Doing Experiments and Revising Rules with Natural Language and Probabilistic Reasoning

Authors: wasu  top piriyakulkij
cassidy  langenfeld
tuan anh le
kevin  ellis
Venue: NeurIPS_cc_2024_Conference
Published: 2024-05-14 20:30:45

Abstract: We give a model of how to infer natural language rules by doing experiments. The
model integrates Large Language Models (LLMs) with Monte Carlo algorithms for
probabilistic inference, interleaving online belief updates with experiment design
under information-theoretic criteria. We conduct a human-model comparison on a
Zendo-style task, finding that a critical ingredient for modeling the human data is to
assume that humans also consider fuzzy, probabilistic rules, in addition to assuming
that humans perform approximately-Bayesian belief updates. We also compare
with recent algorithms for using LLMs to generate and revise hypotheses, finding
that our online inference method yields higher accuracy at recovering the true
underlying rule, and provides better support for designing optimal experiments.

Main paper: 
 introduction: 
 An important way that humans grow their knowledge of the world is by experimentation and other forms of active learning. This process is most clearly present in the experimental sciences, but similar processes of active inference begin in infancy through early childhood [1, 2, 3, 4, 5] . Within everyday adult cognition, active experimentation helps us quickly learn to use new devices and tools.A basic framework for modeling experimentation is to alternate between conducting a good experiment, and updating one's beliefs based on those experimental results [6] . These beliefs concern a latent hypothesis about the regularity or trend the experimenter is investigating. This leaves open at least two computational questions. First, we need to define a hypothesis space. Second, we need efficient algorithms for belief updates and experiment generation. Such algorithms should reason about probabilistic beliefs-considering many hypotheses and their associated probabilities-in order to find experiments that optimally resolve different competing hypothesis.Here we will introduce a model that represents hypotheses in natural language-even for problems that do not intrinsically involve human language. We do this for two reasons. First, natural language can index many human concepts, and can recursively combine them, giving an expressive hypothesis space. Second, it allows using Large Language Models (LLMs) to aid the inference task of updating beliefs after each experiment, giving tractable, approximate probabilistic inference when we view the LLM as a proposal distribution for a Monte Carlo estimator.We are especially interested in comparing our model to human behavior, given the long legacy of probabilistic modeling within cognitive science [7, 8] . We find a nuanced picture: vanilla LLMs are not humanlike on our active learning tasks (and underperform humans); our full model outperforms humans; but a simple change-switching from deterministic to probabilistic hypotheses-allows matching humans in overall performance, and agreement with humans on more fine-grained metrics.From a technical perspective, our work needs to infer natural-language hypotheses in an online setting, so that it can cycle between experimentation and hypothesis formation. This differs from recent batched approaches for hypothesis formation [9, 10, 11] . To allow online inference, we hybridize LLMs with Sequential Monte Carlo Samplers (SMC-S: [12] ). In SMC-S, one tracks a modest number of hypotheses that serve as (approximate) samples from the posterior. Meanwhile, the LLM focuses the sampler on a small set of candidate hypotheses that it deems relevant, given the data. The resulting sampler facilitates active learning by choosing an experiment which optimally "splits" the candidate hypotheses. With strategies that do not use probabilistic framing, such as tracking a single best-guess hypothesis, the active learner would have little guidance on what experiment to do next.We will focus here on active inference of basic symbolic concepts expressible in natural language, as we believe these are tractable first targets of study. Concretely, we consider tasks in the spirit of the boardgame 'Zendo', a challenging but accessible game where human players actively learn binary rules combining logical and spatial relations [13, 14, 15] , as well as 'Blicket test' style tasks, inspired by studies in developmental psychology [16, 2, 17] that investigate how children learn the causal mechanism behind the activation of a machine. See Figure 1 .We contribute the following:1. An algorithm for probabilistic inference of latent natural language hypotheses. This derives from SMC-S, but uses an LLM proposal distribution to allow tractable inference over natural language strings, essentially using the LLM to suggest ways of revising the belief state.2. Model-Human/Model-Baseline comparisons, finding that (1) we get a better fit to human data using natural language, instead of formal languages;(2) the model can be further made more humanlike by considering fuzzy (probabilistic) rules, and (3) that our online inference also yields better accuracy at the actual task relative to recent work [10, 9, 11] .3. Empirical findings about the ability of LLMs to revise hypotheses and propose experiments. On the domains we consider, we find that LLMs are effective for proposing and revising hypotheses, but do not consistently outperform random guessing when proposing experiments. 
model: 
 We start with standard Bayesian optimal experiment design, which gives a framework for describing both experimentation and hypothesis formation [18, 19] . Our model includes natural-language hypotheses h ∈ Σ * , experiments x ∈ X , and experiment outcomes y ∈ Y. We consider equipping h with real-valued parameters θ: For example, if the hypothesized rule is fuzzy (noisy), then θ would control the noise level. As new experiments are proposed sequentially, we index experiments and outcomes with subscripts, i.e. x t and y t for the t th experiment and outcome, respectively. The objective is to identify ground-truth h * , and to accurately predict the outcome of future experiments.The machine makes sound when Hypothesis 1: at least one of them is a yellow object Hypothesis 2: more than three objects are presentThe machine makes sound when Hypothesis 1: at least one of them is a yellow object or a cylinder Hypothesis 2: there are at least two objectsThe machine makes sound when Hypothesis 1: at least one of them is a yellow object or a cylinder that is not red Experimentation Hypothesis Revision The joint distribution over hypothesis h, θ and outcomes y 1:T , given experiments x 1:T , isEQUATIONwhere the prior p(h) favors shorter or simpler hypotheses. From eq. ( 1) the posterior isEQUATIONwhere we assume the above integral is tractable, because θ is low-dimensional. Ultimately, the purpose of the hypothesis is to make predictions on new experiments. Given a test experiment x test , an ideal learner predicts an outcome y test distributed as follows:EQUATIONThe optimal experiment for identifying h maximizes the following information gain [20] :EQUATIONThe above computations are intractable because they involve considering the infinitely large set of all hypotheses and experiments. We next describe our