Cumulative Reasoning with Large Language Models

• Current LLMs struggle with intricate reasoning and robust logical inference despite broad application progress.Studies cited in the paper also report difficulties on high school mathematics problems.Source: §1 · p. 1
• Prior CoT and ToT methods guide sequential or hierarchical reasoning but often lack flexible mechanisms for storing, verifying, and cumulatively leveraging validated intermediate results.CR targets these limitations through iterative validation and cumulative construction.Source: §1 · p. 2
• CR coordinates proposer, verifier(s), and reporter roles to iteratively generate, validate, and compile reasoning steps.Its dynamic DAG stores validated reasoning steps so later deductions can build on a problem-specific verified knowledge base.Source: §1 · p. 2
• 98.04% accuracy on curated FOLIO and up to 9.3% improvement in logical inference tasks demonstrate CR’s advantage in logical reasoning.Source: §1 · p. 2
• CR achieves 98% accuracy in Game of 24, a 24% improvement over ToT, and improves MATH performance by 4.2% overall and 43% relatively on level 5 problems.Source: §1 · p. 2

2.1 Logic

• Propositional logic uses atomic propositions, logical connectives, truth values, and Boolean laws such as absorption, distributivity, and complementation.Source: §2.1 · pp. 2–3
• First-order logic extends propositional logic with universal and existential quantifiers for reasoning about collections of objects.The section also notes that higher-order logic quantifies over functions and predicates.Source: §2.1 · p. 3

2.2 Illustrative Example

• The illustrative FOLIO-style problem asks whether Rock has wings from premises about monkeys, animals, birds, flight, and Rock’s properties.Source: §2.2 · p. 3
• The derivation concludes that Rock has wings by inferring it is not a monkey, then a bird, then able to fly, and finally winged.Source: §2.2 · p. 3
• The apparently linear derivation is more naturally represented as a DAG whose edges encode individual, interdependent inferences.This structure captures the cumulative nature of the reasoning process.Source: Figure 1 · p. 3, §2.2 · p. 4

3 Cumulative Reasoning

• CR builds solutions by incrementally accumulating and verifying intermediate reasoning steps rather than producing a single linear chain of thought.The approach is motivated by iterative refinement, constructive proofs, and human problem-solving processes.Source: §3 · p. 4
• CR assigns Proposer, Verifier(s), and Reporter roles to generate, validate, and synthesize reasoning steps within a shared DAG.Only verified steps enter the evolving reasoning structure, while the Reporter concludes once sufficient validated information has been gathered.Source: §3 · p. 4

3.1 Comparison with CoT and ToT

• CoT risks error propagation, whereas ToT explores branches but may prune them without consolidating verified knowledge into a reusable structure.CR instead retains historically validated reasoning steps for later inference.Source: §3.1 · p. 5

3 Cumulative Reasoning

• CR derives new propositions from any combination of validated DAG nodes, creating a more interconnected context than linear chains or limited tree paths.Its acyclic structure supports forward progression while avoiding repeated exploration of invalid paths.Source: §3.1 · pp. 5–6, Figure 2 · p. 5

3.1 Comparison with CoT and ToT

• Under a near-perfect-verifier assumption and equal search breadth, the theorem states PCoT-SC ≤ PToT ≤ PCR.The analysis models CR’s next-step probabilities as conditioned on its accumulated history of validated states.Source: §3.1 · pp. 7–8
• Sequential decomposition with intermediate verification improves Game of 24 accuracy in the conceptual experiment.The experiment separates selecting an intermediate number from forming the final expression and repeats each case 1000 times.Source: §3.1 · p. 8, Table 1 · p. 8

4.1 FOLIO Wiki

• On FOLIO-wiki, CR improved over Direct prompting, CoT, and CoT-SC by up to 8.62%, with GPT-4 reaching 87.45% versus 85.02% for GPT-4 with CoT-SC.The testable FOLIO-wiki collection contains 534 examples after exclusions for few-shot prompts and missing source labels.Source: §4.1 · p. 9, Table 2 · p. 9

