Learning to Reason via Mixture-of-Thought for Logical Reasoning
Authors: Tong Zheng
Lichang Chen
Simeng Han
R. Thomas McCoy
Heng Huang
Venue: ICLR 2026 Poster (2026)
Keywords: Logical Reasoning
Self-evolving Training
Large Language Models
Parallel Scaling
Test time scaling
Primary area: foundation or frontier models, including LLMs

Abstract: Human beings naturally utilize multiple reasoning modalities to learn and solve logical problems, i.e., different representational formats such as natural language, code, and symbolic logic. In contrast, most existing LLM-based approaches operate with a single reasoning modality during training, typically natural language. Although some methods explored modality selection or augmentation at inference time, the training process remains modality-blind, limiting synergy among modalities. To fill in this gap, we propose Mixture-of-Thought (MoT), a framework that enables LLMs to reason across three complementary modalities: natural language, code, and a newly introduced symbolic modality, truth-table, which systematically enumerates logical cases and partially mitigates key failure modes in natural language reasoning. MoT adopts a two-phase design: (1) **self-evolving MoT training**, which jointly learns from filtered, self-generated rationales across modalities; and (2) **MoT inference**, which fully leverages the synergy of three modalities to produce better predictions. Experiments on logical reasoning benchmarks including FOLIO and ProofWriter demonstrate that our MoT framework consistently and significantly outperforms strong LLM baselines with single-modality chain-of-thought approaches,
achieving up to **+11.7pp** average accuracy gain.
Further analyses show that our MoT framework benefits both training and inference stages; that it is particularly effective on harder logical reasoning problems; and that different modalities contribute complementary strengths, with truth-table reasoning helping to overcome key bottlenecks in natural language inference.

Published as a conference paper at ICLR 2026
LEARNING TO REASON VIA MIXTURE-OF- FOR LOGICAL REASONING
Tong Zheng'; Lichang Chen' Simeng Han", R. Thomas McCoy', and Heng Huang'
Department of Computer Science, University of Maryland, College Park, MD, USA
Dept. of Computer Science, Yale University, New Haven, CT 06520 Dept. of Linguistics, Yale University, New Haven, CT 06520
ABSTRACT
Human beings naturally utilize multiple reasoning modalities to learn and solve log- ical problems, i.e., different representational formats such as natural language, code, and symbolic logic. In contrast, most existing LLM-based approaches operate with a single reasoning modality during training, typically natural language. Although methods explored modality selection augmentation at inference time, the training process remains modality-blind limiting synergy among modalities. To fill in this gap, we propose Mixture-of- -hToughhh (MoT), a framework that enables LLMs to reason across three complementary modalities: natural language, code, and newly introduced symbolic modality, truth-table, which systematically enu- merates logical cases and partially mitigates key failure modes in natural language reasoning. MoT adopts a two-phase design: (1) self-evolving MoT training, which jointly learns from filtered, self-generated rationales across modalities; and (2) MoT inference, which fully leverages the synergy three modalities to produce better predictions. Experiments on logical reasoning benchmarks including FOLIO and ProofWriter demonstrate that our MoT framework consistently and signifi- cantly outperforms strong LLM baselines with single-modality chain-of-thooe approaches, achieving up to +1 1.p.pp average accuracy gain. Further analyses show that our MoT framework benefits both training and inference stages; that it is particularly effective on harder logical reasoning problems; and that different modalities contribute complementary strengths, with truth-table reasoning helping to overcome key bottlenecks in natural language inference.
INTRODUCTION
Large language models (LLMs) have demonstrated remarkable progress in logical reasoning tasks, especially propelled by methods like Chain-of-Thoughee (CoT) prompting (Wei et al. 2022). However,
these CoT approaches predominantly rely single reasoning modality, i.e., natural language, even when employing ensemble methods (Li et al.] [2023] Wang et al., 2025 2022 Brown et al. 2024
Snell et al. 2025 Liang et al. 2023). Here we refer modality as distinct thought paradigm (e.g.
natural language, symbolic, or code), which differs in representation and inference process. On the
other hand, neuro-symbolic methods (Pan al. 2023 Olausson et al. 2023 Ryu et al. (025) utilize
LLMS as translators and delegate reasoning to external symbolic solvers. Recent work combines CoT with symbolic reasoning via either selecting a single modality per instance (Xiong al., 2024) or augmenting one modality with the other while keeping reasoning confined to symbolic (Xu et al. 2024a) or natural language (Liu et al. 2024p. These methods combine modalities only during inference and ignore the synergy of different modalities during training, thus failing to fully exploit the complementary strengths of different reasoning modalities.
This limitation contrasts sharply with human cognition: Humans naturally employ multiple reasoning modalities, flexibly switching among natural language explanations, code-based procedural thinking,
"Co-first authors.
'ee use the terms thought paradigm and reasoning modality interchangeably.

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Published as a conference paper at ICLR 2026
40
Solve Rate (%)
30
Truth Table
Solve Rate (%)
80
Error Type
CS
20.5
NL Col
FM
20.5
20
70
MB
29.6
10
Code
IC
36,4
60
FOLIO ProofWriter
FOLIO
ProofWriter
20
40
# Benchmark
# Benchharrk
Percentage (%)
(a) Uniqueness Ratio
(b) Coverage Upper-Bound
(c) Error Distribution of NI mode
Figure 1: (a) 1eee--2---B--nttneee solves ~20% of FOLIO and =355 of ProofWriter exclusively per paradigm.
(b) MoreeNNnttuuttbbble yields higher upper- bound coverage than code+NL alone (Xiong et al. 2024). (c) In
NL modes, invalid converee (IC) and missing- branch (MB) emrors comprise =66% of failures (CS: commonsense injection; FM: factual misquote). Percentages sum types. We provide illustrative examples in Appendix H.T
missing-branch
than 100% hecause some cases exhibit multiple error
and formal symbolic manipulation, both when learning complex logical skills and when solving novel problems (Newell et al.) 1972. Gentner, 1983 Larkin & Simon 1987 Goldin 1998). This cognitive versatility, the ability to represent and process information in diverse formats, is crucial for robust reasoning. Current LLMs, largely confined to single-modality training and inference, lack this flexibility. It raises a critical question: Can LLMs achieve more robust and versatile logical reasoning by explicitly learning to operate across multiple complementary reasoning modalities?
Addressing this question requires tackling two challenges: 1) It is still unclear which reasoning modalities should be included; the selected modalities must be complementary to make joint learning worthwhile. 2) Teaching an LLM with multiple modalities is non-trivial, as large aligned reasoning trajectories scarce. Without those high-quality reasoning trajectories, cannot teach LLMs to reason via multiple modalities. Our investigation reveals crucial insights for modality selection.
Natural language bottleneck. Figure [ (c) shows that nearly two thirds of CoT errors arise from missing branches and invalid converse, i.e., poor exhaustive enumeration and complex inference (See examples in Appendix H.I.. Truth-table reasoning, which systematically lists all possibilities, naturally complements this weakness; therefore, we incorporate symbolic truth-table paradigm.
Code-NL complementarity. Inspired by HybridMind (Yue et al. [2024] Xiong et al. 2024), where they show preliminary results that a code paradigm could complement NL reasoning, we also incorporate code one reasoning modality into framework.
Paradigm overlap & uniqueness. Figure (a-b) shows that 35.8% of ProofWriter items and 16.7% of FOLIO items are solved by exactly one paradigm, while the union of three reasoning modalities covers up to 85% of all instances -eineeee-eeddeneee that combining NL, code, and truth-table reasoning is necessary, outperforming the simple combination of code and NL (Xiong et al.) 2024).
Building on these insights, we propose Mixture-of Thoughh (MoT), a human-inspired framework that enables L.LMs to reason via three complementary reasoning modalities: NI., code, and symbolic; an example is shown in table illustrate each modality. Notably, we introduce a new truth-table- symbolic reasoning where LLMs ground propositional variables, construct partial truth table by pruning assignments that violate any premise, and infer the final answer by checking the truth table. Our MoT consists of two parts. One part is training: we propose self-evolving MoT training algorithm, which improves the model's reasoning ability in each modality through joint iterative optimization (Figure (a)). Another part is inference, where we generate responses under each modality and leverage a voting mechanism to produce the final answer (Figure (b)). This simple strategy allows the model combine diverse perspectives and make robust predictions.
Empirically, we show that across three base Gereel--GGmmm2222B--1 Gemma-2-99-HB and Qwen- 2.5-7B-Instruct- MoT consistently surpasses the CoT baseline on Proo/Writer (Tafjord et al. 2021] and FOLIO (Han et al. 2024), with an average accuracy gain of up to +11.7pp. Notably, our B-pprameter MoT matches the results of GPT + Logic-LM on FOLIO. Additional analyses show that 1) MoT training outperforms single-though training; 2) Mixture-of of-Thought sampling yields a higher oracle upper bound than ungle-thought sampling under the same inference budget 3) The gains grow with problem difficulty: MoT helps most depth-5 and other harder problems; and 4) A fine- ain--grained error study reveals a key natural-language bottleneck, i.e., missing branches and frequent invalid converse errors, while the truth-table paradigm help resolve some cases of exactly these types.
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Published as a conference paper at ICLR 2026
Table 1: Illustration the three complementary reasoning modalities, i.e., natural languagg CoT. code-based
reasoning, and rutthttbbl reasoning. We provide the corresponding outputs of LLMs in section
Peter Parker eitber superhero or civilian. The Hulk wakes up when he is angry. If be wakes he will break a bridge. If destroyer breaks a hridge, Peter civilian. Peter wears uniform when he a superhero. Thor god. god not destroyer. Thor break bridge when he is huppy. Thar huppy, the Hulk angry. Question: If Thor is happy, does Peter Parker wear a uniform?
Premise:
Options: (A) True (B) False (C) Uncertain
Natural Language Reasoning
Code-Based Reasoning (Abstruct)
Truth Table Reasoning
<pa__co>>
<cods>> class Bulk:
<eruth_tabic>
Step given peamises Ther huppy, Hulk
Let: happy, Hulk angry, wakes up. B = bridge breaks, C' Peter is civilian, S supethern U = wean uniform.
is anary"" and "The Hulli wakes up when he nn- ary we can know "If Ther happy. then hulk
det -nitt_ andry):
sot wakos_up(celet
wakes up."
daf breaks_ (solr):
class Thor:
Step given premise "Peter Purker cither supethero or civilian"" and derivasion "If Thor happy, then Peser Purker
set _init_
happyl:
SVC. A.A B C.
B.
dof broakka_bridge
civilian", can know Thor happy, then Parker supechaso."
class PazeeParker:
del _init_ (welf, 1epxapezbero]
Logical
(usomme Truel:
T = H o A
Step 6 given premise "Poter Parker wears a uni- form when
sef : ...
B = -C is C = False
superhero." and derivation "If is supethero.".
def apply_preniswa (Llior,,, bulk, pecoe]:
VVC 8 True U True
Thar happy, then Peter we can know "If Thor is happy, then Peter Parker
"un_innferoncetther 201s, peter): Truth Table:
wears a unifoo""
<end_o [_nt__ot>
def thesi_oonacuuscnnt...
T H A
5
<ansxor>
ther Thor (hqppy-Trus) bulk Hulk (aiggssyyeF)alao)) pater
Answer: (A)
True True True True False Tree True
<end C_nnswe2
PotorPazkez (...)
reaull check pomec__ooocleiint....
sond_of_truth__ <enswer>Ansern ott_on__o___oecE___
<end at code> Censver>Aneerr (A)kend of
2 THE MIXTURF-OF-THOUGET FRAMEWORK
In this section, we introduce 1) three complementary reasoning modalities for logical reasoning (Sec. 2.T : 2) our self-evolving training framework that jointly improves these reasoning modalities (Sec. inference strategy that combines diverse but complementary reasoning paths derive robust final predictions (Sec.
2.2]: and 3) our mixture-of-thowee
2.1 HUMAN-INSPIRED COMPLEMENTAR REASONING MODALITIES
Drawing inspiration from human cognition and error analysis in Figure [ we argue that no single reasoning modality suffices for all logical challenges. Therefore, we equip a single model with three complementary modalities: natural language CoT, code CoT, and truth table COT. Specifically, because natural- language CoT often misses branches or makes invalid-convery errors, we design a truth-table approach that explicitly enumerates truth assignments and thus complements these weaknesses. Table T illustrates how the three modalities solve representative problem.
The model explains its reasoning in plain natural language, decomposing the problem into hiep-by-ster justifications. This format is flexible and interpretable.
Natural Language CoT:
Code CoT: The model first transforms logical problem to PYTHON code and then derives the answer based on the PYTHON code. We do not execute the code; instead, we treat it as a way to describe logic in structured form.
Truth Table CoT: The model first explicitly generates a truth table by defining predicates based on the premises and conclusion, then enumerating possible truth assignments, and finally checking which ones satisfy the conclusion.
These complementary modalities are jointly exploited in our self-evolving training (Sec. 2.2 and majority-vote inference (Sec. 2.3 We now detail the design of the Truth Table CoT approach.
