Evaluating the Inductive Abilities of Large Language Models: Why Chain-of-Thought Reasoning Sometimes Hurts More Than Helps
Authors: Haibo Jin
Peiyan Zhang
Man Luo
Haohan Wang
Venue: NeurIPS 2025 poster ()
Keywords: Large Lauange Model
Inductive Abilities
Reasoning
Primary area: evaluation
TL;DR: CoT prompting does not always help inductive reasoning in LLMs—we show it can amplify errors and propose structured interventions to address this.

Abstract: Large Language Models (LLMs) have shown remarkable progress across domains, yet their ability to perform inductive reasoning—inferring latent rules from sparse examples—remains limited. 
It is often assumed that chain-of-thought (CoT) prompting, as used in Large Reasoning Models (LRMs), enhances such reasoning. 
We investigate this assumption with creating four controlled, diagnostic game-based tasks—chess, Texas Hold’em, dice games, and blackjack—with hidden human-defined rules. 
We find that CoT reasoning can degrade inductive performance, with LRMs often underperforming their non-reasoning counterparts.

To explain this, we present a theoretical framework that reveals how reasoning steps can amplify error through three failure modes: incorrect sub-task decomposition, incorrect sub-task solving, and incorrect final answer summarization. 
Based on our theoretical and empirical analysis, we introduce structured interventions that adapt CoT generation according to our identified failure types. These interventions improve inductive accuracy without retraining. Our findings suggest that effective (CoT) reasoning depends not only on taking more steps but also on ensuring those steps are well-structured.

Evaluating the Inductive Abilities of Large Language Models: Why Chain-of-Though Reasoning Sometimes Hurts More Than Helps
Haibo Jin
Peiyan Zhang
School of Information Sciences University of Illinois at Urbana- Champaign Champaign, IL 61820 haibo@illinois. edu
Computer Science and Engineering HKUST
Clear Water Bay, Kowloon, Hong Kong
pzhangao@connect. ust.hl hk
Man Luo
Haohan Wang"
Research Scientist, Intel Labs Santa Clara, CA 95054 luoman. wwtgmail.c..oo.o.cmmmm.........l com
School of Information Sciences
University of Illinois Urbana-Cham -haaaa-Champaign
Champaign, IL 61820 haohanv@illinois edu
Abstract
Large Language Models (LLMs) have shown remarkable progress across do- mains, yet their ability to perform inductive reasoning boooiiingiiffrrinng latent rules from sparse caamples-remainn limited. It is often assumed that chain-of-thought (CoT) prompting, used in Large Reasoning Models (LRMs), enhances such reasoning. We investigate this assumption with creating four controlled, diagnostic game- game-based tasks casks-chess, Texas Hold' em, dice games, and blackjack-win hidden human- defined rules. We find that CoT reasoning can degrade inductive performance, with LRMs often underperforming their non-reasoning counterparts.
explain this, we present a theoretical framework that reveals how reasoning steps can amplify error through three failure modes: incorrect sub-task decomposition, incorrect sub-task solving, and incorrect final answer summarization. Based on theoretical and empirical analysis. introduce structured interventions that adapt CoT generation according to our identified failure types. These interventions improve inductive accuracy without retraining. Our findings suggest that effective (CoT) reasoning depends not only on taking more steps but also on ensuring those steps well-structured.
Introduction
Inductive reasoning- ioniiinn-iffrrinng latent rules from sparse examples-is key capability underlying generalization. While Large Language Models (LLMs) [[2] have made significant advances, they
continue to struggle in tasks that require structured inference under uncertainty [.. These models
often rely on surface-level pattern recognition and static prompt formats [4, making them brittle when confronted with novel structurally complex problems
To address this limitation, recent models incorporate explicit reasoning mechanisms mechanismss-particllaay chain-of-though (CoT) prompting to enable multi-step inference. These Large Reasoning Models (LRMs) 7 8 have shown improved performance in coding, logical inference, and scientific
"Corresponding Author
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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tasks [0] and CoT has been widely adopted under the assumption that structured reasoning en- hances inductive performance [2 13. However, emerging evidence complicates this assumption: Wu et al. [4] observe that longer reasoning traces can degrade accuracy, suggesting non-monotonic relationship between reasoning depth and performance.
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Examples illustrating inductive reasoning on gameplay transcripts. (a) Games begin with both Normal and hidden Special Rules, requiring models to infer latent constraints from observed plays. (b) LLMs can induce rules like card legality and win conditions without explicit guidance. but LRMs such GPT-o3 may underperform due to misaligned or noisy reasoning. (c) Reasoning improves when guided at the decomposition, solving, and summarization stages.
Figure I:
In this work, we investigate the inductive performance of LLMs and LRMs (Fig. D. We introduce a set of controlled diagnostic game-based tasks to isolate inductive reasoning behavior in LLMs. In each task, models are presented with a short transcript of gameplay- rules-and must infer the latent constraints governing legal moves and outcomes. Surprisingly, we find that LRMs often underperform non-reasoning LLMs in these settings, suggesting that CoT reasoning may introduce noise rather than clarity. We also develop a theoretical framework that explains this degradation.
access to the underlying
Our work address the following questions: RQ1: How well do LLMS perform inductive reasoning tasks, and has this improved with recent models? (Section 3 RQ2: Why does reasoning sometimes fail-or even hurt-inductive performance? (Section RQ3: How can we guide reasoning to enhance inductive accuracy without model retraining? (Section 5)
Along answer these RQs, we have made the following contributions.
We construct four controlled diagnostic game-based based tasks with hidden evaluate the inductive reasoning abilities of eight leading LLMs, including both non-reasee non-reasoning models and LRMs.
rules to
We provide a theoretical and empirical analysis identifying three failure modes that explain degraded reasoning: incorrect sub-task decomposition, incorrect sub-task solving, and incorrect final answer summarization.
We propose an error-guided intervention method that adapts CoT generation based on predicted failure types, yielding consistent improvements in inductive accuracy without retraining.
2 Related Work
Chain-of-Though Reasoning. Large language models can perform complex tasks more effectively when guided through intermediate reasoning steps, a process known as ofain-of-thought (CoT) prompting [6). Much recent work has aimed to strengthen CoT-based reasoning through better decomposition, exploration, or decision- making strategies. Zhou et al. (II] introduce least-to-most prompting, which breaks problems into simpler subproblems and solves them sequentially to support easy-to-hard generalization. Yao et al. [22 propose Tree of Thoughts, enabling multi-path exploration with backtracking and self-evaluation. Meng et al. [13] introduce Divide-and Moddee-nd--oome Conquer strategy that improves performance by segmenting input into reasoning chunks, and Zhang et al. [13] evaluate when such strategies yield consistent gains. While these methods aim to enhance reasoning capabilities, they largely assume that reasoning depth is beneficial and do not investigate when CoT may fail.
Mechanisms Behind CoT Reasoning. Several studies seek to formalize the mechanisms underlying CoT prompting. Feng et al. [16] use circuit complexity to show that constant-size models can

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simulate deeper computation via CoT derivations. Li et al. [7] explain CoT's benefits in decoder- only Transformers via sequential computation theory, and Cui et al. [8 show that Coherent CoT improves error correction over stepwise prompting but increases sensitivity to intermediate-step failures. Ton et al. [19] apply information theory to quantify stepwise information gain, and Li et al. [20] analyze fast vs. slow thinking during post-training. Ye et al. [2] investigate CoT in controlled math setting, while Wu et al. [14] demonstrate that performance may peak and then decline with longer reasoning traces, hinting at optimal CoT length. However, while these works identify performance degradation empirically, they stop short of explaining why it arises. Our work builds on these insights by offering formal theory of reasoning failure.
Inductive reasoning. Inductive reasoning [22] a core component of human intelligence, enabling rule generalization from examples without prior knowledge. As LLMs reach human-level perfor- mance, many benchmarks have been developed to assess this ability. Banatt et al. [33 propose WILT,, logic induction benchmark where models infer Boolean rules through trial-and-error queries. Li et al. [24] design MIRAGE, a suite of synthetic analogy tasks that test pattern- generalization. Ma et al. [25] introduce KoR-Bench, which emphasizes rule application across logic puzzles, ciphers, and counterfactual reasoning. Xiao et al. [26] present Logic Vista, focusing on visual-spatial rule inference via figure completion tasks. Xu et al. (27] develop LLM-Script, where models must induce latent functional patterns from input-output examples. Yan et al. (28] introduce BeRcBench, the first benchmark for many- inductive reasoning, where models must induce latent functions from diverse outptt exemplars presented in long-contexs settings.
Key Differences. Existing benchmarks are typically static, using simplified symbolic or visual inputs to test whether models can apply hidden rules to new examples. In contrast, our game-based setup asks models to infer the rule itself from minimal gameplay context, without a separate testing phase. This more closely mirrors human induction, which relies on observing outcomes and forming hypotheses rather than answering predefined queries.
3 Inductive Reasoning Evaluation via Gameplay Tasks
To investigate how well LLMs perform in inductive reasoning tasks (RQ1), we introduce four different types of controlled diagnostic game-based tasks designed to challenge and measure their inductive ability to infer hidden rules from gameplay transcripts. These games span diverse range of domains, including chess, Texas Hold' em, dice games, and blackjack. Each game consists two rule components: normal rule component (abbreviated "NR") that mirrors standard gameplay conventions, and a special rule component (abbreviated as "SR") that introduces manually designed hidden constraints. Together, these components form a composite logic that governs legality, winning conditions, or valid moves. Crucially, these rules are not revealed to the models; instead, the models presented with a few legal gameplay examples and are required to induce the underlying rules through observation.
3.1 Game Setup
Models. We evaluate eight language models, including four non-reasoning LLMs GPT- DeepSeek-V3 (deepseek-v3- 203-2025-03 03-25) 25) [29]. Aven2.5-Max and Grok-2 (grok-2- 1212) BI as well as four LRMs: GPT-o3
4o (gpt-40-22) -000-44-2024-0--06 (qwen- 2025-01-220022 -000-maxx-222501-2255 (o3-2025-04-16) DeepSeek-RI (deepseek-r1-32b 77. QwQ (QwQ-32B) (32, and Grok-3-Mini
(grok-3-mini -btta) [33. We run DeepSeek-RI RI and QwQ locally to enable collection of interme- diate reasoning traces for deeper analysis in RQ2, while the remaining models are accessed via their respective APIs.
Implementation Details. To evaluate each model's inductive performance across rule types, we frame the task semantic alignment boblemm-deerrmngg whether the model's induced rule captures the same logic as the ground-truth. This formulation is supported by recent work showing that large language models can reliably perform semantic evaluation and alignment tasks (34. We therefore use GPT-4o queried three times to independently assess whether the induced rule consistent with the ground- truth, and a majority vote is used to finalize the decision. Rule-level accuracy is computed as the proportion of cases judged as consistent. Representative examples are provided Appendix A.T and the prompt for GPT-4o is provided in Appendix A.3
[I as a reference judge. For each case, GPT-4o is
3

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3.2 Rule Design
We evaluate inductive reasoning in LLMs using four game-based tasks: chess, Texas Hold' sold'em,ann dice games, and blackjack. Each task presents models with gameplay transcripts governed by two types of rules: normal (NRs) and special (SRs). The model must infer these rules purely from observation.
Case Study: Chess. We chess detailed example due to its well-defined rules and spatial structure. In each game instance, we define eight types of pieces, each assigned one rule: either from pool of normal rules (NRs) special rules (SRs). Normal rules based standard chess movements, such as moving one square in any direction or diagonally across the board. Special rules designed to be legal but non-obvious, introducing hidden constraints that models must discover from context. For example, a piece may be assigned rule like "move in straight line any number of squares, then shift diagonally by one square." This rule differs subtly from standard movement and cannot be inferred from single move.
Models are never given any rule descriptions they shown short gameplay transcripts (typically 10-12 moves) and asked to determine what rules govern the movements. We randomize the board size (ranging from 8 15), shuffle rule assignments across games, and ensure each piece appears least three times per episode. We construct 225 distinct transcripts by enumerating all combinations of four NRs and four SRs across the six available options. This design provides broad set of inductive scenarios with controlled structure and consistent constraints. Full rules and prompt formats are in Appendix B and D.I
Other Games. We include three additional games to cover a broader range of inductive reasoning scenarios, each characterized by distinct rule structures and abstraction types. Full rule specifications for all games are provided in Appendix
Texas Hold'em em: Tests symbolic ranking under hidden structure. While normal rules follow standard poker hand rankings, SRs redefine hand strength using patterns like "five consecutive prime numbers"'
Dice Games: Adapted from Sie Bo assess reasoning in noisy environments. NRs involve numeric thresholds (e.g., small vs. large totals), while SRs introduce subtle structural overrides -egg. "a triple of prime numbers beats all other hands".
Blackjack: Focuses on threshold logic with rule exceptions. NRs include standard conditions like "busts lose" and "21 wins," while SRs add constraints such "hands with three-card straight flushes win automatically".
3.3 Inductive Abilities Analysis
Fig. shows rule-wise inductive accuracy across eight models four games. Although designed to enhance multi-step reasoning, models with reasoming capabilities consistently perform worse than their non-reasoning counterparts on special rules pattern observed across all domains.
On normal rules, most models exceed 90% accuracy, indicating strong pattern recognition when the rule is surface-aligned structurally obvious. In contrast, performance special rules drops significantly. For example, in chess, non-reasoning models like GPT-4o and DeepSeek- -33 reach 55-65% on SR1, while their reasoning counterparts fall below 25%. Similar gaps appear in Texas Hold' em, Dice, and Blackjack (Fig. 2 Additional comparison results can be found in Appendix
These results show that reasoning models struggle more with exception- based or hidden rules. This suggests that their multi-step traces may not help and can introduce incorrect assumptions misleading intermediate steps. We examine this hypothesis in detail in RQ2 by analyzing the reasoning outputs directly.
RQI: How well do L.LMs perform on inductive reasoning tasks, and has this improved with recent models?
Answer: Non-reasoning LLMs consistently outperform reasoning-enables models on induc- tive tasks with hidden rules, showing that recent reasoning strategies degrade performance on abstraction beyond surface patterns.