4.2 FOLIO Wiki Curated

• 98.04% accuracy was achieved by GPT-4 with CR on the curated FOLIO-wiki dataset after 74 problematic instances were removed.The curated set contains 460 examples, and the reported error rate is 1.96%.Source: §4.2 · p. 10

4.3 AutoTNLI

• On AutoTNLI, LLaMA-65B with CR achieved a 12.8% accuracy improvement over CoT-SC on the first 1,000 table-hypothesis pairs.The study compares Direct, CoT, CoT-SC, and CR using LLaMA-13B and LLaMA-65B; ablations examine components including the verifier and premises random choice mechanism.Source: §4.3 · p. 10, Table 5 · p. 10, Table 6 · p. 11

4.4 Game of 24

• CR achieved 98% accuracy on the Game of 24, improving over ToT’s 74% while exploring significantly fewer states.CR introduces one verified state at a time, whereas ToT generates multiple candidate states during each iteration.Source: §4.4 · p. 11, Table 7 · p. 11

4.5 Solving MATH Problems

• Without code, CR outperformed Complex CoT by 5.4% overall and by 9.7% on Level 5 MATH problems, corresponding to a 43% relative gain.The evaluation used a 500-example subset spanning difficulty Levels 1–5, with competitive iteration counts.Source: §4.5 · p. 12, Table 10 · p. 12
• CR with a Python code environment achieved 72.2% overall accuracy on MATH, a 38.9% relative improvement over PAL and an 18.8% improvement over ToRA.On Level 5 problems, CR with code showed a 66.8% relative improvement over PAL and a 12.8% improvement over ToRA.Source: §4.5 · p. 13, Table 11 · p. 13, Table 12 · p. 13

Reasoning with Large Language Models (LLMs)

• LLM reasoning research also includes synthetic formal-logic corpora and hybrid symbolic systems using code environments, knowledge graphs, and theorem provers.These approaches illustrate the broader use of symbolic or external systems to augment language-model reasoning.Source: §5 · p. 13

Chain-of-Thought Prompting

• CoT, self-consistency, task decomposition, beam search, and related methods improve reasoning by generating, exploring, or refining intermediate reasoning paths.These approaches include CoT, Least-to-Most, Decomposed Prompting, beam search, Complex CoT, and DIVERSE.Source: §5 · p. 14

Advanced Reasoning Frameworks

• ToT explores multiple reasoning paths and self-evaluates choices, whereas CR emphasizes cumulative DAG construction and distinct Proposer-Verifier-Reporter roles.CR can explore branches, such as through parameter b in Game of 24, but primarily constructs one coherent reasoning DAG.Source: §5 · p. 14
• Graph-based approaches represent thoughts as nodes and dependencies as edges, while CR operationalizes this structure through iterative role-based construction and explicit verification.The comparison positions CR as a specific implementation of graph-based reasoning rather than merely a graph representation.Source: §5 · p. 14

Self-Critique and Iterative Refinement

• CR builds a persistent, growing DAG of validated knowledge that informs subsequent reasoning steps, unlike methods that refine one trace or select among alternatives.Its verification is integrated into step-by-step DAG construction rather than applied only after generating multiple candidate paths.Source: §5 · pp. 14–15
• CR’s current Verifier often makes binary validity decisions or relies on external tools, motivating future integration of richer feedback mechanisms.The related-work discussion identifies more detailed linguistic and generative verification as directions for extending CR’s verifier.Source: §5 · p. 15

6 Conclusion

• CR orchestrates proposer, verifier(s), and reporter roles to decompose problems and recompose validated steps into comprehensive solutions.The method uses a structured, iterative process intended to mirror human cognitive strategies.Source: §6 · p. 15
• CR demonstrates superior performance across logical inference, the Game of 24, and MATH problems.The conclusion presents these domains as evidence of CR’s versatility in complex problem-solving scenarios.Source: §6 · p. 15