Truth-Table CoT: Challenges and Design. Two main challenges arise when enabling LLMSs to reason with truth tables: 1) Exponential blow-up: the number of rows grows exponentially with the propositional variables, easily exceeding the context window and compute budget; 2) First-order grounding: practical tasks are given in first-order logic; one must ground variables, select a finite predicate set, and still ensure that the resulting (partial) truth table remains tractable. To address these challenges, we propose a two-step strategy: (i) grounding. which instantiates first-order formulas into finite set of propositional predicates (Clarke et al. 2001; Wittocx et al., 20100), and (ii) reason to prune, which eliminates rows that violate any premise through reasoning via LLMss
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Published as a conference paper at ICLR 2026
Final Output :A
MoT training
Mixture
High. quality data
Majority Yoting
A
Self- Self-Trainin
Finetune
Quality Check
Ni. Got
Code
Trun Tacle
Generate
Rationales
Question
NR. Cat
Code
Truth Tibble
Training Phase
(b)
Figure Illustration of our MoT Framework. (a) Training phase with three key steps: 1) Rationale Generation where given an initial seed dataset, LLM generates rationales across reasoning modalities (NL, Code, and Truth Table); 2) Quality Checking and Merging where generated rationales are checked for correctness and format consistency. then merged into high-quality MoT training data; and Finetuning where the model trained using the MoT data. These steps iteratively repeals, forming a self-evolving training cycle. (b) Inference phase: the trained model generates outputs for each reasoning modality and applies majority voting to yield the final prediction (e.g., A).
Finally, the LLMs derive the final output with the following rule: True it every surviving assignment satisfies the conclusion, False if none do, and Uncertain otherwise. Moreover, assign modality-specifet tags (e.g., <code> ... send_of_code>. to explicitly indicate the format during training and inference. The prompts are detailed in Appendix
keeping partial truth table (see Table T and Appendix G.3
2.2 SELF-EVOLVING MIXTURE-OF-THOUUE TRAINING
Explicitly learning to reason across multiple complementary modalities, such as natural language, code, and symbolic truth tables, is non-trivial. A key challenge lies in the lack of annotated reasoning trajectories for each modality, especially for our newly introduced truth-table approach. Collecting labeled CoT traces for all of these modalities is also costly and often infeasible. To address this, propose a self-evolving MoT training approach, which enables the model to operate across multiple complementary reasoning modalities by iteratively learning from its generated reasoning traces.
Given the policy M, our goal is to maximize the following objective across the problems x and modalities T (NL, Code, TruthTable):
E (xii)) T-T (et (1 RR2, oi ye: t)],
where D (2,, p: is the dataset with problem x; and corresponding ground-truth yı of and of be model-generated reasoning trace/answer with modality t for i-th problem. To elicit the reasoning modality t. we design a small few-shot example set &, for each t, and prepend the exemplar from the set each problem vs. Conditioned on (wi, &x), 2; is sampled from policy M, followed by the prediction the final answer ÿt. is the reward function and the design is detailed in the following.
Reward Function R. In preliminary experiments, we observe mismatch between tags and reasoning traces. This error leads to performance degradation, as different modalities negatively interfere with each other. Notably, this error is especially prevalent in the code modality. We define the reward as:
[1. y= 9 isValid
R(z v:
0, otherwise,
where the Valid function checks the format consistency by two standards: a) each trace should correctly include its modality's 's structural tag (e.g., kendLof.nl.cot for nl) and b) for code traces, ensuring the presence of both valid function definition (def) and a class definition (class); Following (Zelikman et al. 2022), we also filter out the traces with incorrect answer. We do not perform step-level verification of intermediate reasoning, this would require additional external tools, e.g., LLMs judge (Zheng et al., (223)) and greatly slow down training. Instead, we find that simple checks (e.g., modality tags, basic code structures) already yield significant performance gains.
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Published as a conference paper at ICLR 2026
Training. We conduct multiple rounds of self-evolving training until performance saturates. Mn used to denote the policy in the n-th round with trainable parameters 8n. Leveraging the volicy-gradicn (Sutton et al. 1999) trick,
easily obtain the gradient of eq. ① as
[R(=] Uh ys; t) Ve, 1 log Mn- (2 of T. t, E.)]
(3)
In our current setting, the reward is binary (Eq. 2). This reduces the framework to a degenerate case of RL that can be equiva- lently seen as rejection sampling plus su- pervised finetuning, closely related to self- evolving or STaR training (Zelikman et al. 2022). Algorithm and Figure| 2 illus- trate our multi-round training procedure. At round ni, we prompt the model Mn-1 to generate reasoning trace 2 and pre- dicted answer 9J for each x; across all rea- soning modalities E T (Line 4-9). It is worth noting that we use few-shot prompt- ing only in the first round (Line 7); once model has bootstrapped own reason- ing ability, all subsequent rounds run in zero-shot mode without additional exem-
Algorithm 1 Self-Evolving MoT Training
Input: an LLM M; dataset D ((r,, y:)
reasoning modality T = {NL, Code, TruteTable). Sampling times S, few-shot examples E {ENL, & &rruth
Output: Mixture of Thought enhanced model MN
1: Mo + M
2: for n to N do Initialize Dean
3:
all,n +- 0; NL. + 0: Dodee
Code,n
0: DD TruthTaable, <- 0
4:
for all t do
5:
for = 1 to D do
6:
if then
7:
2/ +
Mn (n
else
plars (Line 9). We retain sample only passes the quality filter (Line 11-13)
9:
2/, 5.0% + Mn=1(x;;t;30 end if
10:
and merge all surviving traces into Deca (Line 16). The updated model Mn, which is finetuned from Mn-1 on the filtered
11:
if R(2 0% t= then Disn < D ((x,, (1,)))
12:
13:
end if end for end for
dataset Daen all. (Line 17). Unlike Zelikman
14:
15:
et al. 2022), which restarts from the base model each round, our training proceeds
16:
all.n + Mix NL,n Code,n Table
een Deen
mn-aolicy-leariing from its own validated
17:
M, <- Train( Maain(Mn-a Mn- Deen
all,))
outputs. We demonstrate the effectiveness
18: end for
of this change in Appendix E.5
19: return MN
2.3 MIXTURE-OU-THOUGGE INFERENCE
To leverage the complementary strengths of three modalities, for each problem, we have three outputs corresponding to three modalities elicited by tagging, i.e., <nl_cot>, <code>, and <truth.table> then we apply majority voting over outputs to decide the final answer. In case of ties, randomly pick up the answer from one reasoning modality. We further explore the test-time scaling of MoT, and analyze its effectiveness in section 3.4
3 EMPIRICAL EVALUATIONS
3.1 EXPERIMENTAL SETUPS
To validate the effectiveness of our MoT, we select four widely-used LLMS across different sizes, reasoning abilities and model families: Qwen-2.55-BB-Instree 4B-Instruct (Yang et al. 2025) and models.
Models.
(Yang et al., 2024/Qwen-33 (Team et al. 2024) as base
Baselines. Our approach is kind of chain-of-thoughh approach. To this end, select baselines from two folds: 1) neuro-symbolic approach and 2) chain-of-though approach. In the first category. select Logic-LM (Pan et al.] 2023)
comparison. For the CoT approach, et al. 2022) as a comparison. Since these approaches heavily rely on strong cnatructionfolloowiii capabilities, directly cite their performance results from the original papers based on GPT-4.
select CoT (Wil
Dataset. We select two logical reasoning benchmarks: ProofWriter (Tafjord et al. 202T) and FOLIO (Han al. 2024) for evaluation. For ProofWriter, we select the hardest subset, which consists
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Published as a conference paper at ICLR 2026
Table Accuracy (%) on the FOLIO and ProofWriter benchmarks. Our MoT training consistently improves the performance of each base model. Applying MoT inference further enhances performance across both benchmarks and all models. @3 denotes Self-Consistency approach (Wang et al. 2022) with three votes. We provide full results of extra baselines (e.g,, LoT) in Appendix F.3
E,44 and robust analysis in Appendix
Model
Method Type
Reasoning Modality
FOLIO
ProofWriter
Avg
(A) Prior SOTA Approach/Modch
GPT-4
Logic-IM
78.9
79.7
79.3
CoT (Vamila)
70.6
68.1
69.4
(B) Base Model: Gemmaa-2-B-Ir
Jomma-2-2B-Jr (3-Shot)
Singe-Thought
Best (ul)
42.4
29.8
41.1
Gemman-224 2-2---2-BB- (3-Sho))
Thought
Best (al)
45.3
38.8
42.1
MoT (0 Shot)
Single Thought
Best
61.1
62.7
61.9
MoT @ 3 (D-Shot)
Single Songle-Thoughe Thought
Best
62.1
60.8
61.5
MoT (0-Shot)
Mixture of Thought
All
62.6
65.0
63.8
(C) Base Model: Gemma-2-99B-t
Gemma-2-9B-4 BBBItCC (3-shot)
Single-Thoue Thought
Best
69.5
61.2
65.4
Gemama-2 2-9B (3-xhol)
Thought
Best (nl)
72.9
62.7
67.8
MoT (O-shot)
SingleThougee
Best
76.9
69.5
73.2
MoT 3 (D (0oShot)
Single-Thouge
Best
75.4
70.5
73.0
MoT (0-shot)
Mixture-or Thought
All
78.9
70.7
74.8
(D) Base Model: Qwen2.5-78. 2an22.--B-Instruee
Qwen2. Instttttt (3-shot)
Single- Thought
Best (ul)
71.9
60.5
66.2
Qwen2.5-7B-BB 5-B-Issrrrttt @ 3 (3-shot)
Single-Thought
Best (ol)
73.4
65.8
69.6
MoT (O-shot)
Best
75.9
69.2
72.6
MoT @i 3 (D-shot)
Single-Thouge Thought
Best
77.3
70.5
73.9
MoT (0-shot)
Mixture-off Thaught
All
78.3
71.8
75,1
(E) Base Model: (ween3-4--Istruct--55
Qwen3-4B Instruct-2507 (3-shot) Brrren3----lssruc--22 @ (3-sbot)
Thoglht
Best
83.3
79.7
81.5
Linglo-Thoughh
Best (ol)
85.2
82.2
83.7
MoT (O-shot)
Thought
Best
87.2
83.0
85.1
MoTT 3 (D-shot)
Single Thought
Best
87.2
83.8
85.5
MoT (0-shot)
Mixture-of Mhxuuhh-ofTThouht
89.2
86.0
87.6
of 600 questions with reasoning depths of 5, the same as Pan al. 2023 FOLIO is recognized for its high-quality export-made realistic test cases with more diverse reasoning depths ranging from 1-8. consists of 203 questions. We utilize accuracy and pass @k as metries.
Training/Inferecee Details. For each dataset, we collect 1000 training samples from the training set. We perform 2 or 3 rounds of self-evolving training. In each round, the model is fine-tuned for two epochs using a learning rate of 2c-5 and batch size of 128. During the trajectory collection phase, the temperature, max tokens, and sample count are set to 1.0, 2048, and 10, respectively. Notably, we use high temperature (1.0) to encourage diverse trajectories. We sample each problem 10 times during trajectory collection to maximize coverage. Of all the generated traces, only the first single trajectory that satisfies quality criteria is retained for the final training set. For evaluation, the temperature and max tooens are configured to 0.7 and 2048, respectively. We do not perform hyperparameter tuning, further optimization may yield even better performance. We run all experiments 4 H100 GPUs. We employ vLLM engine (Kwon et al. 2023) to improve inference efficiency.
3.2
MAIN RESULTS
Table 2 displays displayss the results FOLIO and ProofWriter benchmarks. First, our Lixture-of-Thougee (MoT) training with Single-Thought inference outperforms the corresponding base models by an average of 11.7pp (from 41.1% to 61.9% for Gemma-2-2b-I.. from 65.4% 73.2% for Gemma-2-9b- It, from 66.2% to 72.6% for Qwen-2.5-7b- 2.5--b-Insttrucc and from 81.5% to 85.1% for Qwen-3-4b-Instrucc) demonstrating the effectiveness of our training strategy. When we further apply MoT inference. the MoT-trained model yields consistent additional gains of up to 2.1pp. Notably, our 9B model achieves 78.9% accuracy on FOLIO, matching the performance of Logic-LM, which uses an external solver and focuses close-sourced SoTA LLMs. Applying our MoT approach reasoning model, e.g., Qwen3-4B- 1-33-B-IInstruc achieves an average performance of 87 87.6%. Furthermore, MoT inference provides consistent gains applying ceff-consistency to single-thought setting, suggesting that the MoT mechanism offers complementary benefits. We provide a detailed performance of both base models and the corresponding MoT models, well stronger baselines, in Appendix E E.4

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Table Accuracy (%) different training strategies across reasoning modalities (Same Round). Shaded cells denote in do-mimii evaluation, iee. testing on the same modalities during training. Avg. refers to the average performance using three modalities while Ensemble means the majority vote results three modalities. Values underlined indicate that the model did not follow the instruction (e.g,, when asked to use Code, still used NI...