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GPT 46
DenpSeck-V3 DeepSeck-Rt
Groc2
OwenZ.3 5 Max
Grok3
1.0
100
Acturacy (%1
80
Acturacy (K)
80
60
60
40
40
20
20
5R.
58.
58.4
58.5
5R MR
NA.
MR.
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NR.
MR.
58.
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58.5
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(a) Chess
(b) Texas Holldem
1.00
100
Accurecy (%)
80
Accuracy (K)
800
60
60
40
40
20
20
36.2
58
58.3
5R.4
102
KR
5R.1
5A.
5R.3
5R.4
NR.
NR.
N8.
(c) Dice Game
(d) Blackjack
Figure Inductive accuracy on normal rules (NRs) and special rules (SRs) across four games. Each bar shows rule-wise inductive performance for eight LLMss While most models achieve high accuracy on NRs, reasoning models (lighter bars) consistently underperform non-reasoning models (darker bars) on SRs, indicating that current reasoning may hurt inductive abilities on hidden rules.
4 Root Causes of Reasoning Failures
To understand why reasoning sometimes fails to improve-and in some cases even harms-the the inductive performance of LRMs (RQ2), we conduct both theoretical and empirical analyses to identify the root causes of these failures.
4.1 Theoretical Analysis of Reasoning Errors
To explain why reasoning can fail, we present theoretical framework that models reasoning as sequence of discrete operations: posing a sub-task, solving it, and summarizing the final answer. Each step k N is associated with a reasoning state:
T
(mk: sk) E
IR
(NEEDQ, NEEDA, FINISH]
(1)
beliel state
reasoning
where denotes the model's 's current belief about the correct answer y² RRd, and 85 tracks the stage of reasoming. Detailed mode semantics are provided in Appendix C
Evidence model. The model does not observe the true target y" directly. Instead, at each reasoning step k, receives an indirect evidence vector Ik based on its attempted sub-task resolution. We assume the evidence signal follows:
ghe a' (g" mk-1) 1) + Eks
Ek N10,000
with the following interpretation:
Task alignment (a.). The scalar ag [-1,1] represents how well the current sub-task focuses the relevant latent structure. This captures an observed behavior in LLM reasoning: incorrect or ambiguous intermediate questions Negative alignment (agg < 0) corresponds to misleading questions concepts which have been shown to increase failure rates in multi-hop reasoning [35] 36] The
derail downstream answers by misdirecting attention [7 21).
as proposing irrelevant
scalar form abstracts this variable attention quality while keeping the model analytically tractable. Answer noise (εx). The residual ε models stochastic variation in sub-task resolution, including token sampling noise, hallucinations, and unstable decoding paths. Prior work has empirically shown that even when CoT traces well-posed, decoding introduces variability that degrades
reliability 37 38
Belief update and error propagation. by integrating a new evidence signal: my Mp-1 + Tk.9k.
At each reasoning step, the model updates its internal belief'
5

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where my E Rd denotes the model's 's current belief about the target solution y". and E (0. is step-size scalar that controls how much the new evidence shifts the belief. This form is justified by two observations.
Incremental refinement. Empirical analyses show that LLMs often revise answers through local adjustments rather than full restatements (37]. The additive form captures this bounded update behavior, consistent with chain-of ofain-of-ine of-thhuuht reasoning where successive steps refine hypothesis instead of discarding prior context.
Step-size heterogeneity. The weight y reflects the model's implicit confidence, leading to varying step sizes [39 40), as reasoning trajectories should not integrate all steps equally.
Subtracting y² from both sides, define the belief error ek my y² and obtain the recursion:
6g (1 - 71.g)e YeE
Emergence of reasoning errors. Eq. helps us to decompose the error into three error components:
(1) Incorrect sub-task decomposition (Breakdown Error, (r.). Breakdown errors directly corre- spond poor question alignment (ors R 0) or negative alignment (ax 0). In such scenarios, the coefficient (1 (rg) magnifies or maintains the previous error magnitude ||ex-1 Il. thus preventing error reduction or even causing divergence. (Detailed analysis can be found in Appendix (F)
(2) Incorrect sub-task solving (Solving Error, E&). Even under optimal question alignment (aq Rs 1), the inherent answer-generation noise E introduces stochastic deviations at each reasoning step.
(Detailed analysis can be found in Appendix
(3) Incorrect final answer summarization (Summary Error). The third class of reasoning error arises from deciding when to stop the reasoning process and commit to a final answer. This halting decision determines the total number of reasoning steps N, which directly affects the model's prediction CN == m... We analyze the resulting prediction error by computing the expected squared deviation from the true target y" E IRd. Let ex == mN y denote the final error. Then the expected error given by:
N E(N) = EleNN12 bo
N II (1-) --5))++((NN))) +A(N) j=it1
II (1- + 0
(5)
where ||mn y 1| is the initial squared error, 7/ (0, 1) the integration weight step i, and ā == -[ao]] is the expected question alignment. The product terms reflect accumulated error contraction across steps, and A (NN) 2 0 accounts for additional variance due to misalignment variability. A full derivation of Eq. (5) is provided in Appendix G
Theorem 4.1 (Optimal Reasoning Length Exists). Let the expected mean-squared error after N
Answer defined in Eq 5
Then:
1. E(N) is strictly decreasing for N < N* and strictly increasing for N > N*, with
N = min N 2 0: E(N+1>> E(N)};
2. the minimiser No unique. Consequently, ENN is U-shaped when plotted against reason- ing depth N.
Proofs can be found in Appendix G.4 The structure of Eq 5 can be intuitively interpreted to reveal a fundamental bias-variance trade-off. The first term shrinks as N increases: repeated evidence integration reduces the bias. However, both the second term (arising from answer-generation noise) and the third term (from stochastic question quality) grow with N, reflecting variance accumulation. This competition creates a non-monotonic curve in E(N): at small depths, additional reasoning reduces error, but beyond a critical point, further reasoning begins to increase it. Reasoning that terminates at any depth N + N* incurs additional error due to either under- or over-reasoning. We refer this deviation as the Summary Error-a global error that arises not from a single misstep, but from misjudged stopping point over otherwise well-structured reasoning trajectory.

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4.2 Theoretical Guidance for Intervention Design
In this section, we present additional theoretical results that clarify the structure of each component, which serves the foundation of mitigation methods that discussed in Section Full derivations are provided in Appendix G
Controlling sub-task decomposition error (og)). The alignment scalar 0g determines how well the posed sub-question targets the true residual error. Positive values (ax- 0) reduce bias; negative values (aq < 0) increase it exponentially with reasoning depth. Lemma Ce. establiihhs that any improvement in alignment (ax 1) strictly reduces error, while misaligned reasoning (ax < 0) guarantees divergence. This motivates interventions that eliminate unconstrained question posing,
that we will discuss in Section 5
Controlling sub-task solving error (E&). The noise term Ej models stochastic variation in answer generation, which is scaled by the belief integration weight 7k, making 7 a critical amplifier of solving error. Lemma G.6 formalizes this relationship: while alignment (ox) always improves performance, there exists optimal %g that balances the benefit of information integration against the risk of over-amplifying noise. This interaction motivates our intervention at the solving phase. We will reduce effective variance o² by anchoring the model's outputs to structurally valid reasoning traces in Section
Controlling summarization error (global stopping rule). Reasoning must terminate at some depth N, at which point the model commits to its final prediction UN- Theorem provess that expected squared error E(N) is U-shaped in N. with unique optimum N In addition, Lemma providiee closed-form solution for N in the constant-ry case, offering theoretical guidance on setting reasoning length. Accordingly, we constrain model outputs via fixed token budget to discourage excessively long traces and nudges reasoning toward near-optimal depths in Section
4.3 Empirical Analysis of Reasoning Errors
We empirically validate the theoretical taxonomy introduced in Section 4.1 which identifies three primary sources of reasoning failure: (1) Incorrect Sub-task Decomposition (Breakdown Error) arising from incorrect sub-task decomposition, (2) Incorrect Sub-task Solving (Solving Error) caused by noise in sub-task resolution, and (3) Incorrect Final Answer Summarization (Summary Error) resulting from premature or excessive reasoning steps. These failure modes are grounded in the dynamics of the belief update equation 4 where errors propagate via suboptimal alignment (a&), additive noise (E&), and misjudged stopping time N.
Among these, Solving Error dominates across all models and tasks, accounting for over 80% of failure cases. While theoretically modeled additive noise, solving failures often exhibit structured patterns in practice. Based on prior analyses of LLM reasoning drift Based on consistent error patterns we observed in over 100 failed reasoning traces across multiple tasks and models, we classify these errors into three observable subtypes: (1) Math Overuse, where models inappropriately apply arithmetic operations symbolic inputs (c.g., card suits or chess pieces); (2) Overgeneralization. where rules are inferred from few examples without proper validation; and (3) Hallucinated Rules, where fabricated constraints are introduced without support from input observations. Representative examples for each are provided in Appendix !]]
Breakdown Errors are less frequent but still consequential, especially in structurally complex games like Texas Hold'em. These correspond to misaligned sub-task decomposition, where the model fixates on irrelevant features ignores core inductive structure. Summary Errors are the least frequent and occur when models produce overly long or overly short reasoning chains, diverging from the optimal depth N identified in Thcorem 4.1
To ensure reliability, each failure trace was independently reviewed by two annotators using shared labeling criteria (Detailed in Appendix K). While some interpretation was involved-as is typical in reasoning error -agreisenggeeeetttttt was high, and disagreements were resolved collaboratively. Rather than impose rigid boundaries, aim to capture recurring, interpretable failure patterns ob- served consistently across models and games. Table [ the distribution, and Appendix H ] provides representative examples for each error type.
Three key observations emerge: First, Solving Errors dominate, accounting for over 80% of failures in most settings. This aligns with our theoretical model, where errors introduced during sub-task
7

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Table Error rate analysis across different games and models
Error Rate (count total)
Games
Models
Solving
Breakdown
Summary
Hallucinated Rule Overgencralization
Math Overuse
DeepScck-RI 1 5.8% (64/1109) 17.2% (191/1109)
22.3% (247/1 (09)
47.4% (526/1 109) 7.3% (81/1109)
Chess
QwQ
4.3% (52/1217)
16.5% (201/1217)
26.1% (318/1217)
52.1% (634/1217)
1.0% (12/1217)
Grok3
4.1% (55/1333)
14.5% (193/1333)
24.5% (327/1333)
50.7% (676/1333)
6.2% (82/1333)
DeepSeck-RI
9.3% (14/151)
15.9% (24/151)
11.9% (18/151)
58.9% (89/151)
4,0% (6/151)
Texas Hold' cm
QwQ
14.0% (21/150)
15.3% (23/150)
22.0% (33/150)
42.7% (64/150)
6.0% (9/150)
Grok3
13.2% (19/144)
26.4% (38/144)
12.5% (18/144)
40.3% (58/144)
7.6% (11/144)
DeepSeck-RI
5.7% (3/53)
11.3% (6/53)
20.8% (11/53)
60.4% (32/53)
1.9% (1/53)
Diec games
QwQ
10.3% (4/39)
12.8% (5/39)
23.1% (9/39)
48.7% (19/39)
5.1% (2/39)
Grok3
3,4% (2/59)
38.6% (1 (11/59)
10.2% (6/59)
61.0% (36/59)
6.8% (4/59)
DeepSeek-RI
6.7% (4/60)
21.7% (1 (1.360)
13.3% (8/60)
53.3% (32/60)
5.0% (3/60)
Blackjack
QwQ
8.6% (6/70)
20.0% (14/70)
28.6% (20/70)
40.0% (28/70)
2.9% (2/70)
Grok3
8.2% (5/61)
18.0% (1 (11/61)
19.7% (12/61)
47.5% (29/61)
6.6% (4/61)
resolution (5x) propagate through the reasoning chain. Second, within Solving Errors, Math Overuse consistently ranks highest. For example, accounts for 60.4% of all failures Dice Games (DeepSeek-RI) and 53.3% in Blackjack. This suggests recurring bias where models incorrectly apply arithmetic operations to symbolic domains eo--- treating card suits or piece positions as numeric inputs. Notably, this pattern holds across structurally diverse tasks and models, indicating a deeper inductive misalignment rather than specific noise. Third, Breakdown Errors peak in Texas Hold'en
hold'em (up to 14.0%), reflecting its higher structural complexity. In contrast, Summary Errors remain rare across the board (48%), implying that most failures arise during solving rather than early decomposition or final summarization. Together, these results indicate that reasoning fails not from lack of logical capacity, but because early inductive errors-especin Math Overuse-are preserved and propagated across multi-step reasoning chains.
RQ2: Why does reasoning sometimes fail to improve inductive performance in large language models?
Answer. Reasoning propagates error: when sub-task decomposition is misaligned, or solving introduces noise, each reasoning step compounds the mistake. Such failures dominate in practice, making deeper reasoning harmful unless each step is reliable.
5
Improving CoT Reasoning
5.1 Intervention Design
Building on our theoretical framework, we design interventions that directly target each of the three failure modes: incorrect sub-task decomposition, sub-task solving noise, and overextended reasoning. The Appendix [[ shows the detailed prompts for each intervention.
Sub-task decomposition. To address Breakdown Error associated with misaligned sub-task for- mulation (ag, As 0), we replace free-form reasoning with structured decomposition templates. Each template explicitly separates the reasoning into three phases: (i) identifying relevant entities in the input (c.g., cards, pieces, or dice), (ii) inducing candidate rules based on observed patterns, and (ii) verifying whether cases satisfy those rules. This design constrains model output to stay within task-relevant abstractions and improves alignment between sub-questions and the global task objective. Prior work has shown that semantically aligned decompositions improve reliability in multi-step reasoning [1].
Sub-task solving. Solving Error arises when models introduce stochastic variation (Ex) even under good decomposition. The most frequent subtype is Math Overuse, where models impose numeric operations onto symbolic inputs. To mitigate this, we follow Kuo et al. 41 and guide solving with worked examples that avoid numeric extrapolation. These examples anchor model behavior to structurally relevant patterns, reducing variance and discouraging inappropriate generalization.

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Answer summarization. Summary Error occurs when reasoning continues past the optimal step count N causing error accumulation from unnecessary updates. We adopt recent ideas from efficient reasoning 42 43 and impose strict token budget of 1000 tokens per instance. This constraint limits excessive generation and encourages early commitment to accurate conclusions.
Combined intervention. Each intervention targets a distinct error component, but failure modes often co-oceur. We therefore evaluate the combined setting, which integrates decomposition templates, solving-phase examples, and summarization constraints to stabilize the entire reasoning traces.
5.2 Results and Analysis
We evaluate the improving of performance guided by our intervention strategies across all four games, measuring rule-level induction accuracy under each condition. The results can be found in Fig. 3
DeepSreicva
Dwen2.5-MMa
100
100
Accunscy (8)
10-
Accurecy (9)
20
60-
60
SR
SR.5
NA.
5R
(a) Chess
(b) Texas Hold'em
100
1.00
Accurscy 1%6
20
Accuracy
40
40
20
58.6
NR.
16..4
SR.3
5R.
N5.
NR.
10.
16.0
[c) Dice Game
(d) Blackjack
Figure 3: Inductive rule accuracy across different intervention strategies and models for each game domain. Each subfigure corresponds to one game; bars show average rule-wise accuracy under different reasoning-stuu stage interventions. Across all domains, combined intervention (rightmost bars) achieves the highest performance, especially on special rules (SRs), indicating that structured decomposition, guided solving, and summarization control jointly enhance inductive abilities.
We observe consistent improvements under the combined intervention setting, especially on SRs, which involve higher inductive complexity. In most cases, combined intervention leads to clearly higher SR accuracy than any individual strategy. For example, in Chess and Dice, SRI-SR3 accuracy increases by 20-40% when guided chain -o of-thouuhh is applied, suggesting that multiple types of reasoning failures often co-oceur and need to be addressed together. Compared non-reasoning models, the combined intervention also achieves higher accuracy on both SRs and NRs across most games, indicating that improvements come from better reasoning structure rather than longer reasoning steps. Among individual strategies, sub-task decomposition is most helpful in structurally rich games like Chess and Blackjack, while solving-stage guidance contributes more in symbol-heavy domains such as Dice and Texas Hold'en em. Summarization control provides moderate improvements across all tasks by reducing over-generation in the final answer stage. Performance on NR rules remains saturated under all settings, confirming that gains on SRs are not result of trade-off's with simpler rules. However, we also observe that combined intervention does not always outperform the best individual strategy. In some cases, interactions between decomposition and solving stages may limit overall effectiveness, highlighting the difficulty of coordinating multiple reasoning processes. Detailed results for interventions are provided in Appendix M. and we further discuss how these interventions may generalize to more common reasoning problems Appendix M.2
RQ3: How can we improve the inductive performance of reasoning-enabled LLMs? Answer. Inductive performance improves when reasoning is constrained. We achieve consis- tent gains by (i) enforcing structured decomposition, (ii) guiding solving with nom-numeric examples, and (iii) limiting over-generation through token budgets. Combined, these interven- tions reduce error amplification across all reasoning phases.

