QMFOL: Benchmarking Large Language Model Reasoning via Quantifiable Monadic First-Order Logic Test Case Generation

Kailong Wang; Ling Shi; Lorenz Goette; Tianlong Yu; Xinyi Zheng; Yongxin Zhao

arxiv: 2606.20227 · v1 · pith:DJKUWPK3new · submitted 2026-06-18 · 💻 cs.AI · cs.SE

QMFOL: Benchmarking Large Language Model Reasoning via Quantifiable Monadic First-Order Logic Test Case Generation

Xinyi Zheng , Ling Shi , Tianlong Yu , Yongxin Zhao , Lorenz Goette , Kailong Wang This is my paper

Pith reviewed 2026-06-26 17:22 UTC · model grok-4.3

classification 💻 cs.AI cs.SE

keywords deductive reasoningLLM evaluationfirst-order logicbenchmark generationlogical complexitymonadic FOLtest case generationreasoning benchmarks

0 comments

The pith

QMFOL generates monadic first-order logic benchmarks with precise control over reasoning depth, width, and label types to measure LLM deductive performance.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces an automated way to build deductive reasoning test cases from formal monadic first-order logic structures that use conjunction and disjunction patterns to set exact levels of complexity. These structures are turned into natural-language problems by language models and then checked for consistency by sending the output back through an external prover. The resulting benchmark contains 2880 instances spanning 960 configurations, and tests on several large models show accuracy falling and compute rising as logical depth or width increases. Models handle true-labeled cases more reliably than false or unknown ones and remain sensitive to how the same logic is worded. This setup lets evaluators dial complexity without losing logical soundness, addressing gaps in earlier benchmarks that lacked such control.

Core claim

QMFOL constructs formal logical structures in monadic first-order logic using controlled conjunction and disjunction patterns, translates them into natural language via LLMs, and applies round-trip verification with an external prover to guarantee consistency; the resulting QMFOLBench of 2880 instances demonstrates that large reasoning models exhibit declining performance and rising computational cost as logical complexity rises, with better results on true labels than false or unknown and clear sensitivity to semantic wording.

What carries the argument

Formal logical structures built from conjunction and disjunction patterns in monadic first-order logic, translated to natural language and verified by round-trip prover checks, that enforce controllable reasoning depth, width, label types, and distractors.

If this is right

LLM accuracy on deductive tasks decreases steadily as logical depth and width increase.
Models achieve higher performance on tasks whose correct label is True than on those labeled False or Unknown.
Computational cost for solving the tasks grows with added logical complexity.
Changing the natural-language wording of the same logical structure alters model success rates.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Benchmark designers could apply the same pattern-based construction and verification loop to other fragments of logic beyond monadic first-order logic.
Training pipelines that sample tasks at targeted complexity levels might produce models whose reasoning scales more evenly.
The verification step points toward hybrid systems that combine language models with symbolic provers for generating reliable evaluation data at scale.

Load-bearing premise

Round-trip verification by an external prover is enough to ensure that the natural-language versions keep exactly the same logical structure and label as the original formal statements.

What would settle it

An independent prover or human logician finding a natural-language instance whose truth value differs from the formal source after the round-trip check passed would show the verification step does not preserve semantics.

Figures

Figures reproduced from arXiv: 2606.20227 by Kailong Wang, Ling Shi, Lorenz Goette, Tianlong Yu, Xinyi Zheng, Yongxin Zhao.

**Figure 2.** Figure 2: Overview of QMFOL. Blue boxes represent inputs, while pink represent outputs. 𝑁𝑖 is a predicate node, either an atomic predicate 𝑃𝑖 or its negation ¬𝑃𝑖 . For simplicity, 𝑁𝑖 = 𝑃𝑖 in the shown task instance. ❶ Basic Rules Construction. As shown in Algorithm 1, the construction starts from the initial predicate node 𝑁0. • Depth Expansion: (line 4-6) We first introduce 𝐷 new predicate nodes and construct the… view at source ↗

**Figure 3.** Figure 3: Prompt Template for FOL2NL conversion. transformation is illustrated in the middle part of [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 4.** Figure 4: Average answer accuracy (%), time overhead (s), and token overhead ( [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

**Figure 5.** Figure 5: Model answer changes from QMFOLBench subsets without distractors to those with varying numbers of distractors, [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗

read the original abstract

Large Language Models (LLMs) have made significant progress in reasoning, particularly in deductive reasoning, which is crucial for high-stakes decision-making. As models improve, evaluation benchmarks should evolve to keep pace. However, existing benchmarks lack fine-grained control over logical complexity and struggle to balance semantic diversity with logical consistency. To address these issues, we propose QMFOL, an automated framework for generating monadic first-order logic reasoning tasks with quantifiable and controllable complexity. It constructs formal logical structures using conjunction and disjunction patterns, enabling precise control over reasoning depth, width, label types, and distractors. These structures are then translated into natural language via LLMs, with logical consistency ensured through round-trip verification using an external prover. Based on our framework, we build QMFOLBench, a benchmark comprising 2880 instances with 960 configurations across diverse logical and semantic dimensions. Evaluations on six large reasoning models (LRMs) and two LLMs show that performance degrades and computational overhead increases with rising logical complexity. Models perform better on True-labeled tasks than on False or Unknown ones, and exhibit sensitivity to semantic variation. Overall, QMFOL offers a scalable and reliable approach for constructing deductive reasoning benchmarks with controllable complexity, enabling more precise evaluation of reasoning capabilities in modern language models.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

QMFOL adds controllable monadic FOL generation with prover checks, but round-trip verification leaves semantic drift risk unaddressed.

read the letter

The core offering is a generator that builds monadic FOL structures from conjunction/disjunction patterns, lets you set depth, width, label, and distractors, translates them to natural language with an LLM, and runs round-trip prover verification to keep the logic intact. They produce QMFOLBench with 2880 instances over 960 configurations and run it on six LRMs plus two LLMs, reporting that accuracy drops and compute rises with complexity while models favor True labels over False or Unknown.