Training Appreach
Param
Data
Code
CoT
Truth Table Avg. Ensemble
w/o Training
N/A
56.7
69.5
63.3
66.0
Single-Thouee Thoough Training
Tingle/Though (Code)
9B
61.6
59.1
64.0
61.6
70.4
Single-Thought (NL_CoT)
52.7
73.9
73.4
Single-Thought (Truth Table)
9B
53.2
69.0
71.9
64.7
71.9
Single-Thought (Three Models Combined)
3x98
~ 3x
61.6
73.9
71.9
69.1
77.3
Mfxture-offTThoeee Training
Mixture-of- Bhxture-f-Thougght (NL_CoT + Truth Table)
9B
~ 2x
65.5
72.9
69.5
69.3
72.9
Mixture-of- Phtture-f-Thhough (Truth Table + Code)
2x
71.4
72.4
Caature-of-Thuugee (Code NL.CoT)
2x
70.9
70.0
74.4
71.8
74.9
Mixture-of Thought (Default, All)
9B
3x
73.9
76.9
70.0
73.6
78.9
3.3 MIXTURE-OF-THOUEE TRAINING VS. SINGLE-THOUGHT TRAINING
In this section, we try to answer the key question: Does MoT training truly offer benefits over Single-Thought training? ? We have two baselines: 1) models trained single-thought data and 2) models trained on partially MoT data, e.g., Code + NL. We evaluate both in-mode accuracy and cross-mode generalization. To enhance model's format following ability. use 3-shot prompting to make model output the specific reasoning modality. TableBilluse iillustrate the results on FOLIO.
SoT MoT. First, MoT training achieves the highest average accuracy across all three modalities, beating single-thought trained model, which indicates that our MoT training can jointly improve reasoning ability across all modalities. Second, MoT training can further push the performance boundary for each reasoning modality. For example, by using two of the three modalities, i.e. Code and NL.CoT, the trained models outperform all single-thought baselines. This clearly indicates synergy between these three complementary modalities during training. Third, deploying one model for each modality is resource- exxpureive...eeee In contrast, MoT training enables a single model to seamlessly switch among reasoning modalities based on prompts.
Partial MoT MoT. Our default Mixture-of- Thouuhh setting yields the best average performance and achieves the best accuracy by using two combined reasoning paradigms, which indicates that all the modalities are useful. This superiority is further reflected in the ensemble accuracy, where MoT achieves 78.9%. We provide more evidence in Sec. 4.3 and Appendix E.9
Additional Ablations for MoT Training. We further give more analysis to show 1) robust and optimal design of the MoT framework (Appendix [E.) 2) MoT training is better than single-thought training with distillation data (Appendix (.6) and 3) MoT data outperform an equivalent amount of diverse single-hhought CoT data (Appendix E.7 These results underscore the practical and broader
value of our MoT framework.
3.4 TEST-TIME SCALING ACROSS REASONING MODALITIES
We investigate how different single-thought and MoT inference scale with an increased test-time budget. To do this, we first generate 128 responses from each model with each modality. Then we evaluate two sampling strategies: 1) Single-Thought Sampling: We randomly select k responses from the 128 generated responses. and 2) MoT Sampling: Assuming there are Ny reasoning modalities, & responses from each modality (so that the total number of responses is k). We choose k ranging from to 24 and have 10 runs for each setting.
sample Ny
MoT framework vs. Single-thought Baseline. We compare our Gemma- 2--a------------ It-MoT with Gemma-2-99 2-9b-It.. Figure (a) shows our MoT model with MoT sampling consistently outperforms Gemma-2-999 with single-thought sampling. When the sample budget is less than 20, the perfor- mance gap is significant. It suggests that our MoT approach significantly increases the response
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(a) Base (SoT) vs. MoT (MoT)
(b) MoT (SoT) vs. MoT (MoT)
100
95
Pass@l (%)
90
Pass (%)
Gemma-2 2 9--b- (NL,CoT)
MoT (Code)
90
80
Gemma-2-9h-l (Truth Table)
MoT (Truth Tahle)
Gemma-2-96e (Code)
MoT (NL.CoT)
Gemma-2-9h6 2-9--------- (MoT)
MoT (MoT)
70
85
0
10
20
10
20
Sample Budget
Sample Budget
Figure 3: Pass@k vs. Sample Budget on FOLIO. (a) MoT-trained model with MoT sampling outperforms the base model (Gemma-2-9b-It) with SoT sampling. (h) Within the MoT-trained model, MoT sampling yields higher Pass @k than SoT sampling (NL CoT, Truth Table, Code).
diversity, leading more efficient use of inference compute. We observe a consistent phenomenon in terms of averaged accuracy (Appendix E.8 Figure 6).
Comparison of different modalities. We further plot the scaling curves of MoT model (based on Gemma-2-9B-Ic under three reasoning modalities in Figure Here are insights: 1) While NL significantly outperforms the truth-table paradigm low their theoretical upper bounds converge as k increases; 2) The code paradigm exhibits the lowest upper bound among the three; 3) Across all values of k, our MoT framework consistently achieves the highest pass @ k and attains the largest upper bound, indicating the largest potential of MoT trained models in test-time scaling.
4 FURTHER ANALYSIS
4.1 MIXTURE-OF-THOUCGE INFERENCE BENEFITS MORE FOR DIFFICULT PROBLEMS
We further identify the types of problems that ben- efit most from the proposed MoT inference ap- proach. Specifically, we focus on problem diffi- culty, which can be effectively measured by the depth of reasoning. We conduct analysis on FÓLIO and ProverQA (Qi et al. (025).. Figure 4 shows the performance of our MoT model with different reasoning modalities across reasoning depths.
I I Code Truth Table
MoT
90
86.4
85.4
80.6
MoT benefits more for deeper reasoning.
Accuracy (%)
80
78.6
73.0
70
67:067.0
61.0
60
We can see that MoT inference benefits more in solving more difficult logical problems. Our MoT model with MoT inference achieves an accuracy of 73.0% on challenging logical tasks with reasoning depths ranging from 5 to 8, outperforming each modality by a significant margin, with an average improvement of 9% points. However, such per- formance gains turn into slight degradation when dealing with easy problems. Similar phenomenon
1-5
5-8
Reasoning Depth
Figure Performance comparison of differrent thought paradigms across reasoning depths. On FO- LIO, MoT inference exhibits better performance on difficult problems.
be observed in ProverQA, (See Figure
4.2 ANALYSIS ON GENERALIZATION MOT
This section investigates key question: how does the Mfxture-of-Thougee (MoT) framework, primar- ily focused on improving logical reasoning tasks, generalize to non-logic reasoning tasks (OOD)? Our evaluation focuses on two kinds of tasks: GSM8K and StrategyQA. Specifically, mainly focuses on logical reasoning tasks, some of its reasoning modalities may be not suitable for these two tasks. To this end, evaluate our MoT on these tasks via NL reasoning modalities, ignoring other two reasoning modalities. The results are shown in Table 4 We can see the MoT framework exhibits robust generalization stability non-logic OOD tasks when evaluated through the
our MoT
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Table 4: Performance comparison of Baseline vs. MoT (Single-Thought across benchmarks.
Non-Logic (OOD)
Logic (ID)
Model
Setting
StrategyQA
GSM8K
FOLIO
ProofWriter
Baseline
81.7
52.1
42.4
39.8
Gemma-2-2B-Instee
Mo'I (MoT- Base)
81.2
51.5
61.1
62.7
-0.6
+18.7
+22.9
Baseline
93.2
85.2
69.5
61.2
Gemma-2-9B-Instruee
MoT (houle-Thought)
92.3
84.2
76.9
69.5
A (MoT Basc)
-0.9
-1.0
+7.4
+8.3
Baseline
88.1
87.6
71.9
60.5
Qwen-2.5-70-Instree
MoT (Siugle-Thought)
89.5
86.9
75.9
69.2
A (MoT Buse)
+1.4
-0.7
+4.0
+8.7
NL reasoning modality. This is evidenced by the fact that the performance changes (A) are minimal, predominantly remaining within +1.4 pp. This demonstrates that MoT can significantly improve logical reasoning ability without compromising its general NI reasoning capabilities, successfully mitigating the risk of catastrophic forgetting.
Discussions. While we have shown that MoT achieves robust stability with minimal performance degradation, we also want to highlight that MoT is a general framework, which integrates natural language, code, and symbolic modalities into both training and inference. To directly enable the MoT framework work on non-logic reasoning tasks, a promising way to define a suitable instance of each modality for the specific domain. For example, for mathematical reasoning tasks like GSM8K, the symbolic modality can be instantiated as a equation derivations, algebraic simplification steps, enabling precise numerical manipulation.
4.3 COMPLEMENTARY, UNIQUENESS AND ERROR ANALYSIS
In this section, we quantify the complementary and uniqueness of our reasoning modalities and the training dynamics of our self-evolving MoT training. We focus on three metrics: 1) Unique coverage. i.e., examples solved by exactly one modality; 2) Complementarity coverage, i.e., examples solved by least two modalities; and 3) Oracle upper bound, i.e., examples solved by at least one modality.
Figure (b). shows each modality's 's solve rate and oracle upper bound on ProtfWriter and FOLIO. We further give a detailed unique and complementarity coverage and oracle upper bound in Table in the Appendix. First, although our approach slightly reduces unique coverage compared to the baseline, both methods still achieve strong performance in this metric. Second, in terms of comple- mentarity, our method increases the number of examples solved by multiple modalitiss-particulll particularly on ProofWriter worooiiie--dddoosstattiieee demonstrating enhanced synergy. Third, by incorporating the truth-table paradigm alongside Code and NL, our model attains a higher oracle upper bound than prior work using only Code+NL, underscoring the benefit and necessity of the truth-table paradigm.
Bottleneck of NI reasoning modality. We perform human evaluation of model outputs generated by natural language reasoning on the FOLIO dataset. We identify two major error patterns in the incorrectly solved cases: 1) failure to consider multiple cases when handling disjunction operations. such as "cither/or"; 2) failure to utilize the transposition inference rule. For example, given A + B. the model might sometimes incorrectly produce A =B. Motivated by these observations and error types identified in prior work (Han et al., 2024 Olausson et al. 2023). we define four error categories: (i) invalid converse; (ii) missing branch; (ın) factual misquote: and (iv) incorporation of commonsense knowledge and design an automatic pipeline to assess model rationales. Figure Ic) presents the results, showing that invalid converse and missing- branch errors together account for nearly 66% of all errors. These findings underscore the value of the Truth Table thought paradigm. We also provide error analysis after MoT training in Appendix E.2
Scenarios that Truth Table excels in. We manually analyze all 13 examples (Table that were solved only using the truth table paradigm and find that 1) 5 out of 13 problems require transposition; 2) 5 out of 13 problems contain disjunction or similar operations (e.g., 'Rock fly, or is a bird, or cannot breathe') ) and 3) 2 out of 13 problems contain both. This indicates that Truth Table may indeed complement the NI paradigm some extent. We give two examples in Appendix H.2

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Scenarios that Code excels in. We analyze the case that is solely solved by the code modality as shown in Table IT] We find the Code Modality is superior to the other two where high reasoning complexity is required. These examples are characterized by lots of premises and variables and require long inference chains. In this scenario, NL modality suffers from error accumulation or hallucination, while the TT modality faces challenges in building truth tables. The Code Modality fills this critical gap by providing a scalable and formally expressive encoding. However, we acknowledge that for complex problems that cannot be easily transformed into code structures, the Code Modality still falls short. This explains why code modality almost always performs worse when used alone.
RELATED WORK
LLMs for Symbolic Reasoning. Prior work has explored adapting LLMs to symbolic reasoning.
One common approach treats LLMs as nl-to-fol translators. and then use an external symbolic
prover to derive the final answer (Pan al., 2023 Olausson et al. 2023 Callewaert et al. 2025
Ryu et al. 2025). While effective, this pipeline largely bypasses the model's internal reasoning capabilities, which our work seeks to fully leverage. To alleviate this problem, another line of work seeks to directly leverage LLMs' reasoning ability via CoT prompting (Wei et al. 2022) or structured search procedures such as Tree-of-Though (Yao et al., 2023) and Cumulative Reasoning /Zhngg et al. 2023), as well as improving determinacy via premise identification and prioritization (Sun ct al. 2024). However, natural language remains inherently flexible and sometimes insufficient for structured reasoning. To bridge the gap between flexibility and formal rigor, recent work has explored combining natural and symbolic reasoning (Xu et al., 2024a| Liu et al. 2024 Xiong et al. 2024). These approaches often either rely on primary reasoning modality (e.g., symbolic NL), augmented with auxiliary signals from other representations (Xu et al. 2024a: Liu et al.) 2024) or
select one from multiple reasoning modalities (Xiong et al. (224) at inference time. In contrast, our work 1) explicitly defines three kinds of reasoning paradigms covering natural language, symbolic and code-based reasoning. 2) goes beyond modality selection by jointly learning and inferring with all modalities, via a self-evolving MoT training and inference framework.