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Conclusion
In this paper, we investigate the inductive abilities large language models through four designed games. Our analysis reveals that, contrary expectations, maasoning-particullnn in LRMs- not always help; when poorly structured, can even harm performance by introducing irrelevant misleading chains of thought. We theoretically identify three core types of reasoning failure and empirically verify them through human evaluation of model-generated traces. To address these issues, introduce targeted interventions at the decomposition, solving, and summarization stages. Experimental results show that each intervention independently improves inductive accuracy, confirming the importance of structured, well-controlled reasoning.
Acknowledgement
This research was supported by the National Al Research Resource (NAIRR) Pilot NAIRR240283.
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A Details for Judging
A.1 Examples of Judging
To illustrate the evaluation process, we provide examples of GPT-4o judging whether a model's induced rule aligns with the ground-truth. For each case, we show (1) the true rule, (2) the model- induced rule, and (3) GPT-4o's 's binary judgment.
Table 2: Examples of GPT-4o judging model-induced rules. Each row shows the true rule, the model's induced rule, and GPT-4o's 's binary alignment judgment with explanation.
Game
Ground- Truth Rule
Induced Rule
GPT-4o Judgment
Chess
SR6: Move in straight line any number of squares, followed by one-square diagonal shift.
The piece moves straight and then one square diagonally.
Yes, the hypothesis matches the ground-rrut truth. The structure and in- tent are preserved.
Chess
SR4: Jump to the symmetric po- sition across the nearest blocking piece.
The piece teleports to a random mitrored position.
No, hypothesis does not match. The true rule requires specific sym- metry actoss blockers.
Texas Hold' cm
SR3: Alternaing odd-even values mirror hand, heat- ing any straight.
A hand with alternating red and black suits promoted heat any straight.
No, the hypothesis does not match. The ground-truth -ruth is meric parity, not suit color
Texas Hold'cn cm
SR4: Five consecutive num- bers in the same suit count us struight flush.
A hand with five even cards, gardless suit, is treated as a flush.
No, the hypothesis does match. It ignores the suit constraint and downgrades hand.
Dice Game
SRI: If the a prime number. the roll heats all hands including triples.
If the sum prime, the roll
Yes, the hypothesis matches the ground mrnund-rrath. The paraphrasing is correct.
Dice Game
SR4: A pair plus third die dif- fering by exuctly one (e.g., 4-45) beats regular pairs.
If roll contains a pair any third die, beats all totals.
No, hypothesis does not match. omits the "difference one" dition.
Blackjack
SR2: A three-card straight flush is treated as a blackjack.
If hand has three consecutive cards in the same suit, is equiva- lent blackjack.
Yes, the hypothesis matches the ground-truth. The core condition is preserved.
Blackjack
SR3:
pair of different suits
If hand contains pair, loses. automatically.
No, the hypothesis does match. It generalizes beyond the suit con- straint.
causes nutomatic loss.
A.2 Human Evaluation for Judging
To address potential circularity in using GPT-4o as judge, we conducted a human validation study on 100 rule descriptions generated by GPT-o3 across all four games. Three independent human annotators, blind to both the model identity and GPT-4o's original decisions, assessed whether each generated description was semantically aligned with the ground-truth rule. We then compared the majority human judgment with GPT-4o's evaluations and computed Cohen's 's k to quantify agreement.
Table 3: Human validation of GPT-4o's judgments on rule alignment.
Judgment Type
Human Majority
GPT-4o
Overall Accuracy
Cohen's x
Aligned Cases Not Aligned Cases
60
60
1.00
1.00
1.00
Overall
1.00
GPT-4o's assessments matched the human in all 100 cases (60 aligned and 40 not aligned), yielding an accuracy and к of 1.0. While this perfect agreement may partially reflect the limited sample size, it provides evidence that GPT-4o's 's evaluations are well-aligned with human judgment in this context.
A.3 Prompts for Judging
To evaluate whether model-inferred rule matches the ground-truth, we prompt GPT-4o with the following template. The input includes the game context, the correct rule, the model hypothesis, and fixed binary instruction.
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GPT-4o Judgment Prompt
You are given a reasoning task involving [GAME TYPE].
Below is the ground-truth rule and model's 's hypothesis.
Decide whether the hypothesis matches the ground-truth rule in semantics. Ground-truth rule: [INSERT TRUE RULE]
Model-induced rule: [INSERT MODEL RULE]
Answer with: "Yes, the induced rule matches the ground-truth." or "No, the induced rule does not match." Briefly explain your reasoning in one sentence.
B
Full Rule Specifications for Each Game
B.1
Chess
We define eight types of pieces. Each is assigned one rule from the following pools.
Normal Rules (NRs):
NRI: Move one square in any direction.
NR2: Move an L-shaped pattern: two squares in one direction and one square perpendicular.
NR3: Move any number of squares diagonally.
NR4: Move exactly two squares forward (in the direction of inereasing row).
NRS: Move any number of squares straight (horizontally or vertically).
NR6: Move exactly two squares diagonally.
Special Rules (SRs):
SRI: Move in a straight line any number of squares, then shift vertically by exactly squares. SR2: Move diagonally any number of squares, then two squares in a perpendicular diagonal direction.
SR3: Move exactly three squares in one direction, then move one square downward.
SR4: Jump to the symmetric position across the nearest blocking piece.
SR5: Swap with target piece
occupied square within distance < 3.
SR6: Move in a straight line any number of squares, followed by a one-square diagonal shift.
Each game selects four NRs and four SRs, randomly assigning one rule to each piece. Rule assignments are reshuffled every episode.
B.2 Texas Hold'em
Normal Rules (NRs):
NR1: A hand with one pair is treated stronger than any high card.
NR2: A hand with three kind treated as stronger than two pairs.
NR3: A straight (five cards in sequential rank, any suit) treated as stronger than three of kind.
NR4: A flush (five cards of the same suit, not in sequence) is treated as stronger than any straight.
NR5: Four of kind (four cards of the same rank) treated as stronger than any flush.
Special Rules (SRs):
SRI: A hand containing five consecutive prime numbers (c.g., 2-3-5-7-J) treated stronger than any three-of-a-kind.
SR2: A hand with alternating card colors (e.g., red-black-red-black-ee is treated as a straight regardless of numeric order.
SR3: A hand with alternating odd and even values treated as a "mirror hand" and beats any straight.
SR4: A hand containing five consecutive even numbers in the same suit treated as a straight flush.
SR5: hand with four cards of one parity (odd/even) and one the opposite parity is treated as
"hybrid hand," ranking just below four of kind.
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Games sample two NRs and two SRs per episode, covering 100 combinations (C3 C each with twelve hands (four per rule).
B.3 Dice Games
Normal Rules (NR)):
NRI: A total sum between and 10 (inclusive) is "small total."
NR2: A total sum between 11 and 17 (inclusive) is a "large total."
NR3: roll containing any pair is treated as stronger than small or large totals.
NR4: A triple (three identical dice) is treated as stronger than any pair or total.
Special Rules (SRs):
SRI: :: If the total sum is prime number, the roll beats any hand including triples.
SR2: If all three dice are prime numbers (2, 3. 5), the roll beats all hands except SRI.
SR3: If the dice alternate in parity (odd-even-odd or even-odd-eveny the roll beats all hands
except SR 1/R22/triples.
SR4: If the roll contains pair and the third die differs from the pair by exactly one (e.g., 4-4-5), the roll beats any regular pair or total.
36 rule combinations (C1 C1), 12 observations per episode, 18 appearances per rule.
B.4 Blackjack
Normal Rules (NRs):
NRI: A hand totaling exactly 21 is a "blackjack" and wins.
NR2: Any hand exceeding 21 is a bust. If both bust, the closer total to 21 wins.
NR3: If neither busts nor hits 21, the hand with the higher total wins.
NR4: An ace can be counted either 11 to optimize the hand.
Special Rules (SRs):
SRI: If the total is prime number, the hand wins regardless of bust.
SR2: A three-card straight flush is treated as a "blackjack" regardless of total.
SR3: hand with exactly one pair of different suits special loss.
SR4: A hand with three non non-conseeutive values where the middle equals the average of the other
(e.g., 3-6-9) is an automatic win.
Each hand contains five cards. Rule combinations: C x C = 36, 12 hands per episode, 18 per rule.
C Mode Semanties and Admissible Actions
At each step k, the model is in mode 8 E (NEEDQ, NEEDA, FINISH) with transitions:
NEEDQ
a% Ask(gu), Sk-1 NEEDA
Sg NEEDA
Oą E (Answer, Finish), 8&41 E (NEEDQ, FINISI]
FINISII
halt; output ÿ m&
This structure enforces cognitively meaningful control flow and localizes reasoning failure by step type.
D Prompts for LLMs to Induce Rules
D.I
Prompts
The following prompt is used to instruct the model to induce latent rules from gameplay observations. Each instance provides full episode actions and outcomes. The model must identify the underlying rule that best explains the observed behavior.
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Prompts for LLMs
(Background Information)
You are an agent tasked with discovering how certain game works. The rules of the game are unknown to you in advance. However, you are given a full gameplay transcript from a single episode, showing the actions taken and whether they resulted in success or failure. The rule used in each episode is internally consistent, but may differ across episodes. Your goal is to infer this rule by observing what patterns emerge in the examples. (Instruction) Carefully analyze the episode shown below. Use only the observed behavior and outcomes to deduce the underlying rule. Be precise and concise. Your response should summarize the rule as clearly as possible, in natural language. Do not speculate beyond the given examples. Do not list the examples again. Do not include uncertainty statements ("it might be...". "perhaps..."). Just state the rule that best explains the episode. (...insert game transcripts here...)
(Output Format) Your response should follow this format: Induced Rule: <your concise natural language description>
D.2 Transcripts Each Game
To support rule induction, each model is provided with a structured transcript capturing the observable behavior of single game episode. These transcripts serve as the sole source of information available to the model they do not include rule annotations symbolic summaries, Instead, models must infer the governing logic by analyzing regularities in the recorded actions and outcomes.
Each game type has customized transcript format reflecting its structural characteristics. For example, chess episodes log initial piece placements, move sequences, and capture events; Texas Hold'em em transcripts include player hands, public cards, and final winners; dice games record raw rolls and outcome labels; blackjack tracks hands, point totals, and bust states.
Table f summarizes the key fields recorded for each domain, along with representative examples. These transcripts constitute the model's input during the rule induction task described in Section D.
Table Gameplay transcript fields and structured examples for each game domain.
Game
Recorded Fields Episode
Example Entry
Board size (e.g.. 8x8, 15x15)
Board: 15x15
Initial piece placement (o.g.. .. Red King m14)
Red King @ m14. Black Queen @
Chess
Round-wise moves: source target
Round 1:
Red: m14 vo13;
Black:
Events: captures, illegal moves
k2-k0
Final outcome (optional)
Red captures Black Bishop
Player hole cards (2 per player, with suits) Community cards (flpp turn, river)
Player 24, 4
Player B: 34 3
Texas Hold'
Winning player and hand rank
Board: 46, 5W, 6é, 96, Q
Hand type comparison
Winner: Player B (Pair of Threes)
Roll result: list of 3 dice
Roll I: [4. 4. + Win
Outcome: win/loss
Roll 2: [2. G] - Win
Dice Game
Optional: derived features (e.g.. parity, sum, triple)
Roll 3: [1, 6] Lose
Player hand cards)
Player: Se, 34, A 24.
Total score after adjustment
Total: 20 (A=11). 1)). No bust
Blackjack
Whether bust occurred
Result: Win Dealer (Bust)
Outcome: win/loss/te
E
Additional Inductive Accuracy
E.1
Additional Inductive Accuracy across Model Versions and Sizes
To complement the main paper's results, we further investigate whether the performance gap between reasoning and non-reasoning models may be confounded by differences in model ver-
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sions and sizes. To mitigate these factors, we include additional comparisons using the most recent publicly available reasoning model, SeepSeex-R1-052. and its closest non-reasoning counterpart, heepSeek-V3-O22 (gpt-40-2024-08- as these represent adjacent adjacent releases with minimal temporal and architectural differences. For completeness, we also report the corresponding results from the main paper for fair comparison. Experiments are conducted on the Chess game, and the results are summarized in Table 5
We also compare GPT-o1 (o1-2024-12-177 against 4PT-4o
Table 5: Inductive accuracy on Normal Rules (NRs) and Special Rules (SRs) across different model
versions and sizes on the Chess game.
Models
SRI
SR2
SR3
SR4
SR5
SR6
NRI
NR2
NR3
NR4
NRS
NR6
DeepSeck-VVP (0325)
48.00%
48.00%
14,0015
46.00%
3,00%
19.3356
100.00%
90.00%
88.00%
86.00%
96.00%
95.33%
DeepSeck-RR
34.00%
13.33%
15.335
4.67%
100009%
93.33%
100.00%
94.00%
100,09%
86.005
DeepSeck-R1 (0528)
38.00%
17.33%
18,005
33333
100000
95.33%
100,055
91,334
GPT-40
56,5746
35.33%
28.0095
63.005
18.005
42.00%
10000%
10000%
100.00%
94.00%
100.00%
100,00%
GPT-o1
19.33%
18.57%
8.00%
30.00%8
13.33%
12.00%
10000%
100.00%
98.00%
100.00%
100.00%
97.33%
GPT-03
21.33%
18.009
4.67%
34.67%
13.334
10.67%
100.0%
100.00%
97.33%
100.00%
100,0055
96.009%
We notice that, when controlling for model family and release proximity, aassoning-augmenn models continue to underperform their non-reasoning models on special rule (SR) generalization, while both perform similarly on normal rules (NR). This aligns with our core claim that reasoning can impair inductive generalization under hidden rule settings.
E.2 Comparison Against Other Reasoning Methods
To further test inductive accuracy under different reasoning settings, we conducted additional experi- ments applying Chain-of-Thought (CoT) style instruction to GPT-4o:
"Let's think step by step. First, extract the rule from the transcript. Then explain your reasoning." "
This instruction was applied to each rule inference task in the Chess game, and we compared model performance with and without CoT prompting; the results are in Table
Table Inductive accuracy of GPT-4o on Normal Rules (NRs) and Special Rules (SRs) with and without Chain-of-Ti Thhou-tf-Thought (CoT) prompting the Chess game.
Methods
SR1
SK2
SR3
SR4
SR5
SR6
NR1
NR2
NR3
NR4
NRS
NR6
Call
56,67%
35.3355
28.00%
68 18.00%
18.0%%
42.00%
100.0%%
100,00%
100.00%
94.00%
100,00%
100.00%
58.00%
100.00%
97.33%
A (Change) +1.33%
-13%
-0.67%
+0.00%
+2.00%
-0.00%
+0.00%
+0.00%
+0.00%
+3.33%
+0.00%
+0.00%
Overall, CoT prompting did not lead to consistent improvements. On SR rules, performance was largely unchanged or slightly decreased, suggesting that added reasoning may not help and can sometimes distract -fro-ffoommom inferring hidden rules. On NR rules, performance was already saturated.
It is worth noting that CoT prompting here is fundamentally different from models like GPT-o3, which are explicitly trained or prompted to generate structured, multi-stage reasoning traces (e.g.. decomposition - solving - summarization). Our findings indicate that simply injecting CoT into non-reasoning models does not reproduce the same failure patterns, nor does it resolve them.
F
Belief Update and Error Recursion
Intuitive Goal. Reasoning modeled here as an iterative information--nteeration process: each Ask action decides where to look next, and each ensuing Answer returns evidence that nudges the current belief m either closer to, or further from, the true solution y" The agent does not observe y" directly; instead it receives vector signal that is correlated with the residual error (y* m but corrupted by two orthogonal noise sources:
Question-alignment noise (oạ)): cognitive or pragmatic imperfections in how the sub-question is framed. A well-posed question (ax xs 1) elicits evidence that points
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almost exactly along the residual error, whereas a misguided question (ax 2 0) yields useless tangent, and catastrophically wrong question (ag- < 0) returns evidence pointing the opposite way.
2.
Answer-generation noise (E&): stochasticity in the LLM's sampling, retrieval errors, hallu- cinations, token truncation, memory decay.
Evidence Model. evidence vector
Conditioned on the history up to step 1- the Answer action produces the
Ok
Orke
(V' mk-1)
Ck
Ek ~ ~~,,00
(6)
alignment
unswer noise
where or [-1, is an i.idd. scalar with E[oo]] ā (0, 1) and Varlou] 72. Eq. 6 two critical properties:
embodies
1. Directionality: The deterministic component or (v* m collinear with the current error vector. Its amplitude encodes how well the sub-question aligns with the unknown residual structure.
2. Zero-mean perturbations. The random component Eμ is isotropic and unbiased, reflecting that answer noise does not systematically drift the belief any preferred direction.
Upon receiving 9k the agent chooses how aggressively to incorpo- rate it. The step-integration weight > (0, 1) captures resource- cognitive fusion: large * trusts the new evidence, small 7/k hedges against possible error. (learning rate of the belief state)
delief-integratton Weight
Update Rule.
The new belief is therefore
m + "kax(8" mx-1) + TKEk-
"ik Ik Subtracting y" yields the error recursion
mi m
: 17
7 Yk Ek:
(8)
which is linear, time-homogen homogeneous, and driven by independent Gaussian shocks -xactly the form required for the closed-form analysis carried out in Appendix G
Relation to Classical Reasoning Notions.
The pair (ax: 7k) mirrors the cognitive psychology split between question relevance (task focus) and processing depth (how strongly new evidence is admitted into working memory).
Eq. (7) can be interpreted as single-step Bayesian update with a exed-precision prior versus likelihood,
stochastic approximation à la Robbins- Monro.
The recursion (8) is the reasoning- reasoning-theorett theoretic analogue of a Kalman filter on static target with multiplicative observation noise.
G Closed-form Error Expression
G.1 Unrolling the Recursion
Iterating the error recursion:
Ck my - -y" = (1 - 7x)0k) (ea0x)ek- ek-1 - 7k Ck,
(9)
for N Answer steps yields
N
N
eN [ - 7(a; -aa-)aa))a eo
Σ
7/ II -1- (1-7)a;))
j=it1
Denote BN := II (1 ma.)) bo = 1.
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G.2 Taking Expectations
Because the noises E, are mero, independent, and independent of {a; all cross terms vanish. Using [|[[=:|| = x² we obtain
N
E(N) B + 0²
78
II (1 (1-7)00 -)a)))) 5
(10)
G.3
Averaging over Question Alignment
Taking expectation w.r.t. ag and writing BN := I1 7;)),
B[E(N)] By bn + a² 7 B
Σ B +
A(N) order positive term
(11)
where B1+1,
II (1 7aa) and A(N) 2 0 collects all contributions containing 7
Interpretation.
Bias contributor: B3 by decreases monotonically with N.
Variance contributor: the summation term grows strictly with N.
Mus-question variance A(N): additional variance induced by imperfect question alignment, also
monotonically increasing in N.
G.4
Theorem: Inevitable U-shaped Error Curve
Theorem G.1 (U-shape of expected error). Let the expected mean- mean-squared squared error after N Answer moves be
E(N) = by vā))
N
2
+
N
N
(1-)aa))
(1
+ A(N),
(12)
j=++1
where
bo > 0, 0<a<l, <<>s5
Σn % =0,
am1
and where A(N) 2 0 is a non-decreasing sequence (the extra variance induced by question-alignment randomness 7²). Then:
E(N) strictly decreasing for N N and strictly increasing for > N*, with
N = min N 2 0: E(N+1) > E(N)};
2. the minimiser N is unique. Consequently, ((N) is U-shaped when plotted against reason- ing depth N.
Proof.
Write
12)
A((N) = by P(N)?
E(N) A(N) V(N) ANN) with
V(N) -' 2 P(++,,)))
where P(a. b) := To )yā) and we abbreviate P(1 by P(N).
Step 1 (monotonicity of A and V). Because - 7t 1, the product P(N) decreases strictly as N increases, hence A(N) is strictly decreasing. For VNN) observe
V(N+1) = 02 NN
Σ
- P(1-1,
1=1
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since every summand is positive. Hence (NN) is strictly increasing, and ( is non-decreasing by assumption.
Step 2 (existence of finite minimiser). Because E, 7: oc, P(N) - 0 and thus A(N) - 0. Meanwhile V(N) diverges to +oo and does (NN). Therefore (NN) attains its minimum some finite N
Step 3 (uniqueness), Define the forward difference (E == E(N+1) E(N) A(N+1)
A(N)] + (VV V[N)] + [A(N+1) - A(N)]. The first bracket is negative, the latter two are positive. Because A(N) decays geometrically while V(N) grows unboundedly, there is exactly one index where the sign of (E(N) flips from negative to positive; call it N Hence I and follow. [
Remark G.2 (Interpretation of the U-shape). The term A(N) captures residual bias: with every additional Answer step the multiplicative factor (1 7;0k) shrinks, so A( A(N) falls exponentially. Con- versely (NN + A(N) aggregates variance: each step contributes fresh noise and never diminishes previous noise, that component rises monotonically. Their competition enforces an optimum where further reasoning flips from net helpful to net harmful-producing the observed inverted-U curve for accuracy as depth increases.
G.5 Lemma: Closed-form Optimum When N/ 7
Lemma G.3 (Explicit N for constant step weight), Assume the integration weights are constant, 1/s # E 10, and let p = ya (0, 1). Further set 72 that ((N) 0. Then the expected enror 12 reduces to
D2N
Egeam (N) bo 2N 0272
NEN,
(13)
p2
and the unique minimiser N is the smallest non-negative integer satisfying
N 1
pl In bo 1 (1-9²) )
(14)
02n²
Consequently (N) is strictly decreasing for N < N geom and strictly increasing for N >
N geom
Proof. Step 1 (geometric closed form). With % 7, P(N) pN. The variance series in
12
becomes a finite geometric sum:
N-1 V(N) 02n2
02g2 1 2N
Σ 2 k=
p2
yielding
13
Step (continuous relaxation). Treat V real variable and define f(t) = bop2t 0222(11 / - p²) for tort 2 0. Differentiating,
0²n² p2
f'(1) = 2(1n p²1 bo +
Since p < the sign (t) flips exactly where the bracket vanishes, i.e. at 2 ln(bn(1 (0²))))
Step (integer minimiser). Because f(t) is strictly convex in (second derivative positive), its minimum over N is attained at the smallest integer not less than 8*, which gives (14). The claimed monotonicity on either side of geom follows from the sign of ft)...
2N Remark G.4 (Deterministic variance balance). Equation (33 isolates the bias term 2 decaying exponentially with depth, and the variance term 0²²r2 (1 //((1 1/(1-p²), growing monoton- ically from to the asymptote o²~2/(t (1=2) The closed-form optimum (14 makes the bias- variance trade-off explicit: deeper reasoning preferred when the initial misconception by is large or the per-step noise a²~² is small, and vice-versa. In all cases the curve remains U-shaped because the two components have opposite monotonicity with respect to N.
26