A Proofs of Theorems

• The appendix combines Lemma 3.3, Assumption 3.2, and intermediate deductions before concluding the proof.The supplied passages mark these dependencies but do not state their full contents.Source: A Proofs of Theorems · p. 22
• The argument uses ordering relations among expressions involving p_1 and p_2 to establish the needed inequality bounds.The proof states the chain (1 − p_1 − p_2 + p_1 · p_2) ≤ (1 − p_1) ≤ (1 − p_2) ≤ (1 − p_1 · p_2).Source: A Proofs of Theorems · p. 22
• Monotonicity and symmetry of the relevant power-function terms complete the comparison around the midpoint 2 − p_1 − p_2.The proof invokes monotonicity on the specified interval and symmetry of the paired expressions.Source: A Proofs of Theorems · p. 22
• The proof of Theorem 3.4 transforms the target inequality through successive algebraic equivalences.The displayed steps rewrite powers and products involving p_1, p_2, and n into equivalent inequalities.Source: A Proofs of Theorems · p. 22

B.1 More Experimental Results

• Third-party reproduced results are reported for LogiQA, ProofWriter, FOLIO validation, and LD to support fair method comparisons.Implementation details are deferred to the cited work by Sun et al. (2023).Source: §B.1 · p. 22
• Table 13 presents comparison results on LogiQA.The supplied table caption identifies the dataset scope but does not provide cell values.Source: Table 13 · p. 23
• Table 15 presents comparison results on the FOLIO validation set.The supplied table caption identifies the evaluation split but does not provide cell values.Source: Table 15 · p. 23
• Table 17 compares results across different datasets.The supplied caption does not specify the individual values or winning method.Source: Table 17 · p. 23

B.2 Computational Considerations

• 98% accuracy with 14.86 visited states versus ToT’s 74% with 61.72 states shows CR can improve accuracy while exploring fewer states.The comparison is reported for Game of 24 with beam size b=5.Source: §B.2 · p. 23
• CR may require more LLM calls than a single CoT pass because each reasoning cycle can include Proposer and Verifier calls.The number of iterations is controlled by the Reporter or a predefined limit.Source: §B.2 · p. 23
• Symbolic verifiers such as a Python interpreter can reduce verification cost while increasing reliability compared with LLM-based verification.The paper contrasts LLM verifiers in some FOLIO experiments with code-environment verification in MATH experiments.Source: §B.2 · p. 23
• Conditioning on previously validated DAG steps can expand the Proposer’s context and increase token counts.Selecting only the most relevant prior steps is proposed as a way to manage context size.Source: §B.2 · pp. 23–24
• CR uses one underlying LLM for all roles and is presented as suitable for test-time scaling when accuracy justifies extra computation.The architecture distinguishes roles through prompts rather than loading multiple large models.Source: §B.2 · p. 24

C More on Logic

• First-order logic is limited by insufficient expressiveness, translation misalignment with natural language, and undecidability in general truth determination.FOL cannot naturally represent some properties of real numbers, may mismatch predicates during translation, and has an undecidable general truth problem.Source: C More on Logic · p. 24

C.1 Illustrative example on higher-order logic

• Higher-order logic extends FOL through quantification over functions and predicates, generalized quantifiers, modal operators, and attitude or veridical predicates.These constructions represent linguistic and semantic relationships that exceed FOL’s expressive capacity.Source: §C.1 · pp. 24–25
• CR solves difficult higher-order-logic reasoning problems relatively well without relying on or being restricted to symbolic systems.The passage contrasts CR with the difficulty experts and LLMs face when constructing solvable HOL programs and formal theorem-prover systems.Source: §C.1 · p. 25
• The modified FraCas example uses cumulative intermediate propositions to confirm as True that representatives took more than a week to read the report and most valued it.CR derives a total reading time of at least 8 days from sixteen representatives taking at least half a day each, yielding a correct prediction.Source: §C.1 · p. 27

D Appendix for Examples

• The appendix figures show CR using intermediate propositions on FOLIO, while contrasting direct chain-of-thought answers with staged hints, questions, and answers on MATH.The figures cover FOLIO, Game of 24, and MATH examples; the MATH comparison specifically describes CR’s staged reasoning process.Source: Figure 4 · p. 28, Figure 5 · p. 29, Figure 6 · p. 30