LLM-guided approximation methods.We introduce a generalization of the Sequential Monte Carlo Sampler (SMC-S) [12] , an online approximate inference algorithm which tracks a small pool of hypotheses-called particles-that evolve over time as new data is collected. Tracking representative high-posterior particles allows approximate inference (eq. ( 2)) and prediction (eq. ( 3)) by only considering the current particles. This makes the model "boundedly rational" [21] : as the bound on computation (# particles) grows large, the sampler better approximates optimal inference. To the extent that our work offers a cognitive model, we are claiming that humans only consider a small number of hypotheses, which evolve in ways that approximate probabilistic reasoning. This should be seen within the tradition of using approximate inference methods to give mechanistic accounts of human learning [22, 23, 24, 25] .Standard SMC-S tracks n particles at each time point t, written H t = {h(i) t } n i=1 . Each particle has a weight, W t = {w (i) t } n i=1, giving the approximate posterior p(h|x 1:t , y1:t ) ≈ i w (i) t 1 h = h (i) t. Upon observing a new data point, the particles H t are pushed through a forward kernel q t+1 (h t+1 |h (i) t ), which randomly perturbs the particles, to obtain new particles H t+1 . Next, the particles are reweighed to obtain W t+1 . Finally, a resampling step can be executed to prune low-weight particles and multiply high-weight particles. Figure 2 : Sequential Monte Carlo method tracks a small number of hypotheses (called particles), each of which is a natural language rule, represented above by circles. After each experiment, the particles are revised in light of the new data by pushing the particles through the forward kernel. Then, the new particles are reweighed according to how well each explains the data we have seen so far. Resampling prunes low-probability hypotheses while multiplying high-probability ones.Here h is a natural language string, suggesting an LLM should define q t (h t |h (i) t-1 ). For example, an LLM can be prompted with a hypothesis, together with the latest experiment outcome, and asked to revise that hypothesis. But calling an LLM to perturb every single particle is expensive, and unnecessary for hypotheses that already explain the data well.We therefore design a variant of SMC-S whose forward kernel looks globally at the current set of particles and prompts an LLM to revise the worst (lowest-likelihood) particles, while keeping unchanged the best (highest-likelihood) particles. This concentrates the computation on improving bad hypotheses, instead of wasting effort altering what already works. Within the context of LLMs, this can be seen as an online, probabilistic version of hypothesis refinement [9, 26, 10] . Within the context of SMC-S, this mathematically corresponds to defining a forward kernel that conditions on the entire set of previous particles and all seen data points, q t (h t |H t-1 , x 1:t , y 1:t ). 2 Below we formalize our new SMC-S variant, which we call LLM-SMC-S, illustrated in Figure 2 .Procedure: LLM-SMC-S (A.4). Given H t , W t where p(h|x 1:t , y 1:t ) ≈ i w(i) t 1 h = h (i) t :1. Define unnormalized target densities γ(h) = p(h, y 1:t , x 1:t ) and γ ′ (h) = p(h, y 1:t+1 , x 1:t+1 ). 2. Sample h ′ ∼ q t+1 (•|H t , x 1:t+1 , y 1:t+1 ) (i.e., using LLM to revise hypotheses) 3. Compute the weight w ′ for h ′ followingw ′ = A(h ′ , H t , W t ) q t+1 (h ′ |H t , x 1:t+1 , y 1:t+1 )whereA(h ′ , H t , W t ) = 1 n n i=1 w (i) t γ ′ (h ′ )r(h (i) t |h ′ ) γ(h (i) t ) (5)with the reverse kernel r(h|h ′ ) defined as uniform up to strings of a maximum length. 4. Repeat steps 2-3 (sampling/weighing) a total of n times, and normalize the weights. Optionally, resample to generate an unweighted posterior (we always resample). 5. Output: H t+1 and W t+1 , formed from n samples of h ′ , w ′ with w ′ normalized from step 4, which approximate p(h|x 1:t+1 , y 1:t+1 ).The correctness of the above procedure is most easily understood using the following definition:Definition: Proper Weighting [27] . Let γ(h) be an unnormalized target density, which we can evaluate. Let the corresponding normalized target density be π(h) = γ(h) Zπ where Z π = γ(h)dh is the normalization constant. A weighted particle h, w is properly weighted with respect to γ if for any function f ,E[wf (h)] = Z π E π(h) [f (h)]Proposition 1. If H, W input to Procedure LLM-SMC-S is properly weighted with respect to γ, then the output h ′ , w ′ is properly weighted with respect to γ ′ . (Proof in Appendix A.1.)Our active learning works by doing an experiment that maximizes information gain (eq. ( 4)). Experiments may be complex, such as involving putting objects or instruments in specific positions, and there might be combinatorially many possible experiments. For a rich space of experiments, a bounded learner-human or AI-cannot consider all possibilities.We will propose experiments using an LLM, but then reassess those proposals under probabilistic criteria. Particularly, we provide an LLM with the hypotheses tracked by the SMC-S sampler at each iteration, and prompt it to generate experiments that support and falsify each hypothesis. Empirically, this process yields a diverse pool of experiments. We take the best experiment proposed by the LLM, as measured under the approximate posterior from SMC-S: x t+1 =where p is approximated with the weighted particles from SMC-S. 3 2 We note that whether we condition on the seen data points or not does not change the proof. The main novelty of this new variant lies in how q can be conditioned on the entire set of particles Ht-1 3 We let the particles Ht be the support for both distributions so that we can calculate KL divergence.All of our experiments have binary outcomes (y ∈ {0, 1}), and all of our natural language hypotheses correspond to rules that predict whether an experiment succeeds or fails (1 or 0). Although the rules predict hard all-or-none judgments, a learner can relax that constraint by assuming that the underlying rule is fuzzy (noisy). Many natural language facts and rules actually only partly hold, such as birds fly (almost always true), or birds lay eggs (true half the time). To handle the possibility of fuzzy rules, we equipped each hypothesized rule with real-valued parameters θ that control the noise level. The noise parameters decompose into a pair θ = (ϵ, δ) controlling the rate of false-positives/false-negatives:p(y = 1|x, h, ϵ, δ) = δ if h(x) = 1 1 -ϵ if h(x) = 0Under this formulation, hard rules corresponds to p(ϵ) and p(δ) having non-zero probability only at value 1. For probabilistic, fuzzy rules, we use Gaussian priors for p(ϵ) and p(δ), truncated to [0.5,1], and with a bias toward larger ϵ. The prior p(h) is defined as inversely proportional to wordcount, giving a gentle bias toward parsimony. We investigate both hard and fuzzy rules in our experiments.Evaluating h(x) requires checking the natural language string h against experiment x, for which we use GPT-3.5 to translate the natural language h to code which is run on x. We use GPT-4 Turbo to propose hypotheses [30] . Recent studies find a similar breakdown of LLMs works well [9, 10, 11] . 
experimental results: 
