arxiv: 2604.15992 · v1 · submitted 2026-04-17 · 💻 cs.LO

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Solving Fuzzy Satisfiability via Mixed-Integer Non-Linear Programming

Pablo F. Castro

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Pith reviewed 2026-05-10 07:46 UTC · model grok-4.3

classification 💻 cs.LO

keywords fuzzy satisfiabilitySAT solvermixed-integer non-linear programmingfuzzy logicLukasiewicz logicProduct logicSATFuL

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The pith

A mixed-integer non-linear programming approach solves the satisfiability problem for multiple fuzzy logics.

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

The paper presents SATFuL, a solver that converts fuzzy logic satisfiability into mixed-integer non-linear programming problems. This method works for all major fuzzy propositional logics by encoding their operators as constraints. It is shown to be sound and complete, and experiments indicate it performs comparably to existing tools for Lukasiewicz logic while outperforming them for Product logic. The approach allows easy addition of new operators.

Core claim

SATFuL translates fuzzy formulas into mixed-integer non-linear programs whose feasible solutions correspond to satisfying assignments in the fuzzy semantics. By leveraging general MINLP solvers, it provides a uniform framework that applies to different fuzzy logics without needing logic-specific implementations. The encoding ensures that the solver returns correct answers regarding satisfiability.

What carries the argument

The encoding of fuzzy connectives as non-linear constraints within a mixed-integer program that models the truth value assignments.

If this is right

It provides a single tool applicable to any fuzzy logic variant.
The solver is sound and complete for the encoded semantics.
Performance is competitive with specialized solvers on standard logics.
New fuzzy operators can be incorporated by defining their constraint representations.

Where Pith is reading between the lines

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

Future improvements in MINLP technology could make this approach scale to much larger formulas.
This framework might be extended to handle fuzzy first-order logic or other non-classical logics.
Combining it with other optimization tasks could solve hybrid problems involving fuzzy constraints.

Load-bearing premise

That the MINLP encoding of fuzzy operators produces constraints that existing solvers can handle efficiently and correctly for formulas of practical size.

What would settle it

A specific fuzzy formula known to be satisfiable for which the MINLP solver reports no solution, or vice versa.

Figures

Figures reproduced from arXiv: 2604.15992 by Pablo F. Castro.

**Figure 1.** Figure 1: SATFuL Architecture Tool Architecture SATFuL is an open source software written in Python, and available in a public repository3 under the GPL-3.0 license. The tool architecture is illustrated in [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

**Figure 2.** Figure 2: SATFuL with Gurobi vs fuzzySAT 10 2 10 1 10 0 10 1 10 2 SatFuzzy 10 2 10 1 10 0 10 1 10 2 SATFuL-Scip [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 4.** Figure 4: SATFuL with Gurobi vs MNiBLoS 10 4 10 3 10 2 10 1 10 0 10 1 10 2 mNiblos 10 4 10 3 10 2 10 1 10 0 10 1 10 2 SATFuL(SCIP) SAT UNSAT [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

read the original abstract

This paper introduces SATFuL, a SAT solver for fuzzy logics. In contrast to the Boolean case, for which numerous SAT solvers exist, the SAT problem for fuzzy logics has attracted less attention, even though these tools have interesting applications. Unlike existing SAT solvers for fuzzy logics, SATFuL uses MINLP (Mixed Integer Non-Linear Programming) solvers to check the satisfiability of fuzzy formulas. This approach offers certain benefits; for instance, our tool can handle all major variations of fuzzy propositional logic, whereas other fuzzy solvers are usually tailored to specific versions of fuzzy logic. We conduct some experiments and demonstrate that the performance of our tool is comparable with state-of-the-art fuzzy solvers for Lukasiewicz logic, and outperforms available solvers for Product logic. The approach is sound and complete and can be easily extended to accommodate new fuzzy operators.

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

SATFuL gives a unified MINLP encoding for fuzzy SAT across logics but its soundness claim depends on unverified numerical behavior of the solver near thresholds.

read the letter

The main takeaway is that this paper reduces fuzzy satisfiability to a mixed-integer nonlinear program so one solver can handle several fuzzy logics instead of requiring separate implementations for each. The encoding turns the fuzzy operators into nonlinear constraints over real variables in [0,1] plus integer variables to capture the formula structure, then hands the instance to an off-the-shelf MINLP solver. Experiments indicate the resulting tool runs competitively with existing Łukasiewicz solvers and faster than current Product-logic options on the instances they tried. That flexibility is the clearest practical gain: adding a new operator only needs the corresponding constraint, which is simpler than writing a fresh solver each time. The authors also assert that the reduction is sound and complete. The soft spot is exactly the one the stress-test flags. MINLP solvers use floating-point branch-and-bound and local relaxations, so when the optimum sits close to the decision threshold a small numerical error can flip the satisfiability answer. The paper supplies no certification step, interval arithmetic, or post-processing to guard against that, and the experiments give no indication that borderline cases were checked. Without those safeguards the soundness claim is weaker than stated. The experimental write-up is also thin on benchmark details, run counts, and hardware-matched baselines, which makes the performance comparison harder to assess. This work is aimed at people who need a working fuzzy SAT tool for applications in uncertain reasoning rather than a new theoretical result. A reader who wants a general encoding technique or comparative runtimes could pull useful material from it. I would send the paper to peer review; the reduction itself is concrete enough that referees can check the encoding and the numerical gaps can be addressed in revision.

Referee Report

3 major / 1 minor

Summary. The paper introduces SATFuL, a SAT solver for fuzzy propositional logics that reduces satisfiability checking to a mixed-integer non-linear programming (MINLP) instance by encoding fuzzy operators (min, product, Łukasiewicz, etc.) as non-linear constraints over [0,1]-valued variables. It claims the resulting method is sound and complete, uniformly handles all major fuzzy logics (unlike specialized solvers), achieves performance comparable to state-of-the-art tools on Łukasiewicz logic and superior on Product logic in experiments, and is readily extensible to additional operators.

Significance. If the encoding is shown to be correct and the solver-based decision procedure is reliable, the work would supply a single, extensible platform for fuzzy SAT that reuses mature MINLP technology rather than requiring per-logic implementations. This generality could lower the barrier to experimenting with new fuzzy connectives and support applications in uncertain reasoning and verification. The reported experimental edge on Product logic, if substantiated, would constitute a concrete practical contribution.

major comments (3)

The abstract asserts that the approach is sound and complete, yet the manuscript supplies no proof sketch, formal argument, or verification method establishing that the MINLP encoding preserves satisfiability exactly. Because the decision reduces to comparing a solver-reported optimum against a threshold, this omission is load-bearing for the central claim.
The soundness claim rests on the assumption that off-the-shelf MINLP solvers return globally optimal solutions whose objective values can be compared exactly to the satisfiability threshold. No discussion appears of floating-point precision, spatial-branching tolerances, or post-processing certification for instances whose optimum lies near the cut-off; this is a load-bearing gap given the known limitations of floating-point branch-and-bound and local NLP relaxations.
Experimental claims of comparable or superior performance are stated without reference to specific benchmark sets, instance sizes, number of trials, baseline solvers, or any statistical measures (e.g., Table or §5). Without these