E.1 More FOLIO Examples

• In the FOLIO examples, CR correctly predicts False for hyraxes laying eggs, extractive models being not always faithful, no coursing being banned, and correctly resolves the remaining supplied examples.The supplied examples mark chain-of-thought predictions as Unknown for the hyrax, extractive-model, and Allison cases, whereas CR’s corresponding predictions are marked correct; CR also marks the coursing hypothesis False correctly.Source: §E.1 · pp. 32–34
• The example states that everyone at Luis’s dinner party, including Mauricia, has room for dessert.Source: §E.1 · p. 35
• Because room for dessert implies room for broccoli, Allison must have room for broccoli as a dinner-party attendee.Source: §E.1 · p. 35
• The model’s prediction is [True] and is marked correct.Source: §E.1 · p. 35

E.2 Curating FOLIO wiki dataset

• Examples with incorrect labels from the FOL system are removed when curating the FOLIO wiki dataset.Figure 8 illustrates one such example, numbered 679.Source: Figure 8 · p. 36

Type 1 Error: Missing common knowledge or contradictory to common knowledge

• A hypothesis that Croton River water flows to the Bronx is marked Type 1 even though the premises state that Croton River watershed water flows there.The explanation connects the river to its drainage basin and uses the stated watershed relation to support the conclusion.Source: Type 1 Error: Missing common knowledge… · p. 37
• A hypothesis that Bernarda Bryson Shahn was born in Greece is marked Type 1 because the premises place her birth in Athens, Ohio.The explanation also notes that the premises identify people born in Athens, Ohio as Americans.Source: Type 1 Error: Missing common knowledge… · p. 37
• The examples show that Type 1 errors involve either contradiction with stated facts or a conclusion that requires information absent from the premises.The three cases respectively illustrate contradiction, unsupported specificity, and a conclusion linked through the watershed relation.Source: Type 1 Error: Missing common knowledge… · pp. 37–38
• A hypothesis about more income for gate receipts is marked Type 1 because the premises support more income generally but say nothing about gate receipts.The example distinguishes a supported broader conclusion from an unsupported specific qualification.Source: Type 1 Error: Missing common knowledge… · p. 38

Type 2 Error: Overly ambiguous problems failing to provide unequivocal answers

• Type 2 errors occur when premises support a conclusion but leave its truth uncertain, such as whether heat treatment prevents death from a stonefish sting.Similar uncertainty appears when inferring that Inside Out was a punk band and when interpreting P values versus effect sizes in cancer research.Source: Type 2 Error: Overly ambiguous… · pp. 38–39

Type 3 Error: Inherent inconsistencies presented within the premises

• Type 3 errors arise when premises conflict, as William Dickinson is both denied and implied to hold a parliamentary seat.The contradiction follows from supporting the Portland Whigs, whose supporters supposedly did not obtain seats, versus sitting in the House of Commons.Source: Type 3 Error: Inherent inconsistencies… · p. 40

Type 4 Error: Vague premises or typographical errors

• Type 4 errors reflect vague or typographically inconsistent premises, exemplified by the unexplained distinction between “palstave” and “paalstab.”Although the premises establish that palstaves are axes found in Western Europe, the terminology mismatch prevents an unequivocal interpretation.Source: Type 4 Error: Vague premises or… · p. 41

Type 5 Error: Incorrect answers

• Type 5 examples classify both contradicted hypotheses and correctly entailed hypotheses as incorrect answers, including claims about Fort Carillon, New York–Seattle flights, and Michael Crichton.The supplied explanations respectively reject Fort Carillon’s location in Europe, support discomfort on long flights, and support Crichton directing a robot film.Source: Type 5 Error: Incorrect answers · pp. 42–43

F Appendix for Prompts

• The appendix designs role-specific prompts: Proposer generates plausible hypotheses, Verifier assesses validity, and Reporter decides whether reasoning is sufficient to conclude.Prompt templates also constrain logical inference to FOL and direct Game of 24 roles to use valid two-number arithmetic steps and complete expressions.Source: F Appendix for Prompts · p. 44, Figure 9 · p. 44
• The appendix provides a meta prompt for CR with a code environment on solving MATH problems.Source: Figure 16 · p. 49