 Domains. Zendo is a game where a player seeks to infer a hidden binary rule about scenes of colored shapes. Our Zendo games begin with showing the player a positive example scene, followed by 7 rounds of experimentation, where the player builds a scene, and receives feedback on if the scene obeys the hidden rule. After the experimentation phase, players are tested on 8 test scenes, half of which follow the hidden rule. Our setup follows Bramley et al. [13] , but modified for LLMs by presenting scenes as text describing each block by its color, size, orientation, groundedness, and what other blocks it touches and stacks (Figure 3 ).Our second domain, ActiveACRE, derives from The Abstract Causal REasoning (ACRE) dataset [17] , which in turn derives from 'blicket' tests in developmental cognitive psychology [16] . The original ACRE is a causal induction dataset where each task is to figure out what causes the 'blicket' machine to make sounds when multiple objects are put on the machine. We add active learning to ACRE: rather than passively observe examples, our ActiveACRE allows the player to try 7 experiments, after passively witnessing the outcome of one experiment involving eight objects. The player is then tested (without further feedback) on all possible combinations of the original eight objects.Model-Baseline comparisons. Table 1 contrasts the performance of different models, showing that online inference with hard rules outperforms all other models on both datasets, including a ReActstyle baseline [31] (Direct LLM), and batched inference with refinement, an approach advocated for in recent work [10, 9] . To measure accuracy on Zendo, we compute the predictive posterior accuracy summed over the 8 test scenes and averaged over all tasks. Because the test set on ActiveACRE 4a ). With fuzzy rules, the model explains 57% of the variation in this more fine-grained measurement of human accuracy (R 2 = .57). Switching to hard rules drops this to R 2 = .10, suggesting that hard all-or-none rules do not provide as good of an explanation of human behavior, even though hard rules outperform probabilistic ones in terms of accuracy. Doing batch inference instead of online inference degrades fit to R 2 = .05. Having the LLM play Zendo directly (ReAct [31] ) is only loosely correlated with human accuracy patterns (R 2 = .25). We last consider predicting every single human judgment on every single test scene, for every single rule. The online, fuzzy rules model predicts these human judgments at the level of R 2 = .35, and importantly, it is only with combination of online inference with fuzzy rules that gives a significant fraction of explained variance (Figure 5 ), and which assigns the highest likelihood to the raw human data (Table 2 ). We noticed across many rules a significant difference in human accuracy on RF and Not RF test scenes. Whether people finds RF or Not RF test scenes easier depends on the underlying rule. Figure 4b illustrates this phenomenon and compares it against what each model thinks should be the easier condition. Online learning of fuzzy rules successfully predicts the direction of almost all of these trends, unlike the alternative models.Hence, we hypothesize that although the hidden Zendo rules are deterministic, humans might nonetheless infer fuzzy rules. Real-world regularities are seldomly deterministic, so it may be rational for human learners to seek probabilistic explanations, especially when they are uncertain about the underlying rule. However, fuzzy rules on their own do not suffice to explain human judgments: Only by combining with online probabilistic inference do we begin to explain the data.Why reason in natural language instead of a formal language? Many Bayesian models account for human concept learning using probabilistic reasoning over formal languages such as logic [32, 33, 34, 35, 36] . Instead, our model operates over natural language. This helps address two liabilities of formal representations: expressivity and tractability. A handcrafted formal language is often insufficiently expressive, accidentally excluding many human concepts. This expressivity must be limited because, although there exist highly expressive formal languages, in practice, inference in such languages is generally intractable-a tradeoff partly addressed by using LLM proposal distributions.To illustrate these points, we study a new Zendo rule-'the majority of blocks is red'-which is not expressible in the formal language introduced by [13] . We collect new human data in an IRB-approved study. Figure 6 shows that both humans and our model correctly learn this rule 30% -40% of the time. This indicates both the model and humans are able to represent this rule in their hypothesis space, which is unrepresentable in a formal language designed specifically for Zendo. Another reason to use natural language representations is that LLMs, trained on human-generated data, may to some extent capture human bias, judgement, and opinions [37, 38, 39] . Unlike approaches based on estimating probabilities on formal languages, incorporating LLMs into our models might therefore make them display more human-like behaviors-as shown in earlier sections-without access to additional human data. Indeed, Table 2 shows that our best-performing model surpasses [13] 's model on human data log likelihood even though the latter fits their models on both human active queries and predictions, while our model does not perform such parameter fitting. Bounded rationality. To understand the effect of computational cost on the results, we analyze performance and human-model fit while varying the computational budget, as measured by LLM calls. Figure 7 plots human-model fit as compute budget varies (see also Table 5 ). We observe an (inverted) U-shaped curve: Too little budget gives a bad fit, but overshooting also degrades fit. This result aligns with the theory of bounded rationality [21] , which argues for considering human's limited cognitive resources, and with the rational analysis of human processing limitations [23, 40] .What makes good experiments: LLMs, or Information Gain? We first study the importance of the information gain objective (Table 3 ), contrasting three different active learning methods: LLM (prompting with the hypotheses and asking for a good experiment); Random (handcoded random generator), and InfoGain (main method, with LLM proposing experiments). Substituting InfoGain with alternative methods significantly degrades model performance. Reranking LLM proposals with information gain is important, and an LLM-on its own-does not generate experiments that are as effective.Is this explained by the strength of the LLM experiment proposer, or by the strength of the InfoGain objective? While earlier results support LLMs' effectiveness as hypothesis proposers, Table 4 demonstrates that a random proposer, hand-designed under reasonable assumptions, performs similarly to an LLM experiment proposer. This finding is in line with [41] which argues that LLMs may not always produce the most useful set of candidate questions. 
related work: 