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Lemma: Error behavior When a; = a
Lemma G.5 (U-shape with deterministic question alignment), Fix a constant alignment factor E (0, 1) and set T² = 0 so that ( = in 12
Let
(N)
bo II (1 ar:)²
0²
Σ
7 II (1 0ng)3 NeN,
j=it1
where the integration weights satisfy < 9i S and E Li=l vi i==m=:a 0c. Then:
1. There exists a finite N such that Ea (N+1) < & (N), (N) N < N: ( (N( for N 2 N&
2. The minimiser Nº is unique, hence (N) is U-shaped in N.
Proof. Define (a TT an1)) and abbreviate Pa(1,N) by Pa(N). Decompose
(N) Aa(N) + (N) with
N
AN(N) bo Pa()), V.(N) o² 2
Pa(it1,N)
(i) Monotonicity of A.. Because 0 < <1 -1-0n < 1, the product Pa(N) decreases strictly as N grows; hence (N) is strictly decreasing.
(ii) Monotonicity of Va. For any 2 O,
(a(++1) (N) o 1
because each bracket positive. Thus V. (N) is strictly increasing.
(iii) Existence and uniqueness of optimum. Since E, = cc, we have (N + 0, so (N) O; but (N) do because each term is non-negative and least one term grows unbounded. Therefore the difference sequence D(N) := (N - (NV) starts negative (bias dominates) and eventually becomes positive (variance dominates). Striet monotonicity of A, and V, implies D(N) changes sign exactly once, establishing [ and
Remark G.6 varianee interpretation with fixed o)). With deterministic question quality a, every Answer move multiplies the residual bias by 1 ari, shrinking it monotonically. Variance still accumulates additively via the 7 term. Their opposing monotonicities enforce single optimal reasoning depth N beyond which additional steps corrupt more than they correct, regenerating the inverted-U pattern.
G.6 Lemma: Comparative Influence of Question Alignment (a) versus Integration Weight (7)
Lemma G.7 (Marginal sensitivities for constant a and 7). deterministic and time-invariant,
Assume the per-step parameters are
= E 1),
a; = (0,
2 0,
and define p - ary E 1). For a fixed reasoning depth N 2 1 the expected error is (cf. Lemma
G.5
0222 1 2N p2
Eo, bo 2N
Then:
1. Alignment is always beneficial. The partial derivative of (N,(N) with respect to a is strictly negative:
(N) 0, V(a E (0, 1) N 1
da
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2. Integration weight exhibits trade-off. For fixed the map 7 + (N, is U-shaped: there exists a unique YN (0, 1) solving
i
E (N) 0,
(15)
Φy
such that (N, is strictly decreasing (0, 7xx) and strictly increasing on (1t 1).
Proof. (a) Monotonicity in a. Differentiate under the definition p = 1 - ary:
D0 DE Op
(-7)
da
Op da
3p
Because y > 0, only need 00p0p O. Direct calculation gives
30
2N bo pa 2N-1 + 2N p 2N
0p
p²
so the overall derivative is strictly negative, proving 1
(b) U-shape in 7. Fix and again write p ary Differentiate with respect to 7:
9g -2N p 2N-1 20²g 1 p2N 1-
Nayp2N-1 1-
θy
As 7. the first (negative) term dominates, so 0E O; Yt 1 the positive term proportional to o²ry dominates, so the derivative is positive. Because the derivative continuous on (0, 1) crosses exactly once, defining the unique root 7 that satisfies (15) and confers the claimed monotonicity.
Uniqueness of Y/ follows from strict convexity: 0 > 0 on (0, 1) (direct but tedious algebra). ensuring the derivative changes sign only once.
Remark G.8 (Practical reading). Result says that better-focused questions (higher a) always help, regardless of noise or depth. Result (] shows shows that how aggressively one updates (1) must be tuned: too timid (ny & TTy) leaves large residual bias, too bold (1 > (xx inflates variance. Crucially, a influences only the location of the optimum via p. whereas 7 controls both how fast bias falls and how steeply variance grows. Thus, when per-step noise o² is high, improving question aligument the more reliable path to lower error; when noise is low, fine-tuning " around Yiy yields further gains.
G.7 Lemma: Qualitative Regimes of Question Alignment a
Lemma G.9 (Effect of when 74 =)) Let the integration weight be fixed (0,1) and assume 72 = 0. Define p(a) = 1 ory and consider the expected error after 2 Answer moves
(N( = bo p(a)2N + 0272 p(
(16)
1 2
Then the qualitative behaviour splits into three disjoint regimes:
1. Positive alignment (0 < a <t))).
Here 0 < p(a) < 1 and
(N) 0, V
aa
Increasing a always lowers the error.
2. Zero alignment (a = 0). Then p = 1 and Ev(N) = by + No?,2 Bias never decays; variance grows linearly with depth.
3. Negative alignment (-7-1 < a <).
Now p(a) > 1 and (N) 2NN + 02220²2= -1 which increases strictly with N. More- p=1
over 2 Ea(NN 0; making less negative reduces error.
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Proof. Differentiate
16 wrt. a via p(a)):
de
dE
-7
da
0p
a
0292
direct calculation gives
2N p² 2N
bo
Op
p2
Regime 1 0 1. Because p² > 0 we have ho - -0222/( - p²) < 0, hence / < 0
and therefore /ē
Regime 2 p = Take the limit p 1 in (16) to obtain the linear form stated.
Regime > 1. Now pp2² < 0 so by 0²/2/( /(1 (11 p²) > O, making a > and hence a da
0. Moreover (NV increases with N because both bias and variance terms grow when
p
Remark G.10 (Which lever matters more?). An comparative discussion on alignment and integration weight.
Alignment (a) is unconditionally helpful whenever it is positive-each incremental improvement shrinks error regardless of depth, noise, or weight. When alignment turss negative, performance deteriorates catastrophically with depth.
Integration weight (7)) by contrast, exhibits the bias-variance U-shape of Lemma G.6 It must be tuned around its optimal value Tx: deviations on either side hurl.
Therefore, ensuring non non-negative aligument is the first priority no choice of y can compensate for < O-while fine-tuning yields second-order gains once positive alignment has been secured.
H Examples of Breakdown Error
Breakdown Errors occur when the model fails to properly decompose the task, leading misaligned or irrelevant sub-questions.
Breakdown Error Example
Game: Chess
Ground Truth: Move in straight line any number of squares, then shift vertically by exactly
two squares.
Model Reasoning:
Let me break this down. First, I'll look at whether the piece lands on a square with a prime- numbered file or rank. That seems relevant, since many special rules involve numerical properties.
Next, I'll check if the move creates a diagonal alignment across pieces from both teams. so, that might indicate a coordination based rule.
The movement goes from b2 e4. That's (+3 columns, +2 rows), which doesn't match a traditional L-shape or knight move, but perhaps it's coded prime-sum rule?
The file total is (2) e = and + Hmm-maybe only prime-ranked destinations are allowed?
Model Rule Answer: The piece can move to squares where the file and rank add up to a prime number.
Error: The model's decomposition introduces irrelevant sub-questions (prime sums, team diagonals) that do not relate to the actual two-phase movement structure of SRI. Although the reasoning is internally coherent, grounded in misaligned task interpretation. This constitutes Breakdown Error: the model fails to recognize the structural form of the rule and instead explores symbolic numerical patterns unrelated to the task.
I Examples of Solving Error Subtypes
We provide representative reasoning traces for the three empirical subtypes of Solving Errors identified in our study. These traces illustrate how errors emerge during the solving stage, even when the
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decomposition is valid. Each example shows the model's step-by-step reasoning and highlights where inductive drift or inappropriate generalization occurs.
1.1 Hallucinated Rule
This subtype of Solving Error occurs when the model fabricates rule that was never presented, demonstrated, or implied the input data. Unlike overgeneralization wwhic extrapolates too broadly from limited evidence prddence-hallucinateee hallucinated rules entirely invented. These errors often emerge when the model tries to impose structure where none exists, or fills uncertainty with overconfident but unsupported patterns.
Hallucinated Rule Example
Game: Chess
Ground Truth: Move in a straight line any number of squares, followed by a one-square diagonal shift.
Model Reasoning:
(...earlier reasoning omitted...)
I'm thinking through how the piece might combine horizontal and vertical moves, akin to Brooklyn distance, cover large distances effectively in single turn. It's interesting to think if the piece might follow an angular path, allowing it to shift vertically, horizontally, diagonally a single turn, adding dynamic edge the game. (...coniinued reasoning...)
Model Rule Answer: The piece can move in a flexible angular trajectory that combines horizontal, vertical, and diagonal steps, as long as the total displacement approximates a straight path.
Error: The model hallucinates continuous-spar movement rule based blended di- rectional offsets. This rule does not exist any defined NR or SR, and contradicts the stepwise constraint in SR6. The model invents pseudo-geometry (e.g., "angular displace- ment") to rationalize incidental move patterns, reflecting a hallucinated inductive structure that misinterprets discrete board logic as continuous spatial flow.
1.2
Overgeneralization
This subtype of Solving Error occurs when the model prematurely commits to rule based on a small number of supporting examples. Rather than verifying consistency across diverse cases, overextends an observed pattern to structurally dissimilar inputs. In our settings, overgeneralization often arises when models collapse multi-step actions into simplified summaries, or infer category-wide behavior from isolated instances. These errors reflect tendency to form rules that are overly broad, failing to capture the precise structure or constraints intended by the task.
Overgenerallization Example
Game: Chess
Ground Truth: Move exactly three squares in direction, then move square downward.
Model Reasoning:
(...earlier reasoning omitted...)
Looking at this move from g4 to f7, the total shift (-1 column, +3 rows). That's a net movement of three squares diagonally.
It looks similar to previous valid moves that landed three or four squares away from the starting point. So assume this pattern allows 3-step diagonal moves.
Alternatively, maybe it's just a "distance-3" rule, where the end location must be offset by a vector of total length 4, in any direction. That would explain why g4 f7 allowed. I'll with that-this seems to fit with the movement pattern I've seen. (...continued reasoning...)
Model Rule Answer: A piece can move 3 or 4 squares in any direction, including diagonals. Error: The model collapses composite 3+1 directional movement into single displacement vector, overgeneralizing to rule based on total distance. It ignores the two- swo-phase structure and the downward constraint, falsely classifying structurally invalid moves rule-consistent.
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1.3
Math Overuse
This enror occurs when the model incorrectly applics arithmetic numerical reasoning tasks that are fundamentally symbolic, structural, or categorical in nature.
Math Overuse Example
Game: Chess
Ground Truth: Move straight line any number squares, then shift vertically by exactly
two squares.
Model Reasoning:
(...earlier reasoning omitted...
OK, let me see. The piece's moves are quite varied, with differences like (2,8) and 12.11). This suggests they must stay within maximum distance, maybe determined by the difference of steps.
I'm analyzing the Euclidean distance for piece's movement from b1 to i3, with column and row differences, leading to distance of approximately 7.28. This method allows for accurate computation.
(...continued reasoning...)
Model Rule Answer: The piece can move any square within Euclidean distance of approximately from its starting position.
Error: The model misinterprets the piece's movement by applying Euclidean distance as a decision criterion, which is not relevant to the rule-based nature of chess. This reflects a Math Overuse error: numerical metrics are substituted for structured reasoning over discrete directions and board geometry.
Examples of Summary Error
Summary Errors occur when the model's reasoning process is internally consistent and correctly processes the input, yet the final nule outputs incorrect. Unlike Solving Breakdown Errors, the failure here does not lie in intermediate logic or sub-task handling, but rather in the concluding infer- ence step. These errors often manifest as misstatements, mislabelings, or minor structural inaccuracies in the final rule summary despite evidence that the correct pattern was already identified.
Summary Error Example
Game: Chess
The piece follows SR6: "Move in straight line any number of squares, followed by a one-square diagonal shift."
Ground Truth:
Model Reasoning:
Thinking 5m12s
(...eallirr reasoning omitted...)
Step I: The move from c3 to f4 involves a three-square horizontal move (c - f), followed by one-square diagonal shift upward and right (from row 3 to 4 and to g). Step 2: This clearly breaks down into: - A straight- line segment: c3 - f3 (horizontal, squares), Followed by: f3 + g4 squere diagonal shift). The diagonal is exactly one square and continues smoothly from the initial segment.
Step 3: The structure matches previous valid cases. The piece first makes linear move, then pivots diagonally once. There are no blockers. The transition clean. (...reasoning ends..))
Model Rule Answer: The piece moves in a straight line followed by a vertical shift of one square.
Error: The model's reasoning accurately captures the two-stage movement required by SR6, identifying both the straight-ling component and the one-square diagonal shift. However, misstates the final rule, replacing "diagonal shift" with "vertical shift" in its summary. This is a Summary Error: the model reasons correctly but fails in final rule articulation.
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K
Classification of Reasoning Failures
To assess the reliability of our manual classification of reasoning failure types, we conducted an inter-annotator agreement study involving two independent annotators. Each annotator labeled 100 sampled outputs from reasoning models before any discussion resolution of disagreements.
Table 7:
Inter-annotator agreement between two annotators across three reasoning error types.
Failure Mode
Agreement Count
Total Cases
Agreement Rate
Cohen's x
Decomposition Error
36
40
0.900
0.84
Solving Error
38
42
0.905
0,87
Summarization Error
16
18
0.889
0.83
Overall
90
100
0.900
0.86
We computed Cohen's between the two annotators based on their independent annotations prior to reconciliation. Agreement statistics for each error type are summarized in Table The results agreement across all categories, with overall 0.86,
indicate consistently high suggesting strong reliability the identification and classification reasoning failure modes.
L
Intervention Prompt Design
We provide full prompt templates used for each intervention strategy described in Section 5.1 These prompts were designed to isolate or jointly address the three major reasoning failure modes.
L.1 Sub-task Decomposition Constraints
To mitigate Breakdown Errors caused by poorly aligned sub-question formulation, we provide structured prompts that explicitly constrain how models decompose the task. The following template guides the model through three-phase reasoning procedure, with detailed sub-steps for entity extraction, rule induction, and rule verification.
Prompts for Sub-task Decomposition
You are given complex reasoning task. Follow the structured reasoning steps below. Each step includes internal sub-steps ensure clarity and alignment with the task goal.
Step Identify and organize relevant entities.
Break the input into interpretable components. Answer the following sub-questions:
What are the basic elements
input (e.g., cards, pieces, dice)?
What attributes are associated with each element (e.g., suit, number, position, color)?
Are there groupings, repetitions, or orderings that might matter (e.g., same suit, consecutive values)?
Represent the input in a structured, canonical form for downstream rule inference. Step 2: Induce candidate rule(s) from prior context. Based on previous examples or observed patterns, hypothesize a rule that could explain the current past cases. Sub-steps:
Look for shared properties among successful examples (e.g., all include prime number sequence).
Consider combinations of attributes that might define a category (e.g., "all red cards", "adjacent positions". "triplets").
Formulate one or more abstract rules using natural language or logical expressions. If multiple rules seem possible, rank or explain them by plausibility.
Step 3: Verify the inferred rule against the current input.
Apply your proposed rule(s) to this instance. Proceed with the following:
Does the structured input satisfy the rule exactly?
partially satisfied, explain which components match or fail.
If none match, state clearly why the rule does not apply.
Conclude with a binary result (rule matched not matched), and explain how the final decision is reached.
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L.2 Sub-task Solving Constraints
To mitigate Solving Errors- partis-particularly those caused by Math Overuse-we encourage models to reason in a grounded, step-by-step manner. Instead of relying on arithmetic shortcuts, models are guided to verbalize their observations, hypotheses, and decisions in a natural, reflective style. The following prompt mimics internal reasoning without appealing to symbolic abstraction.
Prompts for Sub-task Solving
Getting sense of the setup
I'm looking over the current configuration. There's a set of game elements cards, pieces, liie-arrrange within defined structure. I begin by scanning each entity and noting its type, position, and any immediate groupings. I try to understand what roles these elements might play, and whether any of them marked, repeated, stand out visually. Once have a general grasp, start mapping the layout mentally so can refer back to it during analysis. Spotting initial patterns
As move through the input, some patterns begin to emerge. I see repeated forms-like similar numbers, mirrored types, or alternating colors. In some cases, specific alignments appear intentional, like a row of matching elements or a cluster that resembles a known configuration. note these early signals and consider whether they resemble any previous examples I've worked with. These patterns may not yet define rule, but they give me a starting point.
Tracking how things evolve Now focus on lraages--oveeeee replacements, or newly introduced elements. observe which parts of the structure are dynamic and whether these shifts maintain or break previous patterns. For example,
card swaps position or piece moves diagonally, I check if that action matches others I've seen. I also look at directionality and symmetry: are changes centered around a pivot? Are actions constrained to certain zones? of this helps me refine how the system behaves.
Interpreting the intent
I try understand not just what changed, but why. The observed actions feel deliberate, so begin thinking about what constraints or goals might be shaping them. Perhaps certain moves legal only under hidden conditions, some combinations gain value due an unknown rule. think about whether the system rewards alignment, diversity, or some balance in composition. This lets go beyond just pattern matching I'mm starting infer purpose. Refining the hypothesis
Now compare my current with earlier examples. I'm looking for consistency: do similar setups always lead to the same outcome? check whether specific attributes-like color sequences paired strengthens. If not, I adjust. also check for edge cases that might help distinguish between competing rules. The more refine, the clearer the rule's shape becomes. Committing to a conclusion
reappear under the same conditions. If they do, my hypothesis
With all this in mind, I'm ready to decide. The current setup aligns with the rule I've been building. see enough evidence -ridunce-through repetition, structure, and behavior commit to an answer. There's no need for math here; it's the alignment between elements and rules that matters. finalize my judgment and prepare to apply this same logic again if needed.
L.3 Summarization Constraints
We enforce 1000-token generation limit using stop sequences and explicit model instructions. The prompt also includes final reminder:
33