Previous work diversifies the CoT to further improve reasoning performance. A common strategy is to sample multiple outputs with higher temperatures (Wang al., 2022] Brown et al., 2024). but this cannot guarantee true diversity (Wang et al. (025). To address this, some work uses varied prompts by task type (Wang) jet al. 2025), difficulty (Li et al. 2022), or strategy (Li et al. 2023 He et al. 2024)
Encouraging Diverse Thinking in Chain-of-Thoughtt
agent-
prompting via multi-agent debate (Liang et al. 2023) Hegazy, 2024) or self-reflection {Zhang ct al., 2024) to elicit diverse CoTs These methods diversify within one modality (NL or code). In contrast, we systematically introduce modality-leve diversity- truth table, natural language, and code reasoning which better aligns with the structural requirements of symbolic tasks and complements existing approaches. Recent work has also explored training smaller models on diverse CoT's generated by large LLMS (Ho et al., 2022 Puerto et al., 2024), though these approaches are limited to single-modality supervision and rely on external teacher models. In contrast, our method introduces modality-level diversity and requires no external supervision. We demonstrate that inter- modalttyy diversity yields greater benefits for self-training than intra- midda-ty diversity in Appendix E.7 Concurrent to our work, Shain-of--Resoonig (CoR) (Yu et al. 2025) also studies synergy of multiple reasoning modalities. However, CoR focuses on sequential synergy and targets mathematical reasoning. In contrast, our approach (i) focuses on logical reasoning and introduces TT reasoning modality, (ii) develops a self-evolving MoT training algorithm to bootstrap multi- modaityo capabilities, and (iii) exploits parallel synergy across modalities during both training and inference.
6 CONCLUSION
We presented Mixture-of- Thought (MoT), unified framework for improving logical reasoning by enabling LLMs to reason through natural language, code-based, and symbolic (truth table) paradigms within a single system. Unlike previous work, our approach combines self-evolving training process that fosters cross- pooss-parddigm synergy with an inference-time voting mechanism that aggregates complementary reasoning strategies. Extensive experiments two challenging logical reasoning benchmarks, FOLIO and ProofWriter, demonstrate that MoT substantially outperforms strong baselines, particularly on complex, high-depth problems.
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ACKNOWLEDGMENT
This work was partially supported by NSF IIS 2347592, 2348169, DBI 2405416, CCF 2348306, CNS 2347617, RISE 2536663.
ETHICS STATEMENT
Our Mixture-of- of-uue-- Thought (MoT) framework is designed to improve logical reasoning by integrating multiple complementary modalities (natural language, code, truth tables). We do not foresee any risks related to ethics issues.
REPRODUCIBILITY STATEMENT
We include detailed information about our approach in Section 2 and include experimental settings in Section and Appendix [D] We have give details in Section and Appendix Do reproduce our results and we will open-source the code as soon as we collect our scripts for an easy way to reproduce our results.
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Wenqi Zhang, Yongliang Shen, Linjuan Wu, Qiuying Peng. Jun Wang. Yueting Zhuang, and Weiming Lu. Self-contrast: Better reflection through inconsistent solving perspectives. In Lun-Wer Ku, Andre Martins, and Vivek Srikumar (eds.), Proceedings of the 62nd Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pp. 3602-3622, Bangkok, Thailand, August 2024. Association for Computational Linguistics. doi: 10. (0.1863/v1/2024, acl-long. 197. URL htttps://accattttoto
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Yifan Zhang, Jingqin Yang, Yang Yuan, and Andrew Chi-Chih Yao. Cumulative reasoning with large language models. arXiv preprint 2308::330800377/ 2023.
Lianmin Zheng, Wei-Lin Chiang, Ying Sheng, Siyuan Zhuang, Zhanghao Wu, Yonghao Zhuang, Zi Lin, Zhuohan Li, Dacheng Li. Eric Xing, et al. Judging auddas---judge with mt-bench and chatbot arena. Advances in neural information processing systems, 36:46595-46623, 2023.
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Tong Zheng, Hongming Zhang, Wenhao Yu, Xiaoyang Wang, Xinyu Yang, Runpeng Dai, Rui Liu, Huiwen Bao, Chengsong Huang, Heng Huang, et al. Parallel-rl: Towards parallel thinking via reinforcement learning. arXiv preprint arXiv:2509.0798/ 07980, 2025.
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A THE USE OF LLMS
We only use L.LMs to polish the paper writing.
B LIMITATIONS AND FUTURE WORK
While our Mixture-ofe of -ae---o-Tohhhe Thought (MoT) framework demonstrates strong performance on logical rea- soning tasks, we have not evaluated its effectiveness on other types of reasoning tasks, such as mathematical or commonsense reasoning. Additionally, our test-time scaling experiments suggest promising directions- such as dynamic mixture-of-thougee sampling under budget constraints but our current work still has not fully explored the benefits of complementary reasoning modalities. Further exploring these aspects could be important to further push the performance boundary of open-source models reasoning.
We plan to further explore them in the two aspects:
work cannot directly applied reasoning tasks out of logical reasoning. This is because the reasoning modality we define our work is specific for logical reasoning, e-g., Truth Table. Therefore, reasoning modality that can be applied to more general broader of reasoning tasks and further show how our MoT framework can further improve performance of reasoning tasks beyond logical reasoning.
Extended to boarder tasks: currently
plan to define more general but complementary
Adaptive parallel thinking via mf An interesting question is How can we fully leverage the benefits of complementary reasoning modalities during inference? Recent work has shown that reinforcement learning instill parallel thinking in LLMs (Zheng al. 2025). enabling models to dynamically activate and coordinate multiple reasoning paths. This opens up natural opportunity to integrate such adaptive parallel thinking with our MoT framework: treating each reasoning modality as an atomic unit, and allowing the model to dynamically trigger, combine, prune modalities during inference. We believe that this unified view, blending MoT's modality-level diversity with dynamic parallelism, can lead to more flexible and efficient reasoning systems.
C BROADER RELATED WORK: SELF-EVOLVING TRAINING.
Self-evolving training techniques have been widely adopted to improve reasoning ability in LLMs, especially when there is lack of reasoning trajectories. Notably, Zelikman et al. (2022) propose bootstrapping framework that iteratively generates and verifies reasoning trajectories based on the derived final answer, then fine-tunes the model on these self-labeled examples improve reasoning performance with minimal human supervision. Following this idea, several works adapt self-evolving
training wider range of tasks (Hosseini al., 2024 Xiong al., 2024 Zelikman et al. 2024b
Wang et al.) 2024: Lin et al. 2024: Zelikman et al.) 2024a; Poesia et al. 2024 Guan et al. 2025
Li et al. 2025). Additionally, researchers also explore improving the high-quality of rationales during STaR algorithm (Poesia et al., 2024: Guan et al. 2025 PENG et al.) 2025). incorporating techniques such as formal verification, monte carlo tree search, and abstract-to co-cccrete prompting. While previous work primarily focuses generating higher-quality reasoning paths within single modality, our work explores complementary direction: how to jointly evolve and coordinate reasoning across multiple thought paradigms.
D DETAILED EXPERIMENTAL SETTINGS
D.I DATASETS
In this work, adopt three logical reasoning datasets: 1) FOLIO (Han et al. 2024). 2) ProofWriter (Tffoord et al. (021), and 3) ProverQA (Qi et al. 2025). to evaluate the effective- ness of MoT framework.
FOLIO (Han et al. 2024). FOLIO provides both the training and validation subsets, consisting of 1003 and 203 samples, respectively. There subsets with different difficulties: 1) HybLogicc:
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contains 100 complex logical problems (5 8 reasoning steps) and 2) WikiLogic: contains 103 simper logical problems (1 -5 reasoning steps). In this work, we sample 1000 training samples from the FOLIO training set seed dataset for our self-evovling MoT training and evaluate both baselines and our trained model on the FOLIO validation set.
ProofWriter (Tafjord et al., 2021). ProofWriter synthetic dataset designed for evaluating the logical reasoning abilities of language models. It consists of multiple subsets, each containing logical reasoning problems of varying reasoning depths from depth (direct inference) up to depth (requiring multi-step logical deductions). Following(Pan et al. (202)), we select the most challenging subset (reasoning depth 5) to construct our training and test data. Specifically, we sample 1,000 instances from the training set provided by Pan et al. (2023) as training data and adopt their original test set directly for fair evaluation.
ProverQA is recently proposed logical reasoning benchmark, notable for its large scale, high quality, and diversity. It consists of three subsets, each corresponding to a different reasoning difficulty level (i.e., reasoning depth). We select these subsets evaluate the performance of our MoT framework across varying levels of reasoning complexity.
ProverQA @t al., 2025).
D.2 TRAINING DETAILS
We conduct all experiments on 4 H100 GPUs with Alignment Handbook (Tunstall et al.). For each dataset, we sample 1,000 training examples and perform 2-3 rounds of self-evolving training. In each round, the model fine-tuned for 2 epochs with a learning rate 2e-5 and a batch size of 128. We do not perform hyperparameter tuning. Further tuning may lead to better performance. All experiments are run with fixed seed, ie., 42, for reproducibility.
D.3 INFERENCE DETAILS
We employ vLLM (won al. 2023) for efficient inference. During trajectory collection, we generate 10 reasoning traces per example using temperature 1.0, max max_tokens 2048, and sampling count 10. To maximize coverage while ensuring quality, we retain only the first generated trace that passes our quality check. For evaluation, we set the temperature to 0.7 and max tokens to 2048. All experiments are run with a fixed seed, i.e., 42, for reproducibility.
D.4 DETAILS OF "REASONING TO PRUNE"
The "Reasoning Prune" step is the critical component in the Truth-Table CoT modality that resolves the computational inefficiency of enumerating the full 2" truth table space. This process does rely iterative filtering: rather, harnesses the LLM's few-shot reasoning capabilities to perform Constraint-Aware Deductive Reasoning, which implicitly prunes the logically inconsistent assignments.
Implementation Through Few-Shot Prompting We leverage few-shot prompting to guide the LLM into executing specific, goal-oriented logical deduction. This approach compels the model to output only the assignments that constitute a logically consistent world, effectively generating the minimal required truth table row. The mechanism begins with Grounding and then operates via three integrated deductive processes:
1. Grounding and Symbolic Mapping: This initial phase involves the LLM performing
symbolic mapping, instantiating the first-order problem's components into a finite set of propositional predicates. For example, the model establishes variables such "Let: 7 = Thor happy. = Hulk angry, = wakes up." This step formalizes the input for subsequent symbolic reasoning.
2. Logical Chain Induction: The LLM constructs multi-step Logical Chain by identifying and utilizing the transitive properties present within the set of premises. For example, given premises P A = B and Pa B C, the LLM is prompted to perform the deduction + B + C. This inductive process inherently establishes logical dependencies between the variables.
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Table Accuracy (%) on the FOLIO ProofWriter benchmarks under three seeds (2025, 123 42). Our MoT training consistently improves the performance of each base model. Applying MoT inference further enhances performance across both benchmarks and all models, @3 denotes Self Self-Consistency approach (Wang)
al. 2022 with three votes.
Model
Method Type
Reasoning Modality
FOLIO
ProofWriter
Prior SOTA Approach
Logic-LM CoT CoTT(Von (Vanilla)
78.9
79.7
79.3
GPT-4
70.6
68.1
69.4
(B) Base Model: Gemma-2-2B-Ir
Gemma-2-21B--1 (3-Shot)
Single-Thoughh
(nl)
Gemma-2-22 2-22B- (3-Shot)
Sougle-Thought
Best (al)
42.7 + 2.5
38.8 = 1.1
40.8
MoT (0-Shot)
Single-Thought
Best
60.6 + 1.8
61.4 = 18
61.0
(0-Shot)
Mixture-of- ohoThhght
-:
(C) Base Model: Gemma-2-99BIt
Vemmm---9BBII (3-shot)
Single-Thought
Best (nl)
70.5
0.3
65.3
Gemma-2-99 It (3-shot)
Thoggtt
Best (nl)
73.4
62.5 = 0.2
68.0
MoT (0-shot)
Single-Thoughe
Best
77.7
68.4
73.1
MoT (0-shot)
Mixture-of-Thouue
79.3
71.0
75.2
(D) Base Model: Qwen2.5-7B-Instruce
Qwen2,5-7B. 1een2.5-7B-Insttuee (3-shot)
Single-Thought
Best (nl)
1.9
60.8 + 1,9
66.6
Owen2.5-7B-IIsee (3-shot)
Single-Thought
Best (nl)
73.1
65.2 = 0.8
69.2
(0-shat)
Single- Togle-Thouuhh
MoT (0-shor)
Mixture-of of Thouhhhh
All
79.0 :: 0.6
72.2 002::00 0.3
75.6
3. Implicit Constraint Pruning: The deduction is inherently constraint-aware. As the LLM sequentially derives the truth value for each variable (e.g., establishing H = True), this assignment is immediately checked for consistency against all initial premises and all previ- ously derived assignments. The LLM's goal is to derive and output the unique (or minimal) set of variable assignments that is logically sound with the entire premise set. By only outputting this consistent set, the process implicitly prunes all potential 2 combinations that would have violated any premise, transforming the task from a search problem into a direct generation problem.