 Bayesian Concept Learning in Cognitive Science. Bayesian models of few-shot learning of concepts and categories has a long legacy [32, 34, 42, 43, 44, 13, 14, 15, 33, 35, 45, 46, 47, 36] . On Zendo, [13, 14, 15] also engineers a probabilistic context-free grammar to define the Bayesian model to explain Zendo human data; inference is intractable, and they study various approximate inference algorithms on the task. Our work opts for natural language as the hypothesis space for its expressiveness and leverages LLMs to help with approximate inference.Inductive Reasoning with Large Language Models. There are a number of recent works on inductive reasoning with LLMs [10, 9, 56, 11, 57] . [10, 9] study the inductive reasoning ability of vanilla LLMs and propose a simple refinement algorithm to improve performance. [11] explicitly treats LLMs as importance samplers and shows that their model, with prior learned from human data and importance sampling as their inference algorithm, is human-like on some domains. Our work frames many of these works as batch inference-in the spirit of importance sampling, from a probabilistic view-which is compared in our results. Additionally, we model the full life-cycle of experimentation and hypothesis revision.Active Learning with Large Language Models. The problem of performing active learning with the help of LLMs has been studied under the task of asking better questions with LLMs [58, 59, 60, 41, 61] . GATE [58] directly prompts LLMs to ask open-ended, informative questions. Other works [59, 60, 41] use LLMs to help propose several candidate questions, and then use expected information gain to select the question to be asked. Following these previous works, we investigate the relevance of classic criteria from active learning such as expected information gain.Probabilistic Inference with Large Language Models. The framework of probabilistic inference has been applied to LLM-based algorithms. Language-model cascades [62] provides a unifying framework for seeing recent inference-time LLM algorithms [63, 64, 65] as reasoning with probabilistic programs. Other work [66] fine-tunes CoT models by formulating the problem as maximum marginal likelihood where the marginalization over the latent chain of thought is done via probabilistic inference. We use an LLM as an aid for SMC-S, but others have explored using SMC as an aid for LLM decoding [67, 68] , which although technically very different, is conceptually complementary. 
limitations and next steps: 
 The work presented is limited in important ways that suggest next steps. Most immediately, many of the the hypotheses we consider are simple and stereotyped in form, and much of the promise of using natural language is that it exposes a rich, expressive representation for combining and creating new ideas [70] . More ambitiously, hypothesis generation, both in science and in everyday thinking, often involves conjecturing the existence of unseen objects, not just unknown regularities, and incorporating this abductive thinking-which is absent from our model-could open many directions.Although the work here is most directly an account of human behavior within the context of the game Zendo, our model is also more broadly inspired by the mental activities of experimental scientists as they build theories and models, weigh hypotheses, and design experiments. Two basic features of our approach reflect scientific experimentation and theory building. First, scientific theories apply only within a particular regime. For example, many equations in physics only apply when objects are moving slowly, and many thermodynamic equations only apply at equilibrium. Outside this regime, the theory is no longer predictive. Similarly, a variant of a model like ours could hypothesize fuzzy, probabilistic rules which either predict a category with high confidence, or outside the regime in which the rule applies, can fail to make a decisive prediction.The second way in which this work reflects the practices of experimental science is that it builds its hypotheses via an incremental evolutionary process. In this way, our model is best thought of as performing what is sometimes called 'normal science' -where one works within an existing paradigm, and considers piecemeal evidence -and does not model paradigm shifts [71] (what a scientific theorist might pursue) or deeper conceptual changes [3] (what happens during child development), both of which require deep reanalysis of a broad batch of past data, rather than online incremental revision.A.1 Proof for the weight update of LLM-SMC-S Proposition 1. If H, W input to Procedure LLM-SMC-S is properly weighted with respect to γ, then the output h ′ , w ′ is properly weighted with respect to γ ′ .Proof. Let Z π ′ = γ ′ (h ′ )dh ′ be the normalizing constant of γ ′ and π ′ (h ′ ) = γ ′ (h ′ )Z π ′ be the normalized target. We want to showE[w ′ f (h ′ )] = Z π ′ E π ′ (h ′ ) [f (h ′ )].Following arguments similar to [29] , we have LHS ( 7)EQUATION= Z π E π(h) [g(h)] (apply the proper weighting property with test function g)EQUATIONEQUATIONEQUATIONCode and data available at https://github.com/topwasu/doing-experiments-and-revising-rules/Zendo. Zendo is a game where a player seeks to infer a hidden binary rule about assemblies of colored blocks. The game starts by providing the player with a positive scene that follows the hidden rule. Then, the player queries an oracle as to a particular scene follows the rule or not, or makes a guess about the secret rule. The game ends when they guess correctly. [13] introduces a 2D version of the Zendo game shown in Figure 3 . The scenes consist of blocks, each with its own color (red, blue, green) and size (small, medium, large). The blocks can have different orientations and positions in a 2D scene. They may and may not touch each other. The game starts with an initial phase where a rule-following scene is given, followed by 7 rounds of active learning phase where the player gets to query an oracle for ground truth prediction. At the end, the player enters a prediction phase where they are asked to give predictions for 8 test scenes (4 rule-following and 4 not rule-following). Bramley et al. ( 2018) study human gameplay on 10 rules, collecting data from 30 participants who each play Zendo 10 times (once per rule). They use a cover story of an alien planet where some arrangements of blocks emit radiation, and the task is to figure out a rule predicting radiation emission.Zendo is most naturally framed as a visual-physical concept learning problem. For our model, however, we will work with discrete symbolic descriptions of scenes. This makes that problem more compatible with the language-of-thought paradigm, and also allows using LLMs to operationalize the language of thought. We therefore modify Bramley et al. ( 2018)'s version of Zendo by associating each block in a scene with discrete attributes instead. The five attributes are color (red, blue, green), size (small, medium, large), orientation (upright, left, right, strange), groundedness (grounded, ungrounded, stacking), and touching (which blocks it touches / stacks). While this natural language version of the game removes continuous attributes, such as x, y position and orientation in 2D space, from its scene representations, these five attributes still maintain the complexity of the game and are sufficient for all 10 Zendo rules. 