton Heston, acted in, Chiefs (miniseries)); (Stephen Collins, acted in, Chiefs (miniseries)); (Paul Sorvino, acted in, Chiefs (miniseries)); (Danny Glover, acted in, Chiefs (miniseries)). Constraints: (blank 1, acted in, True Lies); (blank 1, acted in, Chiefs (miniseries)). Options: blank 1: A. Bill Paxton, B. Charlton Heston, C. Paul Sorvino Answer: Candidate answer: blank 1: A. Bill Paxton; Filled constraints with candidate answer: (Bill Paxton, acted in, True Lies); (Bill Paxton, acted in, Chiefs (miniseries)); Verification: Does error occur in filled constraints with candidate answer? (Bill Paxton, acted in, Chiefs (miniseries)) is incorrect. Candidate answer: blank 1: B. Charlton Heston; Filled constraints with candidate answer: (Charlton Heston, acted in, True Lies); (Charlton Heston, acted in, Chiefs (miniseries)); Verification: Does error occur in filled constraints with candidate answer? No. Therefore, blank 1: A.
Instruction: Pick the correct answer for each blank that satisfies all the given constraints. Knowledge: (Joe Roberts, acted in, Our Hospitality); (Joe Roberts, acted in, Neighbors (1920 film)); (Taye Diggs, acted in, Rent (film)); (Joe Roberts, acted in, Three Ages); (Bradley Cooper, is married to, Jennifer Esposito); (Taye Diggs, is married to, Idina Menzel); (Charlie Chaplin, is married to, Mildred Harris); (Mary, Queen of Hungary, is married to, Jobst of Moravia); (Idina Menzel, acted in, Enchanted (film)); (Idina Menzel, acted in, Rent (film)); (Joe Roberts, acted in, Neighbors (1920 film)); (Joe Roberts, acted in, Three Ages); (Idina Menzel, is married to, Taye Diggs); (Bradley Cooper, is married to, Jennifer Esposito); (Mary, Queen of Hungary, is married to, Jobst of Moravia); (Charlie Chaplin, is married to, Mildred Harris). Constraints: (blank 1, acted in, blank 2); (blank 1, is married to, Idina Menzel); (Idina Menzel, acted in, blank 2); (Idina Menzel, is married to, blank 1). Options: blank 1: A. Kelly LeBrock, B. Napoleon, C. Taye Diggs blank 2: A. Halloweentown High, B. Magnolia (film), C. Rent (film) Answer: Candidate answer: blank 1: C. Taye Diggs, blank 2: B. Magnolia (film); Filled constraints with candidate answer: (Taye Diggs, acted in, Magnolia (film)); (Taye Diggs, is married to, Idina Menzel); (Idina Menzel, acted in, Magnolia (film)); (Idina Menzel, is married to, Taye Diggs); Verification: Does error occur in filled constraints with candidate answer? (Taye Diggs, acted in, Magnolia (film)) is incorrect. Candidate answer: blank 1: C. Taye Diggs, blank 2: C. Rent (film); Filled constraints with candidate answer: (Taye Diggs, acted in, Rent (film)); (Taye Diggs, is married to, Idina Menzel); (Idina Menzel, acted in, Rent (film)); (Idina Menzel, is married to, Taye Diggs); Verification: Does error occur in filled constraints with candidate answer? No. Therefore, blank 1: C, blank 2: C.
Instruction: Pick the correct answer for each blank that satisfies all the given constraints. Knowledge: (Andy García, acted in, Smokin’ Aces); (Andy García, acted in, Ocean’s Thirteen); (Andy García, acted in, The Untouchables (film)); (Andy García, acted in, The Pink Panther 2); (Jeremy Piven, acted in, Smokin’ Aces); (Joe Roberts, acted in, Our Hospitality); (Joe Roberts, acted in, Three Ages); (Joe Roberts, acted in, Neighbors (1920 film)); (Virginia Madsen, acted in, Scooby-Doo! in Where’s My Mummy?); (Jeremy Piven, acted in, Scooby-Doo! in Where’s My Mummy?); (Mindy Cohn, acted in, Scooby-Doo! in Where’s My Mummy?); (Grey DeLisle, acted in, Scooby-Doo! in Where’s My Mummy?). Constraints: (Andy García, acted in, blank 1); (blank 2, acted in, blank 1); (blank 2, acted in, Scooby-Doo! in Where’s My Mummy?). Options: blank 1: A. Things to Do in Denver When You’re Dead, B. Smokin’ Aces, C. Beverly Hills Chihuahua blank 2: A. Ron Perlman, B. Casey Kasem, C. Jeremy Piven Answer: Candidate answer: blank 1: B. Smokin’ Aces, blank 2: C. Jeremy Piven; Filled constraints with candidate answer: (Andy García, acted in, Smokin’ Aces); (Jeremy Piven, acted in, Smokin’ Aces); (Jeremy Piven, acted in, Scooby-Doo! in Where’s My Mummy?); Verification: Does error occur in filled constraints with candidate answer? No. Therefore, blank 1: B, blank 2: C.
Instruction: Pick the correct answer for each blank that satisfies all the given constraints. Knowledge: (Miriam Margolyes, acted in, Harry Potter (film series)); (David Thewlis, acted in, Harry Potter (film series)); (John Cleese, acted in, Harry Potter (film series)); (Richard Harris, acted in, Harry Potter (film series)); (Joe Roberts, acted in, Three Ages); (Maureen Lipman, acted in, A Little Princess (1986 TV serial)); (Nigel Havers, acted in, A Little Princess (1986 TV serial)); (Miriam Margolyes, acted in, A Little Princess (1986 TV serial)). Constraints: (blank 1, acted in, Harry Potter (film series)); (blank 1, acted in, A Little Princess (1986 TV serial)). Options: blank 1: A. Miriam Margolyes, B. Maggie Smith, C. Emma Watson Answer: Candidate answer: blank 1: A. Miriam Margolyes; Filled constraints with candidate answer: (Miriam Margolyes, acted in, Harry Potter (film series)); (Miriam Margolyes, acted in, A Little Princess (1986 TV serial)); Verification: Does error occur in filled constraints with candidate answer? No. Therefore, blank 1: A [ZERO-SHOT prompt]

heading: Future Work
text: We envision the study of such performance gain and the application of the insight as possible future directions. Application of geometric knowledge reasoning Despite the difficulty of the task, LLMs do show preliminary geometric reasoning ability over incomplete fact network. While such ability still has a long way to achieve perfection, this finding opens up the possibility of using LLMs as flexible relational databases and accessing the parametric knowledge with prompts similar to SQL (structured query language). Same prompting approach with different LLMs While GPT-3. 5-TURBO does not benefit from
Hyperparameter Value
degreec 5, 7, 9 sG 6, 7, 8, 9, 10, 11 sB [ 1 4 · sG, 12 · sG] mr 1. 1, 1. 2, 1. 3 mb 1, 1. 1