E ADDITIONAL. EXPERIMENTAL RESULTS
E.1 ROBUST ANALYSIS ON TABLE WITH THREE SEEDS: 42, 123 AND 2025
Table 2 reports single-run results, which is the standard evaluation setting in prior logical reasoning works such as LINC (Olausson et al. 2023). Logic-LM (Pan et al.) (023). and LoT (Liu et al. 2024). To further access the robustness of findings. we perform three runs under three different seeds,
e.g., 42, 123 and 2025. We present the results in Table We can observe consistent trend as that in Table This confirms the reliability of our main claim.
E.2 POST-MOT-TRANI
ERROR ANALYSIS
To complement the pre-training analysis in Figure Ic), we conduct post-MoT-training error-type analysis to examine whether MoT training also improves the NL modality itself. We evaluate four representative error categories: Invalid Converse (IC), Missing Branch (MB), Commonsense Injection (CI), and Factual Misquote (FC). Following the same annotation protocol as before, we compare the NI modality of the base model with the MoTT-rainedd model.
Table 6: Post-MoT error-type counts for the NL modality.
Model
IC
MB
CI
FC
Qwen-2.5-7B. I-Strutt
22
18
12
12
21
Overall, MoT training reduces all error categories, with notable improvements in CI and FC. However, IC and MB remain the most structurally challenging cases, indicating that NI. alone cannot fully
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resolve these errors. This highlights the importance of MoT-Inference, where NL, Code, and TT modalities collaboratively mitigate such difficult cases.
E.3 EVALUATING LLM PERFORMANCE ACROSS REASONING MODALITIES ON FOLIO AND PROOFWRITER
Table 7: Performance three models across reasoning modalities on FOLIO and ProofWriter.
FOLIO
ProofWriter
Model
Code
Truth Table
Code
Truth Table
Gemma-2-2B-H
42.4
38.4
36.5
39.8
40.8
37.5
MoT training
61.1 (18.71)
61.1 (22.7f)
58.6 (22.1f)
62.7 (22.9f)
61.7 (20.9f)
60.2 (22.7f)
Gemma-2-995 2-B-I-
69.5
56.7
63.6
61.2
39.5
55.8
MoT training
76.9 (7.41)
73.9 (17.21)
70.0 (6.47)
68.5 (7.31)
69.5 (30.01)
66.7 (10.9f)
Qwen-2.5-7B-Instte Instruct
71.9
62.1
69.0
60.5
42.3
53.0
MoT training
75.9 (4.01)
68.5 (6.41)
71.9 (2.91)
69.2
66.7 (24.41)
64.3 (11.3f)
Table 7 displays detailed results of baselines across reasoning modalities on FOLIO and ProofWriter. We can observe that LLMss owns uneven ability across these reasoning modalities. This also highlights the necessary of our self-evolving MoT training, which can equip LLMs with three complementary reasoning modalities. After self-evolving MoT training, all modalities show joint improvements. This effect is especially significant in smaller models, i.e., Gemma-2-2B-II achieves up to more than 20% increase accuracy on average.
E.4 COMPARISON WITH MORE BASELINES FOLIO AND PROOFWRITER
Table 8: Comparison with more baselines on FOLIO and ProofWriter
Method
Base Model
FOLIO (Acc %)
ProofWriter (Acc %)
HybridMind (Xiong et al. 2024
GPT:33,5
76.6
LINC (Olaussop o die 4026
GPT-4
72.5
Symbone
2024b
GPT-4
83.3
82.5
Segic-of-Thouggee @
GPT-3.5
81.5
65.9
Logic-of-Thooee Thoughts
88.2
72.0
MoT
Gemma 2-26- 2b-It
62.6
65.0
MoT
Gemma-2-9b-a
78.9
70.7
MoT
Qwen2.5 Itstuurtuutt
Instruct
78.3
71.8
Table 8 presents a comparison between our approach and prior state-of ofute-of-the-art systems. It demon- strates that our open-source MoT models nearly match the performance leading closed-source prompting methods (e.g., GPT-3.5 and GPT-4). This indicates that enabling L.MSs to learn comple- mentary reasoning modalities a promising direction.
Table 9: Ablation studies on (1) policy strategy; and (2) mixing strategy.
FOLIO Accuracy (%)
Ablation
Setting
NL
Code
Truth Table
MoT
Off- Off-policy MoT On-policy MoT (default)
55,2
54.7
53.7
56,7
1. Policy Strategy
61.1
61.1
58.6
62.6
Random single-modality per question
49.8
50.3
48.3
53.7
Mixing Strategy
Direct mixing (default)
61.1
61.1
58.6
62.6
E.5
ABLATION STUDIES
We perform ablation studies three core components: 1) policy strategy, i.e., on-policy off- policy (Zelikman et al. 2022) and 2) mixing approach, i.e., direct mixture vs. mixture by unique conclusion (randomly select single-modality per question).
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Table reports FOLIO accuracies under each setting. We make two key observations:
On-policy training yields consistent gains. Switching from off-policy to on-policy increases single-modality CoT accuracy by approximately 5-6 pp (c.g., NL CoT from 55.2% to 61.1%) and raises MoT's final accuracy from 56.7% to 62.6%. This demonstrates the importance of updating
the model with its most recent outputs.
Direct mixing outperforms random single-modality sampling. Presenting all three modalities together boosts accuracy by 8-10 pp compared to randomly picking one modality per question (MoT: 62.6% vs. 53.7%). This indicates that joint exposure to multiple modalities provides stronger complementary signals than isolated examples.
E.6 IMPACT OF QUALITY OF INITIAL TRAINING DATA: DISTILLATION + SINGLE-MODAL TRAINING VS. RAW DATA + MOT TRAINING
60
Accuracy (%)
50
o4-mini init
self gen NI.
MoT
Evolving Rounds
Figure 5: Accuracy (%) over three self-evolving rounds the FOLIO benchmark for: distilled NL-CoT (first-round only), raw NL-CoT (no distillation), and MoT (no distillation). The performance evaluated with NL-based reasoning.
Intuitively, the first-round data crucial and have strong impact on the efficacy of self-evolving training. Therefore, we are interested in the following question: Can self-evolving single-thought training enhanced by first-round distillation outperform our self-evolving mixture-of-thouge training without any distillation? To answer this question, we compare the following settings: 1) Self-evolving single-thought (nl) training but with distillation data from 04-mini for first round training, which can provide better initialization; 2) our MoT training without any distillation data; and 3) Self- evolving single-thouge thought (nl) training without any distillation data. Figure displays the results of Gemma-2-2b-Ii on FOLIO benchmark.
The key observations are: adding distillation data from stronger LLMs beneficial for improving performance and convergence rate (blue line vs. orange line), but still lags behind our self-evolving MoT training (blue line vs. green line). This suggests the advantages of our self-evolving MoT training: 1) It requires no reliance on stronger-often more Expensive-LLMs 2) It provides a higher upper bound accuracy.
E.7 FINETUNING WITH DIVERSE SINGLE COT vs. FINETUNING WITH MoT
Ho et al. 2022); Puerto et al. 2024) have explored that finetuning LLMS with diverse CoT can further natural question then is: given a fixed budget of training examples, which strategy yields better results? (1) self-training with 3N natural-lane language CoT samples, or (2) self-training with a total 3.N samples composed of N examples from each of three modalities (NL,
improve the performance.
Code, Truth-Table).
We consider two settings to answer this question: 1) Self-evolving training with 3N natural-language CoT samples for 2 epochs per round over 3 rounds. We sample 10 reasoning traces per question with temperature of 1.0 and keep the reasoning traces that satisfy our filtering criteria: 2) Self-evolving training with total of 3N samples comprising N examples from each of the three modalities (NL.,
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Code, Truth- for 2 epochs per round over rounds. We evaluate those trained model with natural language modality on FOLIO dataset.
Table 10: Accuracy (%) of Gemma-2-2b-I under three self-evolving regimes, with budgets of N or 3N training samples. The accuracy is evaluated with NL-based reasoning on FOLIO benchmark. We can see self-evolving training with MoT achieves the best accuracy, demonstrating the benefit of modality- diversity.
Setting
Training Samples
Accuracy (%)
NL.CoT
N
54.7
2 CLT
3N
57.1
3
MoT data
3NV
61.1
Table 10 shows the results. We can have the following observations: 1) finetuning with diverse NL CoT can indeed improve the performance (#1 #2), which is consistent with findings from Ho et al. (2022): : Puerto et al. (2024). 2) Finetuning with MoT data is more efficient than finetuning with same amount of diverse NL CoT data (#2 vs. #3). This indicates that the diversity of single-mddality CoT data obtained by sampling with high temperature is not sufficient. By contrast, our MoT data, which leverages the complementarity of truth table, code and can produce more diversity, and therefore improve the training efficiency.
E.8 ADDITIONAL RESULTS TEST-TIME SCALING ACROSS REASONING MODALITIES
MoT With Different Thought Paradigms Table (b) illustrates the scaling behavior of our MoT model across different thought paradigms under varying sample budgets. We observe that code-based reasoning consistently lags behind all other paradigms, indicating its relatively poor performance and limited scalability.
Another interesting phenomenon that natural language-based reasoning achieves relatively strong performance when the sample budget is small (e.g., k < 20), outperforming the truth table-based paradigm in this regime. However, as the sample budget increases (e.g., k 20), truth table reasoning begins to match even outperform NL-based inference resources are available.
boauning-highlihhtinng its greater potential when more
Notably, our MoT (ALL) approach offers favorable trade-off between these two paradigms: it achieves strong performance under low-budget conditions, while delivering better performance when the sample budget is large.
Accuracy vs. Sample Budget Figure presents seuuracy-v--ample-buddee curves across different reasoning paradigms. We find that our MoT (ALL) model-trained and inferred under the mixture-of- thought contiig-- achieves the highest accuracy, outperforming all other approaches re- MoT
80
Accuracy (%)
60
gardless of budget size. Additionally, model can benefit better from increased sample bud- get compared wiht all other approaches. Among individual paradigms, NL-CoT performs best un- der majority voting, while truth table reasoning is more stable but shows limited improvement with increased budget. Rode-based reasoning remains the least effective. These results reinforce the value of our MoT framework.
Gemma-2-9h-nn (NL)CoT
Gemma 2-96-It (Truth Table)
Gemma-2-966 2--9hII (Code)
Gemma-2 9bT (MoT)
0
10
15
20
25
Sample Budget
Figure 6: Accuracy vs. Sample Budget for dif-
ferent modes
E.9 DETAILED COMPLEMENTARY, UNIQUENESS ANALYSIS
Across both ProofWriter and FOLIO benchmarks, our Mf (MoT) model shifts away from single-paradigm reliance and toward multi-paradigm collaboration. First,
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NL
Code
Truth Table MoT
65.88
62.2
Accuracy (%)
60
58.6
with deeper reasoning,
MoT also belps
48.4
46.0
43.4
39.3
3-5
6-9
Reasoning Depth
Figure 7: Performance comparison of different thought paradigms across reasoning depths. On ProverQA, MoT inference exhibits better performance difficult problems.
the number of examples solved exclusively by the NL paradigm drops by over 50% (ProofWriter: from 109 to 55; FOLIO: from 18 to 8), and "Only TT correct" cases likewise decrease, indicating that MoT reduces brittle, single-mode reasoning.
Second, pairwise overlaps (NL n Code, NL n TT, Code n TT) all increase Substantially-NNL n Code ProofWriter rises by 76% (172 + 304), and similar gains appear on FOLIO-showing that MoT effectively combines different reasoning formats on the same instance. Finally, the over- all coverage (Code U NL U TT) improves from 511 544 (+6.5%) on ProofWriter and from 174 to 181 (+4%) on FOLIO, demonstrating that MoT recovers difficult cases missed by the baseline. The consistent trends across two datasets confirm that encouraging multi-paradign synergy yields more robust and comprehensive logical reasoning performance.
Table 11: Prediction Category Distribu- tion on Two Benchmarks (Qwen-2.5-77B Instruct vs 2wen-2.5-7B-BB (-sTB---struut-MooT))
ProofWriter
FOLIO
Category
Bascline Our Baseline Our
Single-paradige only
Only NL correct
109
18
8
Only Code correct
6
Only
13
Pairwise overlap only
NL Code only
304
109
125
NI. TT only
210
289
117
125
E. PERFORMANCE
Cude C TT only
170
297
110
112
vs. DIFFICULTY ON PROVERQA (QI ET AL. 2025)
Code NL
445
511
163
168
Code U UL
511
544
174
181
Figure displays the results.
F FULL PROMPTS FOR MoT
The full prompts we utilized in this work are illustrated as follows:
Full prompt used for Mixtureoof-Though
You are a rigereus and logically precise AI assistant. Your task is to answer logical reasoning problem strictly fol fowlowing one of three modes, explicitly specified the input. Only one mode will present the input. Follow that mode excl exclusively.