4The data is licensed under CC-BY 4.0.ActiveACRE. We convert the originally visual tasks into symbolic version of the tasks, similar to [9] . While the ground truth rule always has the structure that the blicket machine produces noises when one or more "blicket" objects (each object is either a blicket or a non-blicket) is placed on the machine, in contrast to [9] , we do not hint the learners that the ground truth rule is of this form, which means the learners are free to think that the rule may have to do with colors, number of objects, etc. We further modify the task to incorporate elements of active learning, making the logistics similar to Zendo: the game starts with 8 relevant objects, described with color (gray/red/blue/green/brown/cyan/purple/yellow), material (metal/rubber), and shape attributes (cube/sphere/cylinder), placed on the blicket machine which causes the machine makes sounds and follows by 7 rounds of query. The prediction phase tests the models on all possible combinations of the eight objects. We call this resulting domain, ActiveACRE. Figure 1 partially shows what a gameplay of simplified ActiveACRE looks like.To obtain the 8 initial objects, we sample uniformly from the three attributes to get an object and keep doing this until we achieve 8 unique objects. This can be done with a simple code, without external data.For all methods, unless specified, the number of LLM calls used per each iteration is 5 with each call batch-sampling 15 natural language hypotheses. Batch Inference. For batch, fuzzy model, we cap the number of unique hypotheses considered to 30, otherwise we would have too many hypotheses considered, since all fuzzy hypotheses have non-zero posterior probability, making inference very compute intensive.Batch Inference with Refinement. We set the number of refinement to 2 (we have tried increasing the number of refinement to 4 but didn't see any improvement). Following [9] , this method works as follows: (1) it first batch-samples many hypotheses with LLM, (2) select the best hypothesis (in Algorithm 1 LLM-SMC-S algorithm Let (x 1 , y 1 ) be the first data point we observe h(1) 1 , ..., h(n) 1 ∼ q(h|x 1 , y 1 ) w (i) 1 ← p(x1,y1,h (i) 1 ) q(h|x1,y1) for 1 ≤ i ≤ n ▷ Reweighting H 1 ← Resampling(H 1 , W 1 )▷ Resampling for t = 2, ..., T doThe active learning algorithm gives (x t , y t ) h(1) t , ..., h (n) t ∼ q(h|H t-1 , x 1:t , y 1:t ) ▷ Rejuvenating A(h (i) t , H t-1 , W t-1 ) = 1 n n j=1 w (j) t-1 p(h (i) t |x1:t,y1:t)r(h (j) t-1 |h (i) t ,x1:t,y1:t) p(h (j) t |x1:t-1,y1:t-1) w (i) t ← A(h (i) t ,Ht-1,Wt-1) q(h (i) t |Ht-1,x1:t,y1:t) for 1 ≤ i ≤ n ▷ Reweighting H t ←H nb = LLM (x t , y t , h i ) ▷ get neighbors (nb) of h i H nb = {h nb ∈ H nb | p(h nb |x 1:t , y 1:t ) ≤ p(h i |x 1:t , y 1:t )} ▷ filter out bad neighbors if |H nb | > m then▷ we want to consider a maximum of m neighbors w(i) nb ← p(h (i) nb |x 1:t , y 1:t ) for 1 ≤ i ≤ n H nb ← Down -Sampling(H nb , p = W nb , size = m) end if result = result ∪ H nb end forreturn result end function numbers of data points accounted) to be refined, (3) use LLM to output a batch of refined hypotheses, and (4) repeat the (2)-( 3) steps until at least one hypothesis fully accounts for all data points.The algorithm for LLM-SMC-S is described in Algorithm 1. For the first iteration, the initial important proposer q(h|x 1 , y 1 ) is defined to be an LLM, similar to batch inference. We define the forward kernel q in the algorithm as follows:EQUATIONThe pseudocode for B can be founded at Algorithm 2. What B is doing is basically look at low likelihood hypotheses, prompt LLM to come up with their neighbors, and filter out bad neighbors and limit the number of chosen neighbors to m. We find that having the down-sampling step to keep the number of neighbors considered low is helpful in practice, but one can remove this step to make B fully deterministic. The LLM function in the pseudocode means prompting an LLM with zero temperature.Batch. 'Blocks must touch at least one other block' is proposed but is immediately falsified by an existing experiment where a scene with no blocks touching is negative.Batch with refinement. 'Blue blocks must not touch green blocks' is proposed and then refined into 'Blue blocks must not touch blocks of any color other than red'. This hypothesis later gets falsified, without an opportunity to refine itself since the model is not online, when a scene with no blocks touching is negative. Online. 'There must be a blue block' is proposed and added to the pool of particles. Since it has higher prior than other particles (has shorter length); it keeps surviving while others get killed, despite some conforming with the data. Upon seeing a scene with a blue touching a green being negative, the particle 'There must be a blue block' is perturbed into 'there must be a blue block touching a red block'.In probabilistic inference terms, both batch with and without refinement correspond to importance sampling p(h|x 1:t , y1:t ) = E p(h ′ |x1:t,y1:t) [1[h = h ′ ]] = E q(h ′ |x1:t,y1:t) [ p(h ′ |x1:t,y1:t) q(h ′ |x1:t,y1:t) 1[h = h ′ ]].The difference in the two baselines lie in how q(h ′ |x 1:t , y 1:t ) is constructed.Batch. q(h|x 1:t , y 1:t ) = U (LLM (x 1:t , y 1:t )) where LLM (...) prompts an LLM to return a list of hypotheses Batch with refinement. q(h|x 1:t , y 1:t ) = U (Ref ined-LLM (x 1:t , y 1:t , N one, 0)) where Ref ined-LLM is defined as follows: First, let s(h, x 1:t , y 1:t ) = 1 t t i=1 1[h(x i ) = y i ].H = LLM -with-h(x 1:t , y 1:t , h) # Prompts LLM to refine h if k = K: return ∅ else if ∃h ′ ∈ H, s(h ′ , x 1:t , y 1:t ) = 1: return h ′ ∈ H|s(h ′ , x 1:t , y 1:t ) = 1 else: h * = argmax h ′ ∈H (s(h ′ , x 1:t , y 1:t )) return Ref ined -LLM (x 1:t , y 1:t , h * , k + 1)where K is the number of refinements allowed. Priors for ϵ and δ p(δ) has a mean of 0.7 and a standard deviation of 0.1, and p(ϵ) has a mean of 0.9 and a standard deviation of 0.01. Both distributions are truncated to remain within the range [0.5, 1]. We found that using different priors for ϵ and δ results in a more human-like behavior as shown in Figure 8 .Prior for h We let p(h) be inversely proportional to the word count of h for Zendo and uniform for ActiveACRE.For Zendo, we consider using a prior that would decay exponentially in length but find that letting p(h) ∝ ( 1 word_count(h) ) 2 already makes the particles become mostly short strings. A prior decaying exponentially in string length would definitely be too harsh on the hypotheses.Computational Cost Analysis Table 5 shows the performance of models with different compute budgets (number of LLM calls per iteration) on Zendo. It turns out that the performance of batch inference models plateaus after just 15 hypotheses (1 LLM call), while the performance of online inference models benefits from being able to sample more hypotheses but also plateaus after 75 hypotheses (5 LLM calls).We also describe here the compute resources required to reproduce the experiments. The main compute cost comes from OpenAI API which we call to prompt GPTs. The models with 1, 5, 10 LLM calls per each iteration uses up roughly $0.5, $1.5, $3 OpenAI API credit to run a Zendo task. For Zendo, one needs to run 50 tasks-10 Zendo tasks on 5 different seeds-to get the performance numbers of a method like we reported. The actual cost, however, could be lower than calculated since one can cache LLM responses.A.9 Human study on 'majority is red' rule details 20 participants from our academic department were recruited via Slack to attempt the rule "the majority of blocks are red". The participants are compensated $10-$20 depending on their performance ($10 base rate + $1.25 bonus for