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Prompts for Final Answer Summarization
You are given a reasoning task. Please approach it step by step, with each step clear and concise. If the answer becomes evident before completing steps, stop immediately and provide the final answer. DO NOT continue reasoning once the question is resolved. Your total output must stay within 1000 tokens. Excessive or unnecessary reasoning beyond this limit will be considered invalid.
M
Additional Interventions Results
M.1 Detailed Results for Interventions on Chess Game
While Section 5.2 and Fig. Billustrate the individual effects of each proposed intervention by showing incremental accuracy gains over the baseline in a compact visual form, some details may be less apparent in that summary. To provide a clearer view, we present below the detailed, table-form results for the Chess game.
Table 8: Detailed inductive rule accuracy (%) across different intervention strategies and models for the Chess game. Numbers in parentheses denote improvements over the non-intervention baseline.
saache clelaity asaee atay da she aamy
beaa ane Lanay E30 (iloiii 1.B (etoma an MEl CEAAN
ant dalaaa
25.00
Hurer' aod
1991
13 201
JAMPS
North (TT
mee (14
(12
tuddl (1979)
(ock))
23 41 Min 5.5 16.339
14.00 (44.79 120 (6:081) VAJ0S (EAPS
16.05 (BUIN) 01 1.3 14.77
Seatartoninn
(ON Vey 110 Niw 1 18
2004
Sciing
BLOS you 271
54 57 141 4TT1
13.00
14.3
130 30T 140 (83)
18 W9 (4:099 05676 GAMET
The solving component accounts for the largest gains in SR generalization across all tested models, while summarization and decomposition offer complementary improvements.
M.2 Intervention Results on Math Benchmark
Our findings for reasoning and proposed interventions can extend to more common reasoning problems, such as mathematical problem solving. Prior work has noted that reasoning-augmente models often generate inflated reasoning traces and redundant self- seff-verifications, leading to inefficient token usage [44 and increased error rates [45]. To test the applicability of our intervention, we evaluated GPT-o3 20 problems from the American Invitational Mathematics Examination (AIME) 2024. We compared its accuracy with and without our combined intervention.
Table 9: Accuracy comparison with and without intervention.
Dataset
w/o Intervention w/ Intervention
AIME 24
45%
65%
As shown in the table 9 the combined intervention improved accuracy by 20%, suggesting its potential to mitigate reasoning-related failures in mathematical tasks.
N Results and Dataset
The comprehensive results and code of our experiment are soun-eerrced on the web. For detailed in- formation, please visit the following link: attp:::/iin-induutireeellllll abilities.... vercel. appp
34