Code Mode (<code> ... <end code> <answer>
end of answer>) of
If the input contains <code>, translate the problem into Python code.
Execute the logic and derive the answer.
Natural Language Chain-of-Thougha Mode (<n1_cot>> end_of_nl_cot> <answer> <end_of_answer>
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If the input contains <nl_cot>, solve the problem step by step in natural language.
Truth Table Mode (streth_table> table> <end orrcoottruttttatee table> answer> send lend_of_answeet [nswee>>
If the input contains struth_table>, construct a truth table and derive the answer from it.
### Rules
Only use the mode specified in the input. Do not switch modes
Generate cutput strictly in the specified mode and format,
with no additional text.
Enclose all reasoning strictly within the corresponding mode tags.
The final answer must be strictly encl osed in <answer> end_of_answers
Do not provide any reasoning or explanations outside the designated mode tags.
The following is the problem you need to solve.
<premises>
(premises)
</premises>
<conclusion>
(conclusion]
</conclusicn>
<question>
Is the fol folicwing statement true, false, or uncertain? {
conclusion)
</question>
<options>
(A) True
(B) False
(C) Incertain </options>
<(tag)>
Full prompt used for Error Detection
"You must determine whether rationale faithfully justifies the truth value of conclusion given set of premises. AnA n"
"Faithful means all and only the steps actually used deriving the concl usi
are grounded in the given premises prior derived steps, An"
apply valid inference rules (no illicit converse or contraposition). \n"
cover every disjunction branch or quantifier case, An" use unstated assumptions, external knowledge, background comnonsense, \n"
and correctly assess whether the conclusion is supported or contradicted by the premises. \nin"
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"You must also diagnose where and how the rationale fails when it is unfaithful, allowing trivial unused zemarks to be overridden, An An
"Error Types:
Missing Branch: Failing to exhaustively consider all
branches of disjunction, conditionals, or quantified cases. \n"
Invalid Converse: Illicitly reversing the direction of a conditional (e.g., mistaking AA -B for 'B +
n"
Commonsense Injection:
Using external background knowledge or commonsense not entailed or implied by the premises.\n"
Factual Mi sisqutt:: Misrepresenting, distorting, or mi misquuting the explicis content of the premises. \n'n"
"Input (JSCN) i'n" (\n"
"premises": "<string>" An
"conclusion": "<string>" An"
"raticnale"; "<strings" An'
"label": "<string>", An
"predict"; "<string> 'n' "Anin"
"Output (JSON): : \n"
(\n"
"faithful"; true talse,\ai werror_type"' "<missing branch | invalid converse commonsense injection factual misquote>" An' "errer_ "error_location"" "ce.g., Step 3, Clause 2>", And "everride": true false,
"analysis"; "<brief summary explsining why the reasoning is faithful unfaithful, citing specific logical failures>"(a' "Anin"
"Notes:\n" * If multiple error types apply, list then all separated by commas. \n"
Always identify the first point in the rationale where the faithfulness failure occurs. An"
-- concise, precise, and consistent in your labeling. \ a\a"
"nnptt \n"
G MORE EXAMPLES GENERATED BY OUR MoT
In this section, we show more examples generated by our MoT with different thought paradigms.
G.1 NATURAL LANGUAGE BASED REASONING
Examples NL-based Reasoning generated by our models.
The following is the problem you need to solve.
<premisest ses>
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All rabbits that can be spotted near the campus are cute. Some turtles can be spotted near the campus.
The only animals that can be spotted near the campus rabbits and squirreis.
If something is skittish, then it is not calm.
All the squirrels that can be spotted near the campus are skittish.
Rockie can be spotted near the campus, and it is calm. </premises>
<conclusion>
Rockic is turtle cute. </conclusien>
<question>
Is the follewing statement true, false, uncertain? Rockie is turtle cute.
</question>
<options>
(A) True
(B) False
(C) Incertain
</options>
<nl_cot>
Step We knew from the premise "The only animals that can be spotted near the campus are rabbits and squirrels." that Rockie, being spotted near the campus, must be either rabbit
squirrel.
Step alse know from the premise "A11 rabbi that can be spotted near the campus are cute" that if Rockie a rabbit, then Rockie is cute.
Step 3: The premise "All the squirrels that can be spotted near the campus are skittish" and "If something is skittish, then it not calm" tell us that Rockie is a squirrel, then Rockie is not calm.
stop 4: The premise "Rockic can be spotted near the campus, and it is calm" contradicts the conclusion from step 3. This means Rockie cannot be squirrel.
Step Therefore, Rockie must be a rabbit.
Step 6: Since Rockie is a rabbit and all rabbits spotted near the campus are cute, we can conclude that Rockie is cute.
<end cot>
<answer>
The final answer is (A) <end_of_answer>
G.2
CODE BASED REASONING
Examples of Code-based Reasoning generated by our models.
The following is the problem you need to solve.
<premi ses? Everyone at the mixer is Grand Slam champion or an Oscar- noninated actor.
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Every Grand Slam champion at the mixer is a professional tennis player.
All Oscar-neminated actors at the mixer celebrities. All professicnal tennis players the mixer are atnietes. If a person at the mixer is a celebrity, then they are well paid.
If person the mixer is athlete, then they famous. mixer live in tax havens. Djokovic at the mixer: if Djckevic is famous athlete, then Djokovic is well-paid.
All well-paid people
</premises>
<conclusion> Djokovic does not live in a tax haven. </ </conclusion>
<question>
Is the following statement true, false, or uncertain? Djokovic does not live in a tax haven.
</question>
<options>
(A) True
(B) False
(C) Uncertain
</options>
<code>
from typing import Optional
class @__grand_slam_champiic Optional [bool] - None,
def iiii (self,
is oacccsccranc escar__nmmiated actor: Optional [bool] None,
is w..profesiinallttt tennnniitteeeii_eeeeeeeee player:: Optional [bool]
None,
is_celebrity: Optional [bool] None, is is_athleto: Optional [bool] = None, is is_wcll_paid: Optional [bool] None, lives_in_tax_ho havenn: Optional [bool] None, is_famcus: [poional[bool]]] None, name: str "Djokovic"):
is_grand sian_champion
self.is self h.as.sa_poteesiiooaLLeeaaaaaaaa player
is_oscar wwwinated actor
is.profassional__cmmml_____llll
self self.is_celebrityy - -ac is_celebrity
self.is_athlete is_athlete
self.is.well =ss is_well_n paid
self.lives_in_taxt haven lives lives_in_tax
sasssinn_txx_aaeee haven
self.is_famous is famous
def apply_premises (person: Person) bool:
changed False
Premise if person. person.is_grand__ aarnonii__ggandssla_cheeeee is oscar wwwinated actor is None:
is None and person.
person.is grand slam champion True changed True
Premise
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if perscn. person.is_grrandsslmmmchaaapppp is True and person.
oeaproffssiioaa__emmsssppppppp is None:
persen. is professional_teniie player True changed True
Premise 3 if person.is_oscar ias__acca__onnateeeaacct is True and person.
is_celebrity None:
persen. is__clebrittyyyyeeeeeeeeeeeee___ True
changed True
Premisc
if persen. persen.is_proffete. tugrrofesssionnlitnssss..eg True and person. ls_athlete is None:
person,is_ad is_athlet - True
changed True
Premise if jennbbbiteeebrtyyy is True and person,is pearsnnissxxell__ well paid is
None: person.is_well_1e True changed True
Premise 6
if person. is_ at_atheete is True and person. is famous is None: persen. sesscnnis_faoore True changed True
Premise
paid is True and person. 1ives tax haven if persen. paid
person. www.el_in_taaa_haeeeeee True changed True
Premise
if persen. name "Djokovic" and person. fanous is True and person.is_well ie__wll_w_edddd is Nonc:
True
changed True
return changed
def run inference (person: Persen)
while apply (person): pass
def check_ccncluusio (person: Person) -> str: run_inference (person)
if perscn.nane "Djokovie" and person. lives in haven True:
return "False"
clse: return "True"
def func():
person Person
"s__grand_slam_champiioo
is_oscar js__sccr__nmiinet www.nateed.ctoorteeeeoooooe
professional tennis player=Nene, te
is us.ccelebriy=NNone
is_athlete-None,
is__ell_paiddeeeee
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1 lives_in_tex_iie jessinn_tax_have-Noneee is_ famcus-None, name="Djokovie"
return conclusioni (person)
if name_ main_'n
result func
print ("Conclusion: Djskovic does net live in tax haven?",
result)
<end_of_code:
<answer>
The final answer (B). cend_of_answer
G.3
TRUTH TABLE BASED REASONING
Examples of Truth Table-based Reasoning generated by our models.
The following is the problem you need to solve.
<premises>
People in this club who pertorm in school talent shows often attend and are very engaged with school events.
People in this club either perform in school talent shows often or are inactive and disintereated community members.
People in this club who chaperone high school dances are not students who attend the school.
All people in this club who
inactive and disinterested menbers of their community chaperone high school dances. All young children and tcenagers in this club who wish to further their academic careers and educational opportunities are students who attend the school.
Bonnie is this club and she either both attends and very engaged with school events and is student who attends the school or is not someone who both attends and is very engaged with school events and is not student who attends the school.
</premises>
<conclusion> Bonnie performs in school talent shows often. </conclusion>
<question>
Is the follewing statement true, false, uncertain? Bonnie performs in school talent shows often.
</question>
coptions>
(A) True
(B) False
(C) Uncertain </options>
<truth_tabe
Let:
T: Bonnie performs in school talent shows often.
E: Bonnie attends and is very engaged with school events.
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S: Bonnie is a student who attends the school.
I: Bonnie is inactive and disinterested community member.
C: Bonnie chaperones high school dances.
Premises:
1. B +8
2. VI
+78
4. +C
5. $ -
6. (E AS)
V(-E A-8)
Logical chain (assuming
Falsc)
From (2): I =Fass -> I True
From (4)
True -> C = True
From (3) C =Trree
True -> -$ True -> False
From (6) =False
True E Fals
From (1): False -> E-4 holds vacuously
-> No contradiction if T False
Logical chain (assuming
True):
From (2)
True
False
From (4) I False -> C False
From (3): C =Fasse -> - = False <> S True
From (5): = ->
True
True
From (6):
True AS True (EAS) True
-> No contradiction if True
Truth Table (remaining consistent rows) C |
True
True True False False False | False | True | True
Palse
<end truth table>
<answer>
The final answer is (C). <ons_of_
Examples of Truth Table-based Reasoning generated by our models.
The following is the problem you need to solve.
<premises>
All fir trees are evergreens.
Some objects of worship are fir trees. </premises>
<conclusi
Some evergreens are //conclusions
not objects of worship.
<question>
Is the following statement true, false, or uncertain? Sone evergreens are not objects of worship. </question>
<options>
(A) True
(B) False
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(C) Uncertain </options>
Struth_tables
Let:
F: An object is fir tree.
E: An object is evergreen.
-: object is an object of worship.
Premises:
1. All fir trees are evergreens.
F +
Some objects worship are fir trees.
Ex (M(x) (x))
Conclusion to Pvaluate:
Some evergreens 3x (x (x)
objects worship.
Logical Chain:
1. From (1), +E.
From (2), there exists some object a such that w (a) (a)
3. Since (()) and F +E, it follows that ((a).
4. Therefore, we have E(a) and (())
5. We need Ex (E(x) A-W(x))
The premises do not provide any information about whether there exists an evergreen that is not an object of worship.
7. Therefore, it is uncertain whether some evergreens are not objects of worship.
Truth Table (Key Possibilities):
F F
E
W
E A-i
True True True
False
True True Falsel True
Falsel
True True
False
Falsel
True
Falsol
True
Rows omitted: any with E= False (irrelevant to EA-W) with True and E= False (violates F-+E).
cend_o__truth_tablee
<answer>
The final answer is (C). answerr
H ERROR ANALYSIS
H.1 EXAMPLES OF ERROR TYPE
Example of Missing Branches
<premises>
All employees who schedule meeting with their customers will go the company building today. Everyone who has lunch in the company building schedules meetings with their customers.
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Employees have lunch either in the company building or at home. If an employee has lunch at home, they are working remotely from home.
Al1 employees who are in other countries work remotely from hone.
No managers work remotely from home.
James will appear in the company today if and only if he is manager.
</premises>
<conclusion> James has lunch in the company. </conclusien>
<question>
Is the follewing statement true, false, uncertain? James has Tunch in the company.
</question>
<options>
(A) True
(B) False
(C) Incertain
</options>
<nl_cot>
Step From the premise \*A11 employees who schedule a meeting with their customers will go to the company building today \" we know that if an employee schedules meeting with their customers, they will be in the company building today.
Step 2: From the premise \"Everyone who has lunch in the company building schedules meetings with their customers, \" can infer that if employee has lunch in the company building, they must schedule a meeting with their customers.