each correct test scence prediction -there are 8 test scenes). Figure 9 shows the web interface displayed to participants. The full instructions given to human participants are displayed at Figures 12 and 13 . In this game you will be learning about an alien planet. This planet is called Zorb.On Zorb, there are these special blocks that look like this: {Figure 9} These special blocks may look the same, but there are many different kinds of special blocks on the planet Zorb. They all have different names, and they all work differently. Sometimes, when the blocks are set up in certain ways, stars will shoot out of them! Every kind of special block has a different rule for making stars shoot out of them.Your job is to figure out each rule for how to make stars come out of all of the different kinds of special blocks! {Figure 10} So there are a lot of things that might make stars come out of certain special blocks! You might get stars from blocks of different numbers, from blocks of different colors, from blocks of different sizes, from blocks facing different directions, and more! We found out that there are 2 more kinds of special blocks on Zorb! Let's call them 'Bemmies' and 'Yoks'. Again, you do not have to memorize the names --we just want to emphasize that different kinds of blocks work under different rules But we don't know the rule for setting up each kind of special blocks so stars will come out of them. Your job is to figure out the two different rules for how to set up each different kind of special blocks so stars come out! Now we're going to watch a video. This video is going to show you how you can move the special blocks around yourself! You must watch the video to continue.Press buttons at the bottom to add blocks Move the blocks around by picking them up with the mouse (left clicking and holding)Turn them using the "Z" (counterclockwise) and "X" (clockwise) keys Right click on them to remove them (command + click if you are using mac trackpad) When you're done moving the special blocks, you're going to test them to see if stars will come out of them. If you set them up in the right way according to the rule, you'll see a bunch of stars appear! Otherwise, nothing will happen. After you move the special blocks around and test them, you're going to see if you can pick out which pictures of the blocks you think will shoot out stars. This video will show you how to do that: {demo video} In the video you can see that this participant thinks that four of the pictures show bemmies that stars will come out of (the ones marked in grey). The right answer could be anywhere between one and seven of the pictures.Adults: You will earn a bonus of $1.25 for each of the pictures in the main task where you guess correctly whether it will shoot out stars (demo task performance does not count). That means, if you get all eight pictures correct in the main task, you will earn a bonus of $10! You must watch the video to continue.Finally, you may guess the rule for how this kind of special blocks works.For example, if it looks like stars only shoot out of the blocks if all of them are green, you would write something like: "all the blocks have to be green"! Warning: Your responses will be checked by a human before HIT approval. Nonsensical or copy-pasted answers will lead to your HIT being rejected. If you truly have no ideas about a rule, please just write "I do not know".You will look at 2 different kinds of special blocks (including one demo task for learning the game) that will shoot out stars if they are set up in certain ways.You must figure out the rule for how each kind of special blocks works.You will set up the special blocks and test them to see if stars will shoot out of them seven times for each type.Your goal is to figure out which out of 8 new pictures of each kind of special blocks will shoot out stars ($1.25 bonus for each correct in the main task)... ...and to write down your best guess of the rule for that kind of special block! The prompts used in all of our experiments can be found in Tables 7 to 12 .For Zendo, we engineer the prompts for initial importance sampler q(h|x, y) for online inference so that they only output simple rules (see Table 7 ); this approach helps the proposer output hypotheses with higher priors, since our prior is defined by the number of words in the rule. We cannot apply this trick to batch inference because, unlike online inference, it does not evolve simpler rules into more complex ones. Additionally, we also design the importance sampler prompts to avoid proposing negative rules ('there is no ...') (see Table 7) . We found that this leads to a more human-like behavior and also better performance.See Table 6 Guidelines:• The answer NA means that the paper does not include experiments.• The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage. • The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute. • The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper). 
code of ethics: 
 Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines?Justification: The research conducted in the paper conforms, in every aspect, with the NeurIPS Code of Ethics.Guidelines:• The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.• If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics. • The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction). 
broader impacts: 
 Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?Answer: [NA] Justification: Our work is foundational research and has no direct societal impact.Guidelines:• The answer NA means that there is no societal impact of the work performed.• If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact. • Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.• The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster. • The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology. • If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML). 
safeguards: 
 Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?Answer: [NA] Justification: The paper poses no such risks.Guidelines:• The answer NA means that the paper poses no such risks.• Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters. • Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images. • We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.12. Licenses for existing assets Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?Answer: [Yes] Justification: The creators of original data are all properly credited, and the license and terms of use of the data are explicitly mentioned and properly respected.Guidelines:• The answer NA means that the paper does not use existing assets.• The authors should cite the original paper that produced the code package or dataset.• The authors should state which version of the asset is used and, if possible, include a URL. • The name of the license (e.g., CC-BY 4.0) should be included for each asset.• For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided. • If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset. • For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.The 10 rules we use are "there's a red", "all are the same size", "nothing is upright", "one is blue", "there's a small blue", "all are blue or small", "a red is bigger than all non reds", "some touch", "a blue and a red touch" "some pieces are stacked", and "some pieces are stacked" 
 