Reviews:
[{"review_id": "ccDRNvJZs3", "reviewer": "Reviewer_3QQD", "summary": "This paper investigates the performance of Large Language Models (LLMs) in inductive reasoning tasks and finds that the widely used Chain-of-Thought (CoT) reasoning approach is not always beneficial, but instead may degrade model performance when reasoning about hidden rules. The authors designed four game-like tasks (Chess, Texas Hold'em, Dice Game, and Black Jack) containing hidden human rules, systematically evaluated the inductive power of eight mainstream language models, and found that inference-enabled models are often inferior to models without inference when dealing with complex, abstract rules. To explain this phenomenon, the authors propose a theoretical framework that identifies three major failure modes in the reasoning process: incorrect subtask splitting, incorrect subtask solving, and inappropriate answer summarization, which gradually amplify errors in the reasoning chain. Based on the theoretical analysis, the authors further propose three interventions that do not require retraining and significantly improve the model's accuracy in summarization tasks by structurally guiding the reasoning process. This study challenges the common assumption that “more reasoning is better” and emphasizes that well-structured, goal-oriented reasoning is more critical than blindly extending the reasoning chain.", "questions": "Q1. The comparison between reasoning and non-reasoning models may be confounded by differences in model versions and sizes. Further clarification is needed to ensure fairness and validity of the comparisons.\n\nQ2. While the theoretical framework is well-formalized, it lacks intuitive illustrations or concrete examples that would aid comprehension. Including visual aids or specific cases could enhance the clarity and persuasiveness of the theoretical analysis.\n\nQ3. The study does not include ablation experiments to isolate the individual effects of each proposed intervention. Adding such analyses would help verify the contribution of each intervention to the overall performance improvement.\n\nQ4. Could these findings potentially be extended to more common reasoning problems beyond the specific cases studied?", "limitations": "Yes.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths\n- The paper addresses a meaningful and timely problem—whether reasoning via chain-of-thought (CoT) actually helps inductive tasks—and supports it with well-designed, controlled experiments using four diverse game-based benchmarks.\n- It provides a rigorous theoretical framework that identifies and formalizes three distinct failure modes in reasoning, supported by clear mathematical derivations explaining performance degradation.\n- The proposed prompting interventions—structured decomposition, guided solving, and output length control—are simple yet effective, yielding significant accuracy gains without retraining.\n\nWeaknesses\n- The comparison between reasoning and non-reasoning models is potentially confounded by version and size mismatches. For example, DeepSeek-V3 uses the latest version from March 2025, while DeepSeek-R1 is based on an older January 2025 version and derived from an earlier DeepSeek-V3. Differences in model sizes between reasoning and non-reasoning variants may also bias the conclusions and require further clarification.\n- The theoretical framework, while well-formalized, lacks intuitive illustrations or concrete examples. Readers may find it hard to understand the roles of abstract variables like αₖ (alignment) and εₖ (noise) without visual aids or case walkthroughs.\n- The paper lacks ablation studies to isolate the effect of each intervention. Such experiments would help confirm the effectiveness of each strategy and validate their correspondence to the identified failure modes.", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for the constructive feedback. We appreciate your recognition of the problem's importance and the clarity and utility of our theoretical framework in explaining performance degradation and guiding effective interventions. We believe the identified concerns can be adequately addressed.\n\n**W1/Q1. The comparison between reasoning and non-reasoning models may be confounded by differences in model versions and sizes.**  \n**A1.** We appreciate the reviewer’s attention to potential confounding factors in model versioning and size. Inspired by this concern, we tried to minimize these issues by including additional comparisons with the most recent publicly available reasoning model, DeepSeek-R1-0528, and contrasting it with its closest non-reasoning model, DeepSeek-V3-0325. We also compare GPT-o1 (o1-2024-12-17) against GPT-4o (gpt-4o-2024-08-06), as they represent adjacent releases with a minimal temporal gap. All results are reported in percentage format.\n\n| Model              |   SR1 |   SR2 |   SR3 |   SR4 |   SR5 |   SR6 |   NR1 |   NR2 |   NR3 |   NR4 |   NR5 |   NR6 |\n|:-------------------|------:|------:|------:|------:|------:|------:|------:|------:|------:|------:|------:|------:|\n| DeepSeek-V3        |  48.0 |  48.0 |  14.0 |  46.0 |   8.0 |  19.3 | 100.0 |  90.0 |  88.0 |  86.0 |  96.0 |  95.3 |\n| DeepSeek-R1 (0120) |  34.0 |  13.3 |   5.3 |  15.3 |   4.7 |   9.3 | 100.0 |  93.3 | 100.0 |  94.0 | 100.0 |  86.0 |\n| DeepSeek-R1 (0528) |  38.0 |  17.3 |   8.0 |  18.0 |   8.0 |  11.3 | 100.0 | 100.0 |  96.0 |  95.3 | 100.0 |  91.3 |\n| GPT-4o             |  56.7 |  35.3 |  28.0 |  68.0 |  18.0 |  42.0 | 100.0 | 100.0 | 100.0 |  94.0 | 100.0 | 100.0 |\n| GPT-o1             |  19.3 |  18.7 |   8.0 |  30.0 |  13.3 |  12.0 | 100.0 | 100.0 |  98.0 | 100.0 | 100.0 |  97.3 |\n| GPT-o3             |  21.3 |  18.0 |   4.7 |  34.7 |  13.3 |  10.7 | 100.0 | 100.0 |  97.3 | 100.0 | 100.0 |  96.0 |\n\nWe notice that, when controlling for model family and release proximity, reasoning-augmented models continue to underperform their non-reasoning models on SR generalization, while both perform similarly on NR. This aligns with our core claim that reasoning can impair inductive generalization under hidden rule settings. Full results for the other games will be included in the revised version of the paper.\n\n**W2/Q2. While the theoretical framework is well-formalized, it lacks intuitive illustrations or concrete examples that would aid comprehension.**  \n\n**A2.** We apologize for the confusion. For each type of error in our theoretical framework, we have provided illustrative examples with visualizations in **Appendices G, H, and I**. In each case, we highlight the specific sentence corresponding to the error type.\n\n**W3/Q3. Ablation experiments to isolate the individual effects of each proposed intervention.**  \n\n**A3.** We thank the reviewer for prompting a closer look at the effects of each intervention. We have evaluated the individual effects of each proposed intervention, as shown in Figure 3, where each improvement is added separately to the baseline accuracy using different colors. However, the figure’s compact format may have obscured these effects. To address this, we provide a detailed ablation table for the chess game below.\n\n| Model                         | SR1            | SR2            | SR3            | SR4            | SR5            | SR6            | NR1            | NR2            | NR3            | NR4            | NR5            | NR6            |\n|:-----------------------------|:---------------|:---------------|:---------------|:---------------|:---------------|:---------------|:---------------|:---------------|:---------------|:---------------|:---------------|:---------------|\n| GPT-o3 w/o Interventions      | 21.3           | 18.0           | 4.7            | 34.7           | 13.3           | 10.7           | 100.0          | 100.0          | 97.3           | 100.0          | 100.0          | 96.0           |\n| GPT-o3 w/ decomposition       | 22.7 (+1.3)    | 24.7 (+6.7)    | 10.0 (+5.3)    | 44.7 (+10.0)   | 14.0 (+0.7)    | 18.0 (+7.3)    | 100.0 (+0.0)   | 100.0 (+0.0)   | 100.0 (+2.7)   | 95.3 (-4.7)    | 100.0 (+0.0)   | 96.0 (+0.0)    |\n| GPT-o3 w/ solving             | 48.0 (+26.7)   | 42.0 (+24.0)   | 46.7 (+42.0)   | 60.7 (+26.0)   | 27.3 (+14.0)   | 46.0 (+35.3)   | 100.0 (+0.0)   | 100.0 (+0.0)   | 100.0 (+2.7)   | 96.7 (-3.3)    | 100.0 (+0.0)   | 100.0 (+4.0)   |\n| GPT-o3 w/ summarization       | 40.0 (+18.7)   | 26.0 (+8.0)    | 26.0 (+21.3)   | 40.0 (+5.3)    | 21.3 (+8.0)    | 27.3 (+16.7)   | 100.0 (+0.0)   | 100.0 (+0.0)   | 100.0 (+2.7)   | 97.3 (-2.7)    | 100.0 (+0.0)   | 98.7 (+2.7)    |\n| GPT-o3 w/ Combined            | 52.0 (+30.7)   | 52.7 (+34.7)   | 54.0 (+49.3)   | 64.0 (+29.3)   | 31.3 (+18.0)   | 46.0 (+35.3)   | 100.0 (+0.0)   | 100.0 (+0.0)   | 100.0 (+2.7)   | 96.7 (-3.3)    | 100.0 (+0.0)   | 100.0 (+4.0)   |\n| DeepSeek-R1 w/o Interventions | 34.0           | 13.3           | 5.3            | 15.3           | 4.7            | 9.3            | 100.0          | 93.3           | 100.0          | 94.0           | 100.0          | 86.0           |\n| DeepSeek-R1 w/ decomposition  | 36.0 (+2.0)    | 18.0 (+4.7)    | 5.3 (+0.0)     | 17.3 (+2.0)    | 12.7 (+8.0)    | 19.3 (+10.0)   | 100.0 (+0.0)   | 93.3 (+0.0)    | 100.0 (+0.0)   | 94.7 (+0.7)    | 100.0 (+0.0)   | 93.3 (+7.3)    |\n| DeepSeek-R1 w/ solving        | 55.3 (+21.3)   | 43.3 (+30.0)   | 28.0 (+22.7)   | 32.7 (+17.3)   | 23.3 (+18.7)   | 35.3 (+26.0)   | 100.0 (+0.0)   | 95.3 (+2.0)    | 100.0 (+0.0)   | 96.7 (+2.7)    | 100.0 (+0.0)   | 98.0 (+12.0)   |\n| DeepSeek-R1 w/ summarization  | 36.7 (+2.7)    | 16.0 (+2.7)    | 9.3 (+4.0)     | 15.3 (+0.0)    | 8.0 (+3.3)     | 14.0 (+4.7)    | 100.0 (+0.0)   | 94.0 (+0.7)    | 100.0 (+0.0)   | 94.7 (+0.7)    | 100.0 (+0.0)   | 94.7 (+8.7)    |\n| DeepSeek-R1 w/ Combined       | 60.7 (+26.7)   | 44.7 (+31.3)   | 41.3 (+36.0)   | 38.0 (+22.7)   | 27.3 (+22.7)   | 35.3 (+26.0)   | 100.0 (+0.0)   | 98.0 (+4.7)    | 100.0 (+0.0)   | 100.0 (+6.0)   | 100.0 (+0.0)   | 98.0 (+12.0)   |\n| Grok-3 w/o Interventions      | 20.7           | 4.7            | 5.3            | 19.3           | 4.0            | 9.3            | 100.0          | 96.0           | 100.0          | 98.0           | 100.0          | 100.0          |\n| Grok-3 w/ decomposition       | 24.0 (+3.3)    | 7.3 (+2.7)     | 8.0 (+2.7)     | 22.7 (+3.3)    | 5.3 (+1.3)     | 14.0 (+4.7)    | 100.0 (+0.0)   | 96.0 (+0.0)    | 100.0 (+0.0)   | 93.3 (-4.7)    | 100.0 (+0.0)   | 100.0 (+0.0)   |\n| Grok-3 w/ solving             | 51.3 (+30.7)   | 15.3 (+10.7)   | 19.3 (+14.0)   | 36.7 (+17.3)   | 27.3 (+23.3)   | 36.7 (+27.3)   | 100.0 (+0.0)   | 98.7 (+2.7)    | 100.0 (+0.0)   | 98.0 (+0.0)    | 100.0 (+0.0)   | 100.0 (+0.0)   |\n| Grok-3 w/ summarization       | 34.0 (+13.3)   | 10.0 (+5.3)    | 4.7 (-0.7)     | 16.0 (-3.3)    | 20.0 (+16.0)   | 24.7 (+15.3)   | 100.0 (+0.0)   | 96.0 (+0.0)    | 100.0 (+0.0)   | 94.7 (-3.3)    | 100.0 (+0.0)   | 100.0 (+0.0)   |\n| Grok-3 w/ Combined            | 55.3 (+34.7)   | 14.0 (+9.3)    | 20.0 (+14.7)   | 27.3 (+8.0)    | 22.7 (+18.7)   | 38.0 (+28.7)   | 100.0 (+0.0)   | 100.0 (+4.0)   | 100.0 (+0.0)   | 98.0 (+0.0)    | 100.0 (+0.0)   | 100.0 (+0.0)   |\n| QwQ w/o Interventions         | 13.3           | 7.3            | 8.0            | 20.0           | 0.7            | 2.0            | 100.0          | 100.0          | 98.0           | 100.0          | 100.0          | 94.7           |\n| QwQ w/ decomposition          | 9.3 (-4.0)     | 5.3 (-2.0)     | 15.3 (+7.3)    | 21.3 (+1.3)    | 1.3 (+0.7)     | 9.3 (+7.3)     | 100.0 (+0.0)   | 100.0 (+0.0)   | 98.0 (+0.0)    | 100.0 (+0.0)   | 100.0 (+0.0)   | 96.0 (+1.3)    |\n| QwQ w/ solving                | 18.0 (+4.7)    | 16.7 (+9.3)    | 27.3 (+19.3)   | 34.7 (+14.7)   | 13.3 (+12.7)   | 27.3 (+25.3)   | 100.0 (+0.0)   | 100.0 (+0.0)   | 98.7 (+0.7)    | 100.0 (+0.0)   | 100.0 (+0.0)   | 96.7 (+2.0)    |\n| QwQ w/ summarization          | 13.3 (+0.0)    | 12.0 (+4.7)    | 20.7 (+12.7)   | 20.0 (+0.0)    | 6.7 (+6.0)     | 11.3 (+9.3)    | 100.0 (+0.0)   | 100.0 (+0.0)   | 98.0 (+0.0)    | 100.0 (+0.0)   | 100.0 (+0.0)   | 94.7 (+0.0)    |\n| QwQ w/ Combined               | 18.0 (+4.7)    | 19.3 (+12.0)   | 27.3 (+19.3)   | 20.7 (+0.7)    | 13.3 (+12.7)   | 30.7 (+28.7)   | 100.0 (+0.0)   | 100.0 (+0.0)   | 100.0 (+2.0)   | 100.0 (+0.0)   | 100.0 (+0.0)   | 96.7 (+2.0)    |\n\nThe solving component accounts for the largest gains in SR generalization across all tested models, while summarization and decomposition offer complementary improvements. The figure will be revised for clarity in the final version, and full ablation tables for the other games will be included in the appendix.\n\n**W4/Q4. Could these findings potentially be extended to more common reasoning problems beyond the specific cases studied?**  \n\n**A4.** We believe these findings can extend to more common reasoning problems, such as mathematical problem solving. Prior work has noted that reasoning-augmented models often generate inflated reasoning traces and redundant self-verifications, leading to inefficient token usage and increased error rates [1]. To test the applicability, we evaluated GPT-o3 on 20 problems from the AIME 2024 exam, comparing accuracy with and without our combined intervention. The results are reported in percentage format.\n\n| Dataset   | w/o Intervention | w/ Intervention |\n|-----------|------------------|-----------------|\n| AIME 24   | 45               | 65              |\n\nAs shown in the table, the combined intervention improved accuracy by 20 percent, suggesting its potential to mitigate reasoning-related failures in mathematical tasks.\n\n**References**  \n[1]. Chen X, Xu J, Liang T, et al. Do NOT Think That Much for 2+ 3=? On the Overthinking of Long Reasoning Models[C]//Forty-second International Conference on Machine Learning."}, {"review_id": "cLQaFBH8WN", "reviewer": "Reviewer_EhoN", "summary": "The paper investigates the effect of chain-of-thought reasoning on the inductive abilities of LLMs. Using games-based tasks with some hidden rules, the paper observes that reasoning based LLMs often perform worse than their non-reasoning counterparts, particularly on tasks requiring the inference of non-obvious or special hidden rules. The paper then presents a theoretical framework that points to three primary failure modes in the reasoning process: potentially incorrect sub-task decomposition, incorrect sub-task solving, and flaws in final answer summarization. Based on this analysis, the paper proposes and validates a set of structured interventions—such as using decomposition templates and guiding the solving process—that successfully improve the inductive accuracy of LRMs without needing to retrain the models. The central argument is that the quality of inductive reasoning in LLMs hinges not merely on performing more reasoning steps, but on the criteria that those steps are well-structured and accurate.", "questions": "The paper fairly convincingly shows that they are better at inducing special rules, but not why.  Are they using a more robust, implicit form of reasoning, or are they relying on a different form of pattern matching that is less prone to the error amplification seen in multi-step CoT?\n\n\nThe interventions presented in the paper are fair in the games-setup, but do they generalize to other domains at all? \n\nCould the identified failure modes be addressed through targeted fine-tuning instead of prompting?", "limitations": "Yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "The paper is a fairly rigorous paper and the tasks used provide a nice and controlled environment to test inductive reasoning.  The core empirical finding—that LRMs with explicit CoT reasoning consistently underperform non-reasoning LLMs on special rules—is a significant and I think a fairly non-obvious result (based on the general acceptance of these). The theoretical framework is clear and can help in understanding when it will fail reasoning. The decomposition of reasoning errors into three distinct types is also intuitive and powerful. Further, the empirical validation makes it strong. The interventions seem to be directly derived from failure modes. \n\nOn weaknesses: I feel the title is overtly attention grabbing and seems like just reading the title alone oversimplifies the findings of the paper. The paper itself shows that it is specifically poorly structured or unconstrained reasoning that is detrimental, and that well-guided reasoning is highly beneficial. The claim could be more precisely worded to reflect that current CoT mechanisms can be flawed, rather than suggesting reasoning itself is the problem. \nThere are some rigour issues: The classification of reasoning failures was conducted by two human annotators. While the paper states that disagreements were resolved collaboratively, this process is inherently subjective. Without quantitative metrics like inter-annotator agreement scores, the reliability of the error distribution presented in Table 1 is hard to independently verify.  The empirical analysis of failure modes was also performed on three specific LRMs. Although the findings are consistent across these models, it is a significant leap to generalize these specific failure patterns to all current and future reasoning models. This requires a careful discussion about the limitations, at the very least. The primary metric for success—rule-level accuracy—is determined by using GPT-4o as a judge to assess semantic alignment — that’s circular and off. There’s not a lot of careful analyses here.", "was_revised": true, "final_justification": "I agree with the responses from the authors. I also am happy with their new title for the paper which is more grounded in the empirical observations. I will retain my positive score for the paper.", "rebuttal": "We appreciate your thoughtful review and are encouraged by your recognition of the significance and non-obvious nature of our core findings, as well as the clarity and utility of our theoretical framework. We believe your remaining concerns are addressable and respond to them in detail below.\n\n**W1. I feel the title is overtly attention grabbing and seems like just reading the title alone oversimplifies the findings of the paper.**\n\n**A1.** Thank you for sharing your concern about the title. We agree that the current phrasing may risk oversimplifying our findings. In the revised version, we will use a more precise title:  **_Evaluating the Inductive Abilities of Large Language Models: Why Chain-of-Thought Reasoning Sometimes Hurts More Than Helps_**\n\n**W2. The classification of reasoning failures was conducted by two human annotators.**\n\n**A2.** Thank you for raising this important concern. Motivated by your feedback, we computed **Cohen’s κ** between the two annotators based on their independent annotations before resolving any disagreements. We evaluated inter-annotator agreement across three error types—**Decomposition**, **Solving**, and **Summarization**—on 100 sampled model outputs from reasoning models. The results are summarized below:\n| Failure Mode        | Agreement Count | Total Cases | Agreement Rate | Cohen’s κ |\n|---------------------|------------------|--------------|----------------|------------|\n| Decomposition Error | 36               | 40           | 0.90           | 0.84       |\n| Solving Error       | 38               | 42           | 0.905          | 0.87       |\n| Summarization Error | 16               | 18           | 0.889          | 0.83       |\n| **Overall**         | **90**           | **100**      | **0.90**       | **0.86**   |\n\nWe believe κ = 0.86 indicates a high level of reliability, and we will include these agreement metrics and this table in the revised version to enhance the methodological rigor of Table 1.\n\n**W3. The empirical analysis of failure modes was also performed on three specific LRMs. Although the findings are consistent across these models, it is a significant leap to generalize these specific failure patterns to all current and future reasoning models. This requires a careful discussion about the limitations, at the very least.**\n\n**A3.** Thank you for raising this point. We acknowledge that our analysis is limited to three reasoning models, and it is premature to generalize the identified failure patterns to all current or future models. We also recognize that comparisons between reasoning and non-reasoning models may be influenced by confounding factors such as version mismatch or model size differences. We will clarify these limitations in the revised paper.\n\n**W4. The primary metric for success—rule-level accuracy—is determined by using GPT-4o as a judge to assess semantic alignment — that’s circular and off.**\n\n**A4.** We appreciate the reviewer’s concern about potential circularity in using GPT-4o for evaluation. Motivated by your comment, we conducted a human validation study on 100 GPT-o3-generated rule descriptions across all four games. Three independent annotators—blind to model identity and GPT-4o’s decisions—judged semantic alignment with the ground truth. We then compared their majority vote with GPT-4o’s decisions and computed **Cohen’s κ** to measure agreement. The results are summarized below:\n\n| Judgment Type     | GPT-4o vs Human        |\n|-------------------|------------------------|\n| Aligned (60 cases)    | 60 / 60 (100%)         |\n| Not Aligned (40 cases) | 40 / 40 (100%)         |\n| **Overall Accuracy**  | **1.00**               |\n| **Cohen’s κ**         | **1.00**               |\n\nGPT-4o’s assessments matched the human majority in all 100 cases (60 aligned, 40 not aligned), yielding an accuracy and κ of 1.00. While this perfect agreement may in part reflect the limited sample size, it provides strong evidence that GPT-4o’s evaluations are well-aligned with human judgment in this context.\n\n**Q1. The paper fairly convincingly shows that they are better at inducing special rules, but not why. Are they using a more robust, implicit form of reasoning, or are they relying on a different form of pattern matching that is less prone to the error amplification seen in multi-step CoT?**\n\n**A1.** While we cannot directly observe the internal mechanisms behind model decisions, our theoretical analysis and empirical findings suggest that non-reasoning models may rely on a different form of pattern matching that is less susceptible to the error accumulation observed in multi-step CoT reasoning. In particular, we conjecture that the absence of explicit reasoning traces in these models reduces opportunities for compounding errors in sub-task decomposition, solving, or summarization.\n\n**Q2.The interventions presented in the paper are fair in the games-setup, but do they generalize to other domains at all?** \n\n**A2.