The synthesis of monadic restriction, explicit control knobs, and automated verification is new relative to the benchmarks cited in the abstract. The evaluation shows clear trends on sensitivity to semantic variation, which is useful incremental work for anyone building deductive reasoning tests.

The main soft spot is exactly the one in the stress-test note. Round-trip verification assumes that reconstructing a formal sentence from the LLM-generated English will catch any drift in meaning, scope, or implied restrictions. Monadic FOL in natural language can still introduce ambiguities that a prover on the back-translation would miss, especially once distractors or width changes alter phrasing. The abstract gives no quantitative checks on how often this happens or how they validated the translations beyond the prover step, so the controllability claim rests on an assumption that may not hold.

The paper is aimed at researchers who need tunable logic benchmarks rather than fixed suites. It is coherent enough on its own terms to warrant a serious referee who can examine the verification procedure and the actual result tables in detail.

Referee Report

1 major / 1 minor

Summary. The paper introduces QMFOL, a framework that generates monadic first-order logic structures via conjunction/disjunction patterns with explicit control over depth, width, label type, and distractors; translates them to natural language using LLMs; and applies round-trip verification with an external prover to ensure consistency. From this it constructs QMFOLBench (2880 instances, 960 configurations) and reports that six LRMs and two LLMs exhibit degrading performance and rising compute cost with increasing logical complexity, a preference for True labels, and sensitivity to semantic variation.

Significance. If the verification step reliably prevents semantic drift, the approach supplies a scalable route to deductive-reasoning benchmarks whose complexity parameters are inherited from the formal side rather than fitted post hoc. The independent-prover check and the explicit enumeration of 960 configurations constitute concrete strengths that would allow more precise isolation of reasoning failures than existing datasets.

major comments (1)

[Abstract] Abstract (and the verification procedure described therein): the central reliability claim—that round-trip verification with an external prover guarantees that LLM natural-language translations preserve the original logical structure, label, and complexity—rests on an assumption whose sufficiency is not demonstrated. Monadic FOL rendered in English can introduce scope ambiguities, implicit domain restrictions, or pragmatic implications that a back-translated formal sentence would not necessarily expose, especially once distractors or width/depth parameters alter surface phrasing. Because benchmark labels and complexity metrics are taken directly from the formal side, any undetected mismatch directly undermines the controllability guarantee that underpins the paper’s contribution.

minor comments (1)

The abstract states performance trends but supplies no quantitative tables, error bars, or exclusion criteria; the full methods and results sections will need to include these to allow readers to judge robustness.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying a key point about the strength of the verification claim. We respond to the major comment below.

read point-by-point responses

Referee: [Abstract] Abstract (and the verification procedure described therein): the central reliability claim—that round-trip verification with an external prover guarantees that LLM natural-language translations preserve the original logical structure, label, and complexity—rests on an assumption whose sufficiency is not demonstrated. Monadic FOL rendered in English can introduce scope ambiguities, implicit domain restrictions, or pragmatic implications that a back-translated formal sentence would not necessarily expose, especially once distractors or width/depth parameters alter surface phrasing. Because benchmark labels and complexity metrics are taken directly from the formal side, any undetected mismatch directly undermines the controllability guarantee that underpins the paper’s contribution.

Authors: We agree that the round-trip verification establishes structural equivalence (the back-translated formula must match the original for the prover to succeed) but does not constitute a complete guarantee against all possible scope ambiguities, domain restrictions, or pragmatic implications that may arise in the natural-language rendering. The check is therefore necessary for preserving the formal label and complexity parameters yet insufficient to rule out every semantic nuance. In the revised manuscript we will (1) qualify the claim in the abstract and methods section to state precisely what the verification guarantees, (2) add an explicit limitations paragraph discussing the referee’s concerns, and (3) report a small-scale human fidelity study on a subset of instances to provide additional evidence. These changes will be reflected in the next version. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via external verification

full rationale

The paper's central contribution is a constructive generation procedure that builds formal monadic FOL structures with explicit parameters for depth/width/label/distractors, translates them via LLM, and applies round-trip verification against an independent external prover. This verification step is external to the generation process and does not reduce any claimed benchmark property back to quantities fitted from the same outputs. No equations, self-citations, or uniqueness theorems are invoked to force the result; the controllability and consistency guarantees are inherited directly from the formal side plus the prover check, making the method self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, new axioms, or invented entities are stated. The framework relies on standard first-order logic semantics and the correctness of an unnamed external prover.

axioms (2)

domain assumption Monadic first-order logic formulas can be constructed from conjunction and disjunction patterns while preserving decidability and allowing exact control of reasoning depth and width.
Invoked when the paper states that formal logical structures are built using conjunction and disjunction patterns to control complexity.
domain assumption Round-trip translation through an LLM followed by prover verification guarantees semantic equivalence between the formal formula and the generated natural-language sentence.
Central to the claim that logical consistency is ensured.

pith-pipeline@v0.9.1-grok · 5778 in / 1428 out tokens · 30212 ms · 2026-06-26T17:22:30.664525+00:00 · methodology

discussion (0)

Reference graph

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