 Appendix:  


Decision: The paper introduces a novel approach to learn natural language hypotheses and conduct active experimentation for hypothesis refinement. It leverages LLMs for hypothesis generation and evaluation within the framework of Sequential Monte Carlo (SMC) and Bayesian Optimal Experimental Design (BOED). Experiments on two cognitive psychology tasks demonstrate improvements over existing methods and some agreement with human behavior. 

The paper is strong because its method is technically principled and experiments are carefully carried out. 

The main weakness is that the experimental settings are restricted, thus not fully utilizing the power of LLMs. Reviewers have raised questions in this aspect, and called for more analysis. Authors explain that experiments in simple domains are "important stepping stone towards fully wielding the power of natural language." The rebuttal also includes some new results and analysis. k

I think the strengths of this paper overweigh its weaknesses, and thus recommend to accept it. But I strongly recommend the authors to clarify their technical motivation and experiment design, as well as include new results and analysis in the camera-ready.

Reviews: review: paper_summary: paper_summary: This paper describes a model of online construction of rules that explain data, where (a) hypotheses are expressed in natural language, and (b) they are updated by proposing experiments (inputs to the ground-truth rule). The method consists of an extension of SMC, where experiments and hypotheses revision are proposed by an LLMs; the best experiment is chosen by an approximate information gain criterion. The paper considers both fuzzy and hard rules, and finds that hard rules perform best in terms of task completion (and the model outperforms previous methods, especially those that propose a batch of hypotheses rather than inferring them online). Moreover, the fuzzy rules model best fits human behavior.
strength_weakness: strength_weakness: While the domains used here are interesting for a first study, they are still relatively simple. Most compelling would be to find a task with a richer space of hypotheses and experiments, perhaps also making better use of common sense priors embedded in LLMs.

The paper also doesn't provide much qualitative insight into what behavioral differences arise from doing online experiments compared to only doing batched inference, perhaps with refinement. While it might be hard to describe general differences, at least having a few example traces in the appendix (e.g., showing example scenarios where batched inference falls short, or outperforms the online model) would help us at least form some qualitative hypotheses on where the gains are coming from.
questions: questions: * Are any of the 10 Zendo predicates particularly unlikely to be recovered? I'd be curious to see an example of the model recovering 'a red is bigger than all non reds', for instance, since this seems like a quite unlikely hypotheses to propose by itself.
* Is the hardcoded random experiment proposer biased in any useful way? It would be good to describe it at least in the appendix. Again, for most of the Zendo rules, it'd guess that most single random experiments would be highly non informative.
* In the result in Table 4, with a single proposal, there's nothing for the InfoGain reranking to do. So should the first column be interpreted as two random repeats of the same exact model?
* Appendix A1: missing an integral sign in (10)?
limitations: limitations: Yes, adequately addressed.
paper_summary: paper_summary: The paper addresses the problem inferring rules and designing experiments based on them. To do so, 1) they propose representing rules in natural language, generated with LLMs. 2) Using Monte-Carlo algorithms to score them 3) Revising these rules and proposing new experiments with LLMs. They instantiate the problem in the domains of Zendo (binary rules) and ActiveAcre (abstract causal reasoning/ blicket detector style tasks). The authors then compare human inferences with LLM inferences to show what might explain human inferences and how their method can outperform humans in inferring underlying rules.
strength_weakness: strength_weakness: - I think the description of the baselines could be made clearer. Including the prompts in the main or using figures to describe them could be useful.
- While I liked the discussion on formal languages vs natural languages, I would have liked to see a baseline where the model implements rules in a formal language. This need not be a DSL, but a way for the model to reason with an external symbolic solver. Eg: the language could be python (with pymc/stan/pyro), or webppl.
    - Relatedly, [https://arxiv.org/abs/2402.17879](https://arxiv.org/pdf/2402.17879) seems like a relevant paper where rules are inferred and implemented in pymc
- Finally, the two domains, while inspired from psychology experiments, are quite artificial. They did make it possible for the authors to do an in-depth analysis (with human responses and try different variations, analyse failures) but adding a more realistic domain the strengthen the contribution of the paper.
questions: questions: See weaknesses.
limitations: limitations: The authors address limitations.
paper_summary: paper_summary: The authors tackle the problem of learning natural language hypotheses and collecting new experiments to enable hypothesis refinement. They propose a system that integrates LLMs and SMC/BOED to solve this problem; crucially, this method goes beyond prior work on inductive reasoning by allowing for experimentation to guide hypothesis refinement. In brief, they use a LLM to propose natural language hypotheses, SMC to reweight and resample these hypotheses based on how well they explain the data, and then use SMC’s weighted particle approximation to approximate the expected information gain and select experiments. 

