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arxiv: 2605.18155 · v1 · pith:WNEVVWGWnew · submitted 2026-05-18 · 💻 cs.CL

FOL2NS: Generating Natural Sentences from First-Order Logic

Mei Jia This is my paper

Pith reviewed 2026-05-20 10:44 UTC · model grok-4.3

classification 💻 cs.CL
keywords first-order logic translationnatural sentence generationneurosymbolic frameworksynthetic FOL formulasquantifier depthnatural language generationsemantic representation
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The pith

A hybrid framework generates natural sentences from deeply nested first-order logic formulas.

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

This paper describes FOL2NS, a system built to produce synthetic first-order logic formulas and translate them into everyday language sentences. It targets structures with many levels of nesting and different numbers of quantifiers that are uncommon in available data collections. The design pairs rule-based parts for creating the logic with language models adjusted on relevant examples to boost variety and reach. Results from tests indicate good production of correct templates and smooth sentences, although getting the exact meaning right becomes harder with more complex nesting.

Core claim

FOL2NS creates synthetic first-order logic formulas with varying quantifier depths and converts them into natural human expressions by integrating rule-driven modules with fine-tuned language models, leading to enhanced diversity and coverage in the generated samples while showing reliable template and fluency performance in experiments.

What carries the argument

The neurosymbolic FOL2NS framework that merges rule-driven generation of logic structures with fine-tuned language models for producing natural language output.

If this is right

  • Enhanced diversity and coverage of training samples for logic-related NLP tasks.
  • Improved support for downstream applications such as semantic parsing and question answering.
  • Reliable generation of well-formed templates even for formulas with high quantifier depths.
  • Decreased performance in semantic accuracy and naturalness as nesting complexity grows.

Where Pith is reading between the lines

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

  • Large datasets produced this way could train models to better handle logical statements in natural language contexts.
  • Similar combinations of rules and models might apply to translating other formal systems like programming languages.
  • Further experiments could test the framework on logic from real mathematical proofs to check generalization.

Load-bearing premise

Combining rule-driven modules with fine-tuned language models will enhance diversity, coverage, and accurate translation for deeply nested first-order logic structures with varying quantifier depths.

What would settle it

Measuring the rate at which generated natural sentences can be translated back to the original logic formula without loss of meaning, particularly for cases with quantifier depth exceeding three.

Figures

Figures reproduced from arXiv: 2605.18155 by Mei Jia.

Figure 1
Figure 1. Figure 1: Overview of the proposed framework for dataset construction. Module A gets the Train and Validation [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The token frequency distribution of FOL as [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: shows examples of FOLIO, and [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The first 10 examples of defined FOL formulas in Module B. [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Examples of the preprocessed FOL format with selected predicate-variable pairs (FOL2NW). [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Examples of T5-generated translations (T5_FOL2NS) from the FOL2NW stage. [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Part of the model results of inputs (logical form), outputs (candidate) and targets (reference) in the [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Part of the model results of inputs (logical form) and outputs (candidate) in the Test set. [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
read the original abstract

Translating formal language into natural language is a foundational challenge in NLP, driving various downstream applications in semantic parsing, theorem validation, and question answering. In this study, we introduce First-Order Logic to Natural Sentence (FOL2NS), a neurosymbolic framework designed to generate synthetic FOL formulas and convert them into natural human expressions. It handles deeply nested structures with varying quantifier depths (QD), which are rarely captured by existing corpora. By combining rule-driven modules with fine-tuned language models, FOL2NS enhances the diversity and coverage of the generated samples. In our experiments, we systematically evaluate the framework's capabilities through both character-level analysis and overall performance metrics. Experimental results show that FOL2NS can reliably produce well-formed templates and fluent statements, but it faces challenges in achieving precise semantic representations and natural generation as structural complexity increases.

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.

Referee Report

2 major / 2 minor

Summary. The paper introduces FOL2NS, a neurosymbolic framework that generates synthetic first-order logic (FOL) formulas with varying quantifier depths (QD) and translates them into natural language sentences. It combines rule-driven modules with fine-tuned language models to improve diversity and coverage of deeply nested structures rarely found in existing corpora. Experiments rely on character-level analysis and overall performance metrics, with results indicating reliable production of well-formed templates and fluent statements but challenges in precise semantic representations and natural generation as structural complexity increases.