** We believe these findings can extend to more common reasoning problems, such as mathematical problem solving. Prior work has noted that reasoning-augmented models often generate inflated reasoning traces and redundant self-verifications, leading to inefficient token usage [1] and increased error rates [2]. To test the applicability of our intervention, we evaluated GPT-o3 on 20 problems from the AIME 2024 exam. We compared its accuracy with and without our combined intervention. The results are summarized below:\n\n| Dataset   | w/o Intervention | w/ Intervention |\n|-----------|------------------|------------------|\n| AIME 2024 | 45.0             | 65.0             |\n\nAs shown in the table, the combined intervention improved accuracy by 20 percentage points, suggesting its potential to mitigate reasoning-related failures in mathematical tasks.\n\n**Q3. Could the identified failure modes be addressed through targeted fine-tuning instead of prompting?**\n\n**A3.** We appreciate this suggestion. We agree that targeted fine-tuning could be a promising direction for mitigating some of the failure modes we identify. Our current focus is on prompt-based interventions mainly due to their ease of deployment. We view prompt-based and fine-tuning-based approaches as complementary, and we hope our analysis can inform future work in both directions.\n\n**References**  \n[1] Chen X., Xu J., Liang T., et al. *Do NOT Think That Much for 2+3=? On the Overthinking of Long Reasoning Models*. ICML 2025.  \n\n[2] Cuesta-Ramirez J., Beaussant S., Mounsif M. *Large Reasoning Models are Not Thinking Straight: On the Unreliability of Thinking Trajectories*. arXiv:2507.00711, 2025."}, {"review_id": "P7SJ3V4Hnb", "reviewer": "Reviewer_yG8D", "summary": "The paper first proposed four controlled game-based tasks that contains both standard gameplay conventions and newly defined constraints. Then, it shows that chain-of-thought (CoT) prompting can hurt the inductive performance of reasoning models through these tasks and presents a theoretical framework that explains how reasoning steps amplify errors through different types of CoT steps. The paper also introduces interventions that can improve inductive accuracy.", "questions": "- I noticed models like GPT-4o also generates chain-of-thoughts reasoning when using them. Why is “their multi-step traces may not help” for inferring the hidden rules, rather than “the procedure of training the reasoning models may not help”? Maybe it is harder for them to adapt the models to new rules because they were heavily trained to stick on the existing rules? Is it possible that for popular games like blackjack or Texas Hold’em, the reasoning model are trained on them heavily by RL, therefore it is harder for them to adapt to new rules?\n- In equation 3 the $\\gamma_k$ is in (0, 1), given the presence of self-reflections/corrections in these reasoning models, should the $\\gamma_k$ be in (-1,1) instead?\n- In figure 3, where does it reflect the improvements of summarization as indicated in line 332-333? The color is not that easy to tell apart. The only blue part that I see are in Chess for NR4 GPT-4o, but on top of the summarization blue it is still GPT-4o colored blue? I am very confused. Please improve the visual a bit.\n- I don’t think the word “counterparts” is appropriate when contrasting the performance between models like o3 and GPT-4o. Calling them \"counterparts\" implies they're the same model with/without reasoning, since there’s no public evidence that one equivalently trained / based-on another like R1 vs V3.\n- In appendix K, the anonymous link only contains a README file.", "limitations": "No limitations are addressed. Please include the limitations of the problem setup (four game-based tasks), the evaluation bias (only uses GPT-4o), and the hypothesis.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "## Strengths\n\n- The paper embeds hidden self-defined rules in addition to the normal rules in common games, and finds that reasoning models often underperform non-reasoning models on inferring the hidden rules is interesting. This directly contradicts the common belief that \"more reasoning = better performance.”\n- The paper provides a theoretical framework that decompose chain-of-thought reasoning into different operations. They also defines different reasoning errors:incorrect sub-task decomposition, incorrect sub-task solving, and incorrect final answer summarization, using the theoretical framework.\n\n## Weaknesses\n\n- This paper heavily rely on GPT-4o as a judge, potentially introducing bias towards certain types/answer styles. Even small scale of human evaluation can help.\n- The paper has a hypothesis of “the base model has fewer CoT while the reasoning model has more CoT“. However, equation 5 states that either under- or over- reasoning can incur this summary error. Section 4.3 empirically classified such errors, but only for the reasoning models. No analysis of base models' reasoning processes and errors. The hypothesis is stated but doesn’t seem to be proved?\n- There’s no comparison against other reasoning method. A proper comparison could be GPT-4o (or non-reasoning models) + explicit CoT instructions, or GPT-4o + tree-of-thoughts. These setting might match the claims better, as the base model remains the same, only the amount or style of reasoning changes.", "was_revised": true, "final_justification": "The author's rebuttal addresses most of the questions I raised.", "rebuttal": "Thank you for the thoughtful review. We are encouraged that you found the performance gap between reasoning and non-reasoning models on hidden rules to be interesting, and we appreciate your recognition of how this challenges the common assumption that more reasoning always leads to better performance. We believe the mentioned weaknesses and questions can be sufficiently addressed.\n\n**W1. This paper heavily relies on GPT-4o as a judge, potentially introducing bias towards certain types/answer styles.**\n\n**A1.** We appreciate the reviewer’s concern about potential circularity in using GPT-4o for evaluation. Motivated by your comment, we conducted a human validation study on 100 GPT-o3-generated rule descriptions across all four games. Three independent annotators—blind to model identity and GPT-4o’s decisions—judged semantic alignment with the ground truth. We then compared their majority vote with GPT-4o’s decisions and computed **Cohen’s κ** to measure agreement. The results are summarized below:\n\n| Judgment Type     | GPT-4o vs Human        |\n|-------------------|------------------------|\n| Aligned (60 cases)    | 60 / 60 (100%)         |\n| Not Aligned (40 cases) | 40 / 40 (100%)         |\n| **Overall Accuracy**  | **1.00**               |\n| **Cohen’s κ**         | **1.00**               |\n\nGPT-4o’s assessments matched the human majority in all 100 cases (60 aligned, 40 not aligned), yielding an accuracy and κ of 1.00. While this perfect agreement may in part reflect the limited sample size, it provides strong evidence that GPT-4o’s evaluations are well-aligned with human judgment in this context.\n\n**W2. The paper has a hypothesis of “the base model has fewer CoT while the reasoning model has more CoT.” However, Equation 5 states that either under- or over-reasoning can incur this summary error. Section 4.3 empirically classified such errors, but only for the reasoning models. No analysis of base models' reasoning processes and errors. The hypothesis is stated but doesn’t seem to be proved?**\n\n**A2.** We would like to clarify that our paper does not formally claim “base models have fewer CoT.” Our actual hypothesis is: *reasoning models’ multi-step traces may not help—and can introduce incorrect assumptions or misleading intermediate steps* (see lines 157–158). This is supported both theoretically (Section 4) and empirically (Section 5). Base models like GPT-4o, when prompted directly via API, do not produce intermediate reasoning traces and are not considered “reasoning models” unless explicitly instructed. Therefore, our failure mode analysis in Section 4.3 focuses on models that generate such reasoning traces (e.g., GPT-o3, DeepSeek-R1).\n\n**W3. There’s no comparison against other reasoning method.**\n\n**A3.** We appreciate the suggestion to compare against explicit CoT prompting. In response, we conducted additional experiments applying a CoT-style instruction to GPT-4o:\n\n> *“Let’s think step by step. First, extract the rule from the transcript. Then explain your reasoning.”*\n\nThis prompt was applied to each rule inference task in the Chess game, and we compared performance with and without CoT prompting:\n\n| Method         | SR1  | SR2  | SR3  | SR4  | SR5  | SR6  | NR1   | NR2   | NR3   | NR4   | NR5   | NR6   |\n|----------------|------|------|------|------|------|------|--------|--------|--------|--------|--------|--------|\n| GPT-4o w/o CoT | 56.7| 35.3| 28.0| 68.0| 18.0| 42.0| 100.0 | 100.0 | 100.0 | 94.0  | 100.0 | 100.0 |\n| GPT-4o w/ CoT  | 58.0 (+1.3) | 34.0 (-1.3) | 27.3 (-0.7) | 68.00 (+0.0) | 20.0 (+2.0) | 42.0 (+0.0) | 100.0 (+0.0) | 100.0 (+0.0) | 100.0 (+0.0) | 97.3 (+3.3) | 100.0 (+0.0) | 100.0 (+0.0) |\n\nOverall, CoT prompting did not yield consistent improvements. On SR rules, performance was largely unchanged or slightly decreased, suggesting that added reasoning may not help—and can sometimes distract from inferring hidden rules. On NR rules, performance was already saturated.\n\nIt is worth noting that this CoT-style prompting differs from models like GPT-o3, which generate structured, multi-step reasoning (e.g., decomposition → solving → summarization). Injecting CoT alone does not reproduce or mitigate the same failure patterns. We will include these results in the revised version.\n\n**Q1. I noticed models like GPT-4o also generate chain-of-thought reasoning when using them. Why is “their multi-step traces may not help” for inferring the hidden rules, rather than “the procedure of training the reasoning models may not help”?**\n\n**A1.** While models like GPT-4o may occasionally produce CoT-like outputs—especially in chat-based interfaces—these traces do not appear when accessed via API, which is the setting used in our experiments.  \nOur study focuses on the impact of multi-step reasoning traces at **inference time**, independent of the model’s training procedure. We find that the observed failure modes emerge even when reasoning is introduced purely via prompting in open-ended inductive tasks [1]. This suggests the issue lies more in the **structure of the reasoning process itself**, rather than in how the model was trained.\n\n**Q2. Maybe it is harder for them to adapt the models to new rules because they were heavily trained to stick on the existing rules? Is it possible that for popular games like blackjack or Texas Hold’em, the reasoning models are trained on them heavily by RL, therefore it is harder for them to adapt to new rules?**\n\n**A2.** We appreciate this line of reasoning. It is indeed plausible that reasoning-augmented models carry inductive biases from training on well-known games or standard reasoning formats, which may limit their adaptability in unfamiliar or rule-shifted settings.   However, our study does not aim to attribute failure to any specific training regime, as we lack access to the proprietary reinforcement learning or fine-tuning pipelines used by closed models. Instead, our findings isolate a distinct inference-time effect: even when reasoning is introduced via prompting, multi-step traces can constrain model behavior and lead to systematic errors in rule induction. These effects are consistent across tasks and model families, and we would be glad to incorporate both perspectives into the revised discussion.\n\n**Q3. In equation 3 the $\\gamma_k$ is in (0, 1), given the presence of self-reflections/corrections in these reasoning models, should the $\\gamma_k$ be in (-1,1) instead?**\n\n**A3.** In our paper, $\\gamma_k \\in (0, 1)$ is intended to model **bounded rational belief updates** rather than full reversals. Even when reasoning models engage in self-correction, their belief adjustments tend to be **gradual refinements** rather than full negations. Negative $\\gamma_k$ would imply a reversal in belief direction, which is not typical in observed reasoning traces. We clarify this design choice further in **Appendix E**.\n\n**Q4. Where does it reflect the improvements of summarization as indicated in line 332–333?**\n\n**A4.** The blue bar in Chess NR4 reflects GPT-o3, not GPT-4o. In this case, GPT-o3 achieves 100% accuracy without intervention, but drops slightly to 97.33% under the summarization constraint—hence the darker blue segment appears below the baseline, indicating a small decrease rather than a gain from summarization.\nWe recognize that in some cases, the stacked figure may obscure intermediate effects, especially when solving contributes the most visible improvement. For convenience, we include a numeric ablation table below to directly reflect the summarization gains on Chess game.\n\n| Model                   | SR1           | SR2           | SR3           | SR4           | SR5           | SR6           |\n|------------------------|---------------|---------------|---------------|---------------|---------------|---------------|\n| GPT-o3 w/o Interventions | 21.3          | 18.0          | 4.7           | 34.7          | 13.3          | 10.7          |\n| + Decomposition         | 22.7 (+1.3)   | 24.7 (+6.7)   | 10.0 (+5.3)   | 44.7 (+10.0)  | 14.0 (+0.7)   | 18.0 (+7.3)   |\n| + Solving               | 48.0 (+26.7)  | 42.0 (+24.0)  | 46.7 (+42.0)  | 60.7 (+26.0)  | 27.3 (+14.0)  | 46.0 (+35.3)  |\n| + Summarization         | 40.0 (+18.7)  | 26.0 (+8.0)   | 26.0 (+21.3)  | 40.0 (+5.3)   | 21.3 (+8.0)   | 27.3 (+16.7)  |\n| + Combined              | 52.0 (+30.7)  | 52.7 (+34.7)  | 54.0 (+49.3)  | 64.0 (+29.3)  | 31.3 (+18.0)  | 46.0 (+35.3)  |\n\nFull table will also be included in the appendix of the revised version to improve overall readability.\n\n**Q5. Word usage of “counterparts”.**\n\n**A5.** Thank you for the suggestion. We agree that “counterparts” may be ambiguous and will revise the phrasing to use “models from the same time period” instead.\n\n**Q6. Appendix K**\n\n**A6.** We appreciate the reviewer highlighting this issue. The problem with the anonymous link in Appendix K has been resolved, and the full codebase is now accessible.\n\n**L1. No limitations are addressed. Please include the limitations of the problem setup (four game-based tasks), the evaluation bias (only uses GPT-4o), and the hypothesis.** \n\n**A1.** We appreciate the reviewer’s point. We will include a dedicated limitations section to clarify scope and assumptions. Our benchmark is restricted to four symbolic games. While our primary evaluation used GPT-4o, we validated its judgments against independent human annotations. Finally, our study focuses on prompting-based reasoning models, and we agree that the generality of the findings across other reasoning paradigms remains an open question.\n\n**References**\n\n[1] Turpin M, Michael J, Perez E, et al. *Language models don't always say what they think: Unfaithful explanations in chain-of-thought prompting*. NeurIPS 2023."}, {"review_id": "9tqrNeeroj", "reviewer": "Reviewer_KF7a", "summary": "This paper examines the inductive capabilities of LLM by employing four meticulously designed games. The findings reveal that LRM exhibit inferior performance in capturing inductive rules compared to LLM. Furthermore, the introduction of irrelevant or misleading CoT can even negatively impact LRM’s performance. Through human evaluation, the authors identified the primary types of reasoning failure. Consequently, the authors proposed a targeted intervention to effectively address these issues.", "questions": "n/a", "limitations": "n/a", "rating": 3, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strength:\n\n* This is an interesting finding that LRM shows poorer performance compared to LLM  when evaluating the inductive ability through specially designed games\n\n* The authors introduced several temps of reasoning failures through aberration to help identify the issues by decomposing the failures into a few core types. \n\n* The authors proposed solutions to address the issue via targeted intervention. \n\n\nWeakness:\n\n* The authors should provide a bit more examples to motivate audience and justify the value of the work., making it more clear that the issue exists, not due to some specific designed games or data patterns. \n\n* The authors should provide a bit more details, such as a  concrete example, especially about the specifically designed games that help evaluate the performance of the models, so that audience can better understand  the assessment processes, and therefore understand  how the inductive ability is quantitatively measured when reading Figure 2 in Section 3.3 .\n\n* It is better having a table of the notations. In Theorem  4.1,  the minimiser N* was used without proper definition, which can cause some confusion or extra efforts to tell.\n\n* The experimental analysis can be enhanced to show more insights of the proposed techniques.", "was_revised": true, "final_justification": "I appreciate the additional input from the authors that helps partially. Considering the over case, I will retain my score for the paper.", "rebuttal": "Thank you for your thoughtful and encouraging feedback. We are glad that you found our main empirical finding—the weaker inductive performance of reasoning-augmented LLMs compared to non-reasoning models in specially designed games—to be interesting. We have carefully considered your concerns and respond to them individually below.\n\n**W1. The authors should provide a bit more examples to motivate audience and justify the value of the work, making it more clear that the issue exists, not due to some specific designed games or data patterns.**\n\n**A1.**  Illustrative failure cases are provided in **Figure 1(b)**. Additional examples and error breakdowns appear in **Appendices G, H, and I**, spanning a range of task settings.  \n\nAs discussed in **Section 4.3**, similar failure patterns arise across all four games, which differ in symbolic structure and inductive demands. This recurrence suggests that the observed issues are not tied to any single task design.  \n\nWhile these examples are drawn from games—where model behavior can be measured in a controlled setting—related failures have been noted in broader contexts as well, such as those in mathematical problems [1,2].\n\n**W2. The authors should provide a bit more details, such as a concrete example, especially about the specifically designed games that help evaluate the performance of the models.**\n\n**A2.**  We refer the reviewer to **Figure 1**, which presents a concrete example illustrating the benchmark setup, rule definitions, and model inference transcript. This figure is intended to clarify both the structure of the games and the evaluation process. The four games (Chess, Texas Hold’em, Dice, and Blackjack) each include a Normal Rule (NR) reflecting standard gameplay and a Special Rule (SR) introducing a hidden constraint.\n\nFor example, in **Figure 1(a)**, the SR specifies that “*five consecutive primes > three of a kind*,” overriding conventional poker rankings. As shown in **Figure 1(b)**, models are tasked with **inferring each rule from game transcripts**. The accuracies reported in **Figure 2** are computed per rule and reflect the model’s ability to recover these rule structures from limited data.\n\n**W3. It is better having a table of the notations.**\n\n**A3.**  The variable $N^\\star$ is defined at **Line 217** and proved beginning at **Line 620**. For ease of reference, we provide the following **notation table** summarizing all variables used in Section 4, including the formal definition of the optimal reasoning length $N^\\star$ referenced in Theorem 4.1:\n\n| Symbol          | Definition                                                                 |\n|-----------------|----------------------------------------------------------------------------|\n| $x_k$           | Reasoning state at step $k$, composed of belief $m_k$ and mode $s_k$       |\n| $m_k \\in \\mathbb{R}^d$ | Model’s belief about the correct answer at step $k$                    |\n| $s_k \\in \\{\\text{NEEDQ}, \\text{NEEDA}, \\text{FINISH}\\}$ | Reasoning mode at step $k$                        |\n| $y^* \\in \\mathbb{R}^d$ | Ground-truth solution the model is trying to infer                  |\n| $e_k := m_k - y^*$ | Belief error at step $k$                                               |\n| $\\alpha_k \\in [-1, 1]$ | Task alignment scalar — how well the sub-task focuses on the target |\n| $\\varepsilon_k \\sim \\mathcal{N}(0, \\sigma^2 I_d)$ | Answer noise at step $k$                              |\n| $\\gamma_k \\in (0, 1)$ | Step size controlling belief update integration                     |\n| $g_k$           | Evidence vector from sub-task resolution at step $k$                      |\n| $N \\in \\mathbb{N}$ | Total number of reasoning steps                                        |\n| $N^\\star \\in \\mathbb{N}$ | Optimal reasoning length minimizing expected error $\\mathbb{E}[\\|e_N\\|^2]$ |\n| $\\bar{\\alpha}$  | Expected question alignment $\\mathbb{E}[\\alpha_k]$                        |\n| $E(N)$          | Expected squared error after $N$ steps                                    |\n| $\\Delta(N)$     | Additional error due to variance in alignment                             |\n| $b_0 = \\|m_0 - y^*\\|^2$ | Initial belief error   |\n\n**W4.The experimental analysis can be enhanced to show more insights of the proposed techniques.** \n\n**A4.**  Our analysis reveals how each proposed intervention addresses distinct reasoning failure modes: summarization mitigates hallucinated rule statements, decomposition reduces error propagation across subtasks, and solving directly corrects invalid inferences. These effects are quantitatively shown in Figure 3. Together, the components yield complementary gains across tasks and models, with solving producing the most robust improvements on inductive failures. We believe the current analysis offers insight into both the sources of failure and the mechanisms through which each technique improves inductive robustness.\n\n\n**References**  \n[1] Chen X, Xu J, Liang T, et al. *Do NOT Think That Much for 2+3=? On the Overthinking of Long Reasoning Models*. Proceedings of the 42nd International Conference on Machine Learning (ICML), 2025. \n \n[2] Cuesta-Ramirez J, Beaussant S, Mounsif M. *Large Reasoning Models are Not Thinking Straight: On the Unreliability of Thinking Trajectories*. arXiv preprint arXiv:2507.00711, 2025."}]