Step From the premise \Employccs have lunch either in the company building or home, \" know that employee only choose one of these two options for lunch.
Step 4: From the premise '"If an empleyee has lunch at home, they working remotely from home, \ we deduce that having lunch at home means they are not in the company building and are working remotely.
Step 5: From the premise \*All employees who are in other countries work remotely from hone, \" know that employees in other countries do not have lunch the company building.
Step 6: From the premise \NNo managers work remotely from home A" we know that managers not have lunch home and thus do not work remotely.
Step 7: From the premise \*James will appear in the company today if and only it he is manager, \" can infer that James is not manager, he will not appear the company today.
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Step 8: Combining the information from steps 1-7, we can conclude that if James is manager, he will be in the company today. Since James will only in the company today it and only he is manager, and managers do not work remotely, James cannot be having lunch at home. Incorrect. It only considers the case where James is a manger, missing another branch: James is not manger.
Step Therefcre, if James not having lunch at home, he must be having lunch in the company building, and by step 2, he must be scheduling meetings with his customers, which means he will be in the company building today.
<end_of_nl_cctt <answer>
The final answer is (A).
<end_of_answer>
<Label>
(C) Uncertain
</label>
Example 2 of Both Missing Branches and Invalid Converse
<premi
People in this club who perform in school talent shows often attend and are very engaged with school events.
People in this club either perform in school talent shows often or inactive and disintereated community members.
People in this club
chapcrone high school dances are not students who attend the school.
All people in this club who are inactive and disintcrested menbers of their community chaperone high school dances.
A11 young children and teenagers further their academic a careers and educational opportunities are students who attend the school. Bonnie is in this club and she either both attends and is very engaged with school events and is student who attends the school or not soneone both attends and is very engaged with school events and is not student who attends the school.
this club who wish
</premises>
<conclusion> Bonnic performs in school talent shows often. </conclusien>
<question>
Is the follewing statement true, false, or uncertain? Bonnie performs in school talent shows often.
</question>
<options>
(A) True
(B) False
(C) Uncertain </options>
<ot>
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Step 1: Analyze the given premises and the statement about Bonnie.
Premise People in this club who perform in school talent shows often attend and are very engaged with school events.
Premise 2: People in this club either perforn in achool talent shows often are inactive and disinterested community members.
Premise People in this club who chaperone high school dances are not students who attend the school.
Premise 4: All people in this club who are inactive and disinterested members of their community chaperone high school dances.
Premise 5: All young children and teenagers in this club who wish to further their academic careers and educational opportunities are students who attend the school.
Bonnie's statement: Bonnie in this club and she either both attends and is vezy engaged with school events and is a student who attends the school or is not someone who both attends and is very engaged with school events and not a student who attends the school. Step 2: Determine Bonnie's status in the club.
According to Bonnie's statement, she is either very engaged and atterding member or not such member. This statement is disjunction (or) of two conditions, which means Bonnie can only fit into one of the two categories described.
Step 3: Consider the implicatiens for Bonnie based on her status in the elub.
If Bonnie is very engaged and attending member, she must perform in school talent shows often (Premi se 1) Incorrect (Invalid Converse). It incorrect ly utilizes "Bonnie is very engaged and attending member -> perform".
Bonnie is not a very engaged and sttending member, she must be an inactive and disinterested community member Premise 2). Inactive and disinterosted members must chapcrone high school dances (Premise 4), which means they cannot be students who attend the school (Premise 3)
Incorrect (Missing Branches). It fails to enumerate all possible cases stated in Premise.
Step Evaluate the given statement.
The statement claims that Bonnie performs in school talent shows often.
Given the conditions and implications, if Bonnie is not a very engaged and attending member, she must be she would
inactive and disinterested member, which be a chaperone and not student, contradicting Premise 5. Therefore, Bonnie must be a very engaged and attending member.
Step 5: Conclude based the analysis.
since Bennic must be a very engaged and attending member, she performs in school talent shows often.
<end_of_nl_com
<answer>
The final answer is (A)
<end answer?
<Label>
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(C) Uncertain </label>
Example 3 of Missing Branches
<premises>
of Zaha Hadid's design styles that Max adores have interesting geometries.
No brutalist buildings that Max adores have interesting geometries.
Every style that Max adores either Zaha Hadid's s design style or Kelly Wearstler's design style.
All of Kelly Wearstler's design styles that Max adores evocative.
All of Kelly Wearstler's s design styles that Max adores are dreamy.
If a design by Max that he adores has interesting geometries, then the design is brutalist building and evocative. </premises>
<conclusion>
A design by Max is evocative and dreamy. </conclusion>
<question>
Is the follewing statement true, false, or uncertain? A design by Max is evocative and dreany.
</question>
<options>
(A) Truc
(B) False
(C) Incertain
</options>
<ot>
Step Analyze the premise \"AL1 of Zaha Hadid's design styles that adores have interesting geometries. This means if a design by Max adores is a zaha Hadid design, then it has interesting geonetries.
Step Analyze the premise \"No brutalist buildings that Max adores have interesting geometries. This means a design by Max adores is a brutalist building, it does not have interesting geometries.
Step Analyze the premise \"Rvery style that adores is ehe Xaha Hadid's design style or Kelly Wearstler's design style. \" This means all designs by Max adores must be either Zaha Eadid's Kelly Wearstler's.
Step Analyze the premise \"All of Kelly Wearstler's design styles that Max adores are evocative. This means if design by Max adores is Kelly Wearstler design, then it is evocative.
Step Analyze the premise \*All of Kelly Wearstler's design styles that Max adores are dreamy. \ This means if a design
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by Max adores is a Kelly Wearstler design, then it is dreamy.
Step Analyze the premise \"IE design by Max that he adores has interesting geometries, then the design is a brutalist building and evocative. This means if a design by Max adores has interesting geometries, then it is both brutalist building and evocative.
Step 7: From Step 1 and Step 6, if a design by Max adores is a Zaha Hadid design and has interesting geonetries, it must also be brutalist building and evocative.
Step 8: From step 2, if a design by Max adores is a brutalist building, it does not have interesting geometries. So, a design Max adores cannot be both brutal ist building and have interesting geometries.
Step 9: Since design by Max adores cannot have both interesting geometries (which would make it a brutalist and evocative) and not have interesting geometries (which would make cannot have interesting geometries.
brutalist), the design by Max adores
Incorrect (Missing Branches). It fails to enumerate all possible cases stated in Premise 3.
Step 10:
Since the design by Max adores does not have interesting geometries, according to Step and Step 5, it cannot be a Kelly Mearstler design. Therefore, it must be Zana Hadid design.
Step 11:
Since the design by Max adores a Maha Hadid design, it must have interesting geometries (from Step 1), but this contradicts the conclusion that it cannot have interesting geometries (from Step 8).
Step 12: Given the premises, it is impossible for a design by Max adores to be both evocative and dreamy, as it would require having interesting geometries and not having them the same time.
<end_of_ni_cet>
<answer>
The final answer is (B). <end answer>
<Label>
(A) True
</Iabel>
H.2 EXAMPLES UNIQUELY SOLVED BY TRUTH TABLE
Examples uniquely solved by Truth Table
<premises> rSngen Things" is popular Netflix show. If a Netflix show is popular, Karen will binge-watch it.
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If and only if Karen binge-watches a Netflix show, she will download it.
Karen does not download "Black Mirror." "Black Mirrer" is Netflix snow. If Karen chtss a Netflix show, she will share it with bisa. </premises>
cconclusion>
"Black Mirror" is popular.
</conclusicn>
<question>
Is the following statement true, false, or uncertain? "Black Mirror" is popular.
</question>
<options>
(A) True
(B) False
(C) Incertain
</options>
<nl_cot>
Step 1: Given that \"Stranger Things\" is a popular Netflix show and if Netflix show is popular, Karen will binge- watch it, we know that Karen would binge-watch "Stranger Things)" if it were popular. #Correct.
Step However, the popularity \"Stranger Things)" does not provide any direct information about the popularity \" Black Mirror. #Correct.
Step 3: We know that \"Black Mirror\" is Netflix show, but we do not have any information that directly links its popularity to Karen's actions. #Correct.
step 4: The statement that if Karen binge-watches a Netflix show, she will download it, and the fact that Karen does not download "BBlack Mirror, \ imply that Karen does binge-watch \"Black Mirror. #Correct.
Step 5: Since Karen does not binge-watch \"Black Mirror, \" and it is popular, Karen will binge-watch it, cannot conclude that \"Black Mirror)" is popular based on Karen's actions. #ncorreec.. Fail to utilize Transposition.
Step The conclusion \"Black Mirror\" is popular cannot be definitively supported or refuted with the given premises. #Incorrect.
<end_of_nl_cot>
<answer>
The final answer (C). #Incorrect.
<end_of_answer>"
<truth table>
P: \"Stranger Things\" is popular Netflix show.
Q: \"Black Mirror is a popular Netflix show.
B: Karen will binge-watch Netflix show.
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D: Karen will download a Netflix show.
S: Karen will share Netflix show with Lisa. Premises:
P True (Given) #Correct.
If P, then B (Given) #Correct.
If B, then D (Given) #Correct.
If and only if B, then D (Given) #Correct.
\"Black Mirrer\" If B, then (Given) #Correct.
a Netflix show (Given) #Correct.
Conclusion to Evaluate:
Q (Is \"Black Mirror(" popular?) #Correct.
Logical Chain:
From (5), \"Black Mirror\" is Netflix show. #Correct.
From (2) and (3), if \"Black Mirror\" is popular (2), then
Karen will binge-watch it (B). #Correct.
From (4), if Karen binge-watches \"Black Mirror, \" then she will download it (D). #Correct.
However, frem (6), we know that Karen does not download \" Black Mirror, \" so D = False. #Correct.
Since в implies D (from (3) and (4)), and D is False, B must also be False. #Correct.
Since is False (Karen does not binge-watch \"Black Mirror
\"), Q (popularity of \"Black Mirror\") cannot be True. #Correct.
Truth Table (Key Possibilities):
(0: \"Black Mirror\" is popular) B (B: Karen will binge- watch \"Black Mirror\") Mirror\")
(D: Karen will download \"Black
False 1 False | False
True True True
Given that ralse, Q must be False. #Correct.
<end_of_truth__table
<answer>
The final answer is (B). #Correct.
<end answer>
<code>
from typing import Optional
class Show:
def iinit_(seff, name: str, is_popular: Optional [bool] self.name name
None, is_downloaded: Optional [bool] None, is ss_binged:
[otionallbool] None):
self.is_popular is_popular self.is iafFis__donnoaddes is ss_dowwlodded
self self.is_binged is is_binged
def agply_premises Show) -> bool:
changed False
Premise 1: "Stranger Things" is popular Netflix show. if Chow.namt "Stranger Things"; #Correct.
show, inww.is_popular None: shew. hhowiispoopular True changed True
Premise 2: a Netflix show is popular, Karen will binge- watch it.
if show is_popular is True and name !- "Black Mirror": #Correct.
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if show. is_binged is None:
show. www.is_ht is_bingedd True changed True
Premise 3: If and only if Karen bitgeesatches a Netflix show, she will download it.
if show.is_binged is True: #Correct. show. show.is_downloeee show. is wwwwiis_dowee ii__ddddlllddddd True changed True
None:
Premisc 4: "Black Mirror" is Netflix show. #Incorrect. Do not encode Transposition. 2. directly set show.is.popular False,, so even if it set it correctly, as it do not include "ransposition, code approach still cannot work correctly.
if show name "Black Mirror":
show.is thow.is_pepular None:
show.is popular hwwwiis_pppular False
changed True
Premise
Karen binge-watches Netflix show, will share it with Lisa.
if show. shon.is_binged is True: #Correct.
if show. shcw.is_downleset downloaded is True:
if show, show. Neww.is_ppuulas is False:
"Black Mirror":
show.is_populae = True changed = True
changed
def run run_inference (show: Show)
while propiisspemieseddeeeeee (show) (shox) pass
def check_ccnclusionn (show Show) -> str: run_inference (show)
if show name "Black Mirror";
if show. nowwiss_p is__ppullarur is True: return "True"
elif shew. show.is_popular is False: return "False"
else:
return "Uncertain"
else:
return "Uncertain"
def func ():
show Show (name="Black Mirror", popularr-one, is di_dowlloddd--Nors is binged-NooN)) return check_conclusion (show)
if name result func()
mainn'n
print ("Conel usion: ("Conclusion:
BBlck Mirror! is popular?", result)
<end of code>
<answer> The final answer is (C). #Incorrect. oanswrrs
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Examples uniquely solved by Truth Table
<premisess ses>
All of Peter's pets that are mammals are also animals. All monkeys are mammals. Peter's pets are all either monkeys or birds. Peter's birds can fly.
All animals that are Peter's pets can breathe.
If Peter's pet can fly, then it has wings. Rock Peter's pet.
Rock can fly, or Rock is bird, or Rock cannot breathe. //sr
<conclusion Rock a menkey. </conclusicn>
<question>
Is the follewing statement true, false, or uncertain? Rock nonkey.
</question>
<options>
(A) True
(B) False
(C) Uncertain
</options>
<n1_cot>
Step 1: From the premise \MAl1 of Peter's pets that are manmals are also animals, \" we know that any manmal pet of Peter's is animal.
Step 2:
From the premise \*All monkeys are mammals, \" we know that if somothing is a monkey, it is also manmal.
Step From the premise \"Peter's pets all either monkeys or birds, \" we know that Rock, being a pet of Peter's, s, must be either monkey or bird.
Step From the premise \"Peter's birds can fly, \ know that if something a bird, it can fly.
Step 5: From the premise \"All animals that are Peter's a pets can breathe, \" know that any pet of Peter's, whether bird mammal, breathe.
Step 6:
From the premise \"If Peter's pet can fly, then it has wings, \" we know that if a pet of Peter's s can fly, it must have wings.
Step From the premise \"Rock can fly, or Rock is bird, or Rock cannot breathe, \" we have disjunction. We need to evaluate each part of this dis diajunction: An- If Rock can fly, then it must have wings (Step 6), and it must be bird Step 4), because only birds can fly among Peter's pets. An- If Rock is a bird, it can fly (Step 4), and it can breathe (Step 5) An- If Rock cannot breathe, this contradicts the fact that all of Peter's pets can breathe (Step 5) #InCorrect.
Rock can fly, then must have wings (Step 6), and must be a bird (Step 4) this step
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incorrectly utilizes rule fly birds to derive rock must be a bird.
Step 8: Given the disjunction in Step 7, the only consistent option that Rock is bird, because if Rock can fly, must be bird, and if Rock cannot breathe, it contradicts the premises. Therefore, Rock is a bird.An\nSte Rock is bird, and all of Peter's pets are either monkeys birds (Step Rock cannot be monkey.
9: Since
<end_of_nl_cct>
<answer>
The final answer is (8).
<end_of_answer
<truth_table
Let:
M: Rock is monkey
B: Rock is bird
F: Rock can fly
A: Rock is an animal
W: Rock has wings
R: Rock can breathe
Premi ses:
1. A1l of Peter's s pets that are mamma 1s are sl animals.
Not directly about Rock)
2. All menkeys are mammals. (Net directly about Rock)
3. Peter's pets directly about Rock)
all either monkeys birds. (Not
4. Peter's s birds can fly. +F)
5. All animals that are Peter's pets can breathe. (A +R)
6. If Peter's pet
fly, then it has wings. -W)
7. Rock is Peter's pet.
8. Rock can fly, or Rock a bird, or Rock cannot breathe.
(F VB V-R)
Conclusion to Evaluate: Rock is a menkey (M)
Correct. The model correctly produce two cases that lead to di fifferent conclusions, f.e., True and False. Therefore, it finally produces Uncertain the final answer.
Truth Table (Key Possibilities)
(Monkey) (Bird) (Can Fly)
(Animal) W (Wings)
R (Can Breathe)
True False | True True True
True
False | True True True True
True
<end_of_truth_tabl <answer>
The final answer (C).
</end answer>
<code>
from typing import Optional
class Pet:
40

---

Published as a conference paper at ICLR 2026
def init (self,
is ss_menkey: Optional [bool] None,
is_bird: Optional [bool] None,
can_fly: Optional [boo]] None,
has has_wings: Optional [bo0] None,
is_animal: Optional [bool] None)
self.is_menkey is_monkey
self.is_bird is_bird
self.can_fly can_fly
AaF.has_wwingh has sas_wings
self.is_animal is_animal
def apply_premises (pet: Pet) beol:
changed False
Premise se 1: A1l of Peter's pets that are manialls are also animals.
Since all of Peter's pets are either monkeys or birds, and monkeys are manmals,
# this premise does not directly affect the state of the pet
Premise 2: All monkeys are mammals. if pet. iet_is_n is True and pet. antimal is not True: bet.is_animal = True changed True
Premise 3: Peter's s pets are all either monkeys or birds. if pet.is_menkey is None and pet betris_bird is None: pet.is_menke False pet.is_birg False changed True
# Premise 4: Peter's s birds can fly.
if pet.is bird is True and pet.can fly is not True: cat fly True changed True
Premise 5: All animals that are Peter's pets breathe. if pet. pet.is_animal is True and pet.can_fly is not True: can_fly True changed True
Premise If Peter's pet fly, then it has wings. if pet. can #ly is True and pet.has has_igssings not True: pet.has_wings True changed True
Premise Rock is Peter's pet.
This does not change the state the pet but indicates that the pet is Rock.
Premise 8: Rock breathe.
fly, Rock is bird, or Rock cannot
pet.name "Rock": if pet.can_fly fly is None:
pet.can_fly True changed True pet.is bird None: pet.is_ bird True changed True
if pet.can_fly is True and petwhas_wings is not True:
41

---

Published as a conference paper at ICLR 2026
www.haswwigsss = True changed True
Incorrect. The model struggies correctly interpret premises that involve multiple possible branches reasoning, such as disjunctions (V) It often comnits prematurely to one or more branches instead of preserving logical uncertainty.
return changed
def run run_inference (pet: Pet): while apply_premises (pet): pass
def ceces_cccclusionieo Pet) -> str: Sen_inferenceepet
if pet. name "Rock" and pet. is_roney is True: return "True" else: return "False"
def func
pet Pet ( is_menkey=None, is is_bird-Nene, can_fly=None, has has_wings-None,
is is_animal=None
pet.nann "Rock" check_conclusi (pet)
if name - main
result func
print ("Cenclusion: Rock is monkey?", result) <end_of_code <answer>
The final answer is (B).
</end__ff_asswer>
42


Reviews:
[{"review_id": "vCJgJvXNeZ", "reviewer": "Reviewer_6rwT", "summary": "The paper introduces Mixture-of-Thought, a framework that enables large language models to reason using three complementary modalities ( i.e., natural language, code, and a symbolic truth-table format ). Unlike prior methods that train with a single modality and only vary modalities at inference, MoT trains and infers across modalities to exploit their synergy. A self-evolving training phase jointly learns from filtered, self-generated rationales in all three formats, while the inference phase integrates them to produce stronger predictions. The truth-table modality systematically enumerates logical cases, helping mitigate common failure modes of natural-language reasoning. Experiments on logical reasoning benchmarks such as FOLIO and ProofWriter show MoT significantly outperforms strong single-modality chain-of-thought baselines, with average gains up to 11.7 percentage points.", "questions": "1. To better highlight the impact of MoT inference, I suggest also reporting the Self-Consistency with three votes metric for the MoT with Single-Thought inference results in Table 2.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper is well-written and easy to follow.\n2. The proposed Self-Evolving MoT Training method is reasonable and effective.\n3. Extensive experiments and ablations were conducted on various models and showed strong performance.", "weaknesses": "1. The proposed method appears applicable only to specific logical reasoning tasks, and its generalization to broader reasoning tasks or other domains remains uncertain.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "oNcsOzPgo3", "reviewer": "Reviewer_1P9u", "summary": "This paper proposes a framework called Mixture-of-Thought (MoT), which aims to enable an LLM to jointly learn and utilize three complementary reasoning modalities (natural language, code, and a novel truth-table modality) within a unified model via a self-evolving training process. This method significantly outperforms single-modality baselines on standard logical reasoning benchmarks (FOLIO, ProofWriter), particularly by introducing the truth-table modality to address key bottlenecks in natural language reasoning, such as \"missing branches\" and \"invalid converse\".", "questions": "See above.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- The paper's motivation is clear and well-supported. The fusion of three modalities is experimentally shown to effectively enhance model performance. Notably, the design of the truth-table modality is driven by a thorough error analysis shown in Figure 1c, which pointedly addresses key bottlenecks of NL CoT.\n\n- The paper proposes a self-evolving training method that enables a single LLM to jointly learn and synergistically utilize multiple reasoning modalities, addressing the \"modality-blind\" problem of existing methods during the training phase.", "weaknesses": "- The generalization ability of the truth-table (TT) modality is questionable. This modality is currently highly customized for logical reasoning and has only been validated on two logical reasoning benchmarks. I suggest the authors discuss or explore the framework's potential application in other reasoning domains, such as mathematical or commonsense reasoning.\n\n- Experiments in Table 3, Figure 3b, Figure 4, and Figure 7 show that the code modality almost always performs worse when used alone. However, Table 3 also shows that including it in joint training still yields performance gains. It will be better to provide supplementary qualitative or quantitative analysis on: (i) What are the specific advantages of the code modality? (ii) What specific problems or sample types does it solve that the other two modalities cannot?\n\n- It can benefit from further analyzing the application scenarios for each modality. If it's possible to identify which problems are best suited for a specific modality, perhaps a dynamic routing strategy could be employed at inference time instead of performing three separate inferences for all problems (as in MoT-Inference), thereby optimizing the high inference cost.\n\n- Please provide a more detailed explanation of the \"reason to prune\" step within the Truth-Table CoT, either in the main paper or the appendix. The current description, \"eliminates rows that violate any premise through reasoning via LLMs\", is too abstract and lacks sufficient implementation details for reproducibility.\n\n- Including a post-MoT-training error analysis is necessary, similar to the pre-training analysis in Figure 1c. This would help verify whether MoT training not only improves ensemble performance but also enhances the NL modality itself, for example, by reducing the frequency of \"missing branch\" and \"invalid converse\" errors.\n\nSome recent relevant works should be included to discuss or compare with the proposed method: \n\n[1] Tree of Thoughts: Deliberate Problem Solving with Large Language Models\n\n[2] Cumulative Reasoning with Large Language Models\n\n[3] DetermLR: Augmenting LLM-based Logical Reasoning from Indeterminacy to Determinacy", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "MKPQOpi6rg", "reviewer": "Reviewer_yjSj", "summary": "This paper proposes a method to fine-tune LLMs to reason in three different “thought modalities” for solving logical reasoning problems. They train in a self-evolving manner where they generate thoughts and use thoughts which resulted in correct answers as the data for the next iteration. To perform inference, the model responds in each of the three modalities and then takes the majority answer as the final answer. They show that this approach improves in accuracy over using any one of the thought modalities as well as helps on the hardest of reasoning problems.", "questions": "1. Does MoT Single-Thought use majority voting from three samples? If not, I think this is an important baseline as well.\n2. How should one come up with the relevant reasoning modalities? Are there a set of general reasoning modalities that could work for any reasoning problem?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "* The “thought modality” fine-tuning appears to be novel. The exploration of fine-tuning on different “thought modalities” such as coding and CoT has been extensively studied in terms of post-training on different datasets (coding problems, math word problems, etc.), but I am not aware of work which fine tunes on different solution methods for the same dataset.\n* The paper is well written and easy to follow.\n* The problem of improving reasoning performance by increasing “thought” diversity is important and this approach to fine-tuning on different solution strategies can be impactful.", "weaknesses": "* Experiments are only performed on two datasets and results are missing standard deviations.\n* The natural language and code reasoning modalities seem highly general, but the truth table modality seems specific to the two benchmarks used in this paper. If using MoT on a different reasoning dataset, this could potentially harm performance.\n* The experiments are a little hard to evaluate due to the confounder of the number of samples which different methods use and the different models used. For instance in Table 2, the last row MoT method takes the majority answer from 3 samples, but I assume that the line above that (MoT Single-Thought) only uses one sample. Also, Logic-LM is only evaluated with GPT-4.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "0TIodNaAdy", "reviewer": "Reviewer_Fz8C", "summary": "This paper proposes an approach to improve the logical-reasoning capabilities of LLMs by incorporating different reasoning modalities into the decision process. Study aims treat natural language, code and truth table as different reasoning interfaces and apply majority voting to determine model's output to a logical reasoning problem after training the model on multiple reasoning modalities. Performance of the proposed approach has been evaluated on FOLIO, ProofWriter and ProverQA benchmarks, compared with Gemma-2-2B/9B and Qwen-2.5-7B-Instruct.", "questions": "- Include comparisons with models better at reasoning.\n- Additional experiments on the efficiency of truth-table as a reasoning interface would proof its contribution.\n- Evaluation with more comprehensive reasoning benchmarks (e.g. math reasoning) would highlight this approach's capabilities and limitations.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The study harvests different reasoning interfaces and connects NL reasoning with symbolic reasoning in a single framework.\n- Clear performance superiority compared to Qwen-2.5-7B-Instruct, a model capable of reasoning.\n- Proposed approach does not require any external teachers.\n- Uses interpretable reasoning interfaces, provides space for interpretability studies.", "weaknesses": "- Gemma-2-2B/9B models are inherently not proficient in reasoning tasks, their pre-training lacks exposure to used reasoning interfaces (code and truth table).\n- Evaluation domain is restricted to logical reasoning which is not the most suitable domain for harvesting different multimodal reasoning interfaces, specifically code.\n- Although it is symbolic, truth-table's capability and suitability as a reasoning interface is not convincing.\n- Majority voting is a shallow approach for combining the outcomes of reasoning interfaces. It is not suitable for different reasoning problem types.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)