Significance. If the central claims hold under more rigorous semantic evaluation, the framework could aid creation of synthetic datasets for semantic parsing, theorem validation, and question answering by addressing gaps in coverage for high-QD formulas. The neurosymbolic design offers a potential balance between symbolic control and neural fluency, though its advantages over purely neural or rule-based baselines remain to be quantified.

major comments (2)
  1. [Abstract] Abstract and experimental evaluation: The claim that FOL2NS 'reliably produce[s] well-formed templates' rests on character-level analysis and unspecified overall metrics. These track surface-form correctness and fluency but do not measure logical equivalence, scope preservation, or predicate-argument fidelity for nested quantifiers; without back-translation or entailment checks against source FOL, surface success cannot be cleanly separated from semantic accuracy, especially at high QD.
  2. [Experimental results] Experimental results paragraph: The reported 'challenges in achieving precise semantic representations ... as structural complexity increases' are acknowledged but left unquantified. No per-QD breakdown, error typology (e.g., negation scope errors, quantifier misplacement), or comparison against a pure neural baseline is provided, weakening the ability to assess whether the neurosymbolic combination actually mitigates or merely defers the semantic issues.
minor comments (2)
  1. [Abstract] Notation for quantifier depth (QD) is introduced without an explicit definition or example formula in the abstract; a short illustrative example would clarify the range of nesting depths considered.
  2. [Abstract] The abstract states that the framework 'enhances the diversity and coverage of the generated samples' but provides no quantitative comparison (e.g., unique formula count or coverage of predicate arity) against prior synthetic generators.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We address each major comment below and indicate the revisions we will incorporate to improve the evaluation section.

read point-by-point responses

  1. Referee: [Abstract] Abstract and experimental evaluation: The claim that FOL2NS 'reliably produce[s] well-formed templates' rests on character-level analysis and unspecified overall metrics. These track surface-form correctness and fluency but do not measure logical equivalence, scope preservation, or predicate-argument fidelity for nested quantifiers; without back-translation or entailment checks against source FOL, surface success cannot be cleanly separated from semantic accuracy, especially at high QD.

    Authors: We agree that the current evaluation relies on character-level analysis and fluency metrics, which primarily confirm surface-form well-formedness rather than full semantic fidelity such as logical equivalence or scope preservation. This approach was chosen to first establish reliable template generation before deeper semantic validation. In the revised version, we will explicitly qualify the abstract claim to reflect this scope, add back-translation and entailment checks for a subset of high-QD examples, and report the results to better separate surface and semantic accuracy. revision: partial

  2. Referee: [Experimental results] Experimental results paragraph: The reported 'challenges in achieving precise semantic representations ... as structural complexity increases' are acknowledged but left unquantified. No per-QD breakdown, error typology (e.g., negation scope errors, quantifier misplacement), or comparison against a pure neural baseline is provided, weakening the ability to assess whether the neurosymbolic combination actually mitigates or merely defers the semantic issues.

    Authors: We acknowledge that the challenges are described qualitatively without per-QD quantification or detailed error categorization. We will revise the experimental results section to include a per-quantifier-depth performance table, an error typology breakdown (including negation scope and quantifier placement issues), and explicit discussion of these trends. A comparison against a pure neural baseline was outside the original experimental scope, which prioritized the neurosymbolic design; we will add this as a limitation and include preliminary baseline results if space allows. revision: partial

Circularity Check

0 steps flagged

No circularity: framework and evaluation presented as independent empirical construction

full rationale

The paper introduces FOL2NS as a neurosymbolic framework that combines rule-driven modules with fine-tuned language models to generate synthetic FOL formulas and convert them to natural sentences, with explicit focus on handling deeply nested quantifier structures absent from existing corpora. Evaluation relies on character-level analysis and overall performance metrics to report reliable well-formed templates and fluent statements alongside noted challenges in semantic precision at higher complexity. No derivation chain, equations, fitted parameters renamed as predictions, or load-bearing self-citations are present in the provided text; the central claims rest on the described combination of components and experimental observations rather than reducing to definitional equivalence or prior self-referential results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the work relies on standard domain assumptions in NLP about hybrid symbolic-neural methods without detailing free parameters or new entities; the central contribution is the framework itself rather than new postulates.

axioms (1)
  • domain assumption Existing corpora lack sufficient coverage of deeply nested first-order logic structures with varying quantifier depths.
    Explicitly stated as motivation for creating synthetic data in the abstract.

pith-pipeline@v0.9.0 · 5657 in / 1298 out tokens · 48041 ms · 2026-05-20T10:44:51.341083+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Foundation/ArithmeticFromLogic.lean reality_from_one_distinction unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    By combining rule-driven modules with fine-tuned language models, FOL2NS enhances the diversity and coverage of the generated samples... Experimental results show that FOL2NS can reliably produce well-formed templates and fluent statements, but it faces challenges in achieving precise semantic representations and natural generation as structural complexity increases.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages · 1 internal anchor

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