Author rebuttal:

Meta-review: The paper evaluates reasoning and non-reasoning models on certain inductive tasks – identifying secret rules for modified versions of games like chess, black jack etc.. It finds that reasoning models perform worse than non-reasoning counterparts in identifying newly constructed rules, even though they are better on the standard rules. The paper hypothesizes that error accumulation in reasoning models leads to these failures. It identifies three key failure modes and manual inspections of model outputs are used to estimate the failure rates for these. The paper also provides a mathematical formulation for how these can negatively affect thinking models, akin to a mental model. Finally the paper proposes certain interventions that tackle these failure modes, through better prompting, few-shot examples and strict budget forcing. These interventions mitigate the error propagation and lead to better performance on these tasks.

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The reviewers appreciated the finding that thinking models can be worse on the tasks considered, and the controlled experimental setup that helps identify the failure modes. The mathematical formulation, while very simplified, provides a decent mental model for how thinking models might propagate errors. There were few concerns raised in the discussion period, but the author's response seems to have addressed them.

R1: version mismatch in reasoning and non-reasoning models?, intuitive illustrations for theory, ablations for individual interventions, other tasks — *Addressed by re-evaluating on other versions, presenting ablations as a table and showing the benefit of interventions on AIME*

R2: title is misleading, story is more about “guided reasoning > vanilla reasoning”, no inter-rater agreements for annotators reported — *Addressed by changing title and including inter-rater correlation*

R3: heavily rely on GPT-4o as a judge, CoT baseline, unclear visuals — *Addressed by showing agreement between GPT-4o and human annotators, adding CoT baseline*

R4: clearer presentation of results, more examples for motivation, — *Addressed by pointing to figures in the paper*

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There are still some typos in the paper and the main figures are very hard to parse (reviewers pointed this out too). Furthermore it is not clear if these issues are specific to inductive reasoning or other tasks too. The newly proposed title addresses this to some extent. Overall the paper makes a positive contribution in understanding limitations of thinking models and provides some mitigation strategies that could inspire more research. The recommendation is to accept the paper, and the authors are encouraged to change the title to the new one.
Decision: Accept (poster)