The authors then evaluate this approach on two task Zendo and ActiveACRE, showing improvements against previous inductive reasoning methods and showing some agreement between their proposed system and human behavior on these tasks.
strength_weakness: strength_weakness: I think the tasks do a fine job of evaluating the system but I'm not sure if they're the most compelling demonstrations of the value of natural language hypotheses/they might not fully utilize the power of natural language hypotheses. For example, it seems like a restricted DSL might be enough for some of these tasks; I do note that the authors do an analysis to show the benefit of natural language.

Additionally, there are a few ablations of the system that could strengthen the paper. 

* Ablations that illustrate the role of a natural language hypothesis space or at least some additional discussion of more realistic settings where this is a good design choice. 

* Number of particles/number of hypotheses 

* Some analysis of particle degeneracy would be helpful 

* How does a model with vision capabilities perform on this task (e.g., GPT-4 V)? 

I think the presentation can be improved a bit in some places. For example, Figure 4b could probably be replaced with a summary figure. It’s a bit difficult to parse a 6x10 grid of figures. 

Furthermore, it was sometimes hard to understand the details of some of the baselines (without reading the hypothesis refinement/search papers). For example, I think clearly defining the batched inference with refinement baseline (in math) would be helpful in the main text since it's a crucial method to understand.
questions: questions: * Does LLM-SMC-S produce a consistent estimator of expectations of test functions wrt the normalized posterior (this is the guarantee enjoyed by vanilla SMC)? I think this probably follows from the properly weighted property and the fact that ratios of consistent estimators are consistent. It would be good if authors could comment on this in the paper though. 
* Does the procedure suffer from particle degeneracy? It’s fine if it does but since this is a considerable practical challenge of SMC, I think it’s worth commenting on this. 
* The derivation in A.1 has some typos. I think there’s a missing integral sign in line 10. 
* I think I would have liked to see some of the actual hypotheses proposed by the LLM in the main text. 
* I think it would be helpful to see Equation 6 is actually approximated explicitly in math. 
* Figure 2 was a bit confusing for someone who’s used to seeing plots in SMC papers where the circles correspond to particles; here I think the circles represent specific natural language hypotheses (e.g., 3 red balls) but at first I thought they represented particles.
limitations: limitations: The authors discuss limitations.
paper_summary: paper_summary: This paper integrates a Large Language Model (LLM) within a Sequential Monte Carlo (SMC) algorithm, where the LLM functions as both the proposal distribution (revising existing particles) and the evaluator (assessing each hypothesis in light of new data). The authors applied this method to two cognitive psychology tasks: Zendo and ActiveACRE. These tasks involve formulating latent game rules, updating the probability of these rules based on new data, and proposing new experiments to gather further data.

In both domains, the Online, Hard variant of SMC consistently outperforms other SMC variants in identifying the best latent rule, while the Online, Fuzzy variant of SMC more accurately captures human data. However, compared to optimal experimentation (the InfoGain algorithm), none of the LLM-based proposers outperform the InfoGain algorithm. Furthermore, a random proposer outperforms the LLM-based methods, indicating a significant deficit in the LLM’s performance as a proposer.
strength_weakness: strength_weakness: The main weakness of the work is that both the enhancement in problem-solving capability and the correspondence with human data are not convincing. While the Online, Hard variant of LLM-based SMC clearly outperforms ReAct-style LLMs, it employs LLMs in three distinct roles: as a proposer (for revising existing hypotheses), as an evaluator (for translating natural language hypotheses into code and verifying them), and as an active learner (for proposing new experiments to perform next). The performance of the LLM as an active learner, however, is particularly weak, even underperforming a random active learner (see Tables 3 and 4). Replacing the active learner in the Online, Hard method with a random active learner would likely significantly improve performance compared to using the LLM. Consequently, it is unclear which aspects of the pipeline benefit from the inclusion of the LLM and which aspects are detrimental.

Second, the comparison between humans and models is superficial. The best-fitting Online, Fuzzy model clearly includes at least two free parameters, as detailed in Section 2.3. However, the authors did not report the values of these best-fitting parameters. Are these the only two free parameters? Did you add additional decision noise to the model to better capture human data? If so, it would be more informative to report scores such as BIC or cross-validation results rather than just the simple log likelihood of the model.

Figure 7 aims to illustrate a bounded rationality model for human-likeness, suggesting that the Online, Fuzzy method best aligns with human behavior when the number of LLM calls per iteration is neither too few nor too many. However, the evidence for this claim appears weak. The bounded rationality pattern only emerges with the Online, Fuzzy method, and the authors tested only three different numbers of LLM calls per iteration (x-axis). With just three data points, it is difficult to conclusively demonstrate an inverted U-shaped relationship. Additionally, when LLMs were used with methods other than the Online, Fuzzy method, the inverted U-shaped relationship did not appear at all.
questions: questions: Lines 227-229: The authors claim that “Indeed, Table 2 shows that our best-performing model surpasses [13]’s model on human data […], while our model does not perform such parameter fitting.” However, the Online, Fuzzy method does contain free parameters (specifically, the two free parameters that determine the false-positive and false-negative rates for each hard rule) that can be adjusted to better fit human data. Did the authors actually fit these parameters using human data?
limitations: limitations: I believe the paper would be improved if the authors could independently comment on the three separate roles of the LLMs (i.e., proposer, evaluator, and active learner) in their proposed method.

Minor comments:
Line 224: “judgement” —> “judgment”

Comments: title: Acknowledged and maintain my positive support
comment: Thanks for your clarifications! I"ll maintain my score and advocate/support for the acceptance of this paper. 

I only have one presentation comment.

For Proposition 1, consider making it a statement about consistency of LLM-SMC-S? Presumably, consistency is the guarantee that people care about and the properly weighted property is why this guarantee holds (I could be missing something though).

Regarding, the author's response to the first concern. That's reasonable and consider elaborating further in the Discussion (although this is already briefly discussed).
Thank you for your responses and addressing my concerns! I'll keep my current score.
Thanks for the response. I appreciate the clarifications. I think commenting in the paper about what is hard about the two hardest Zendo rules would be useful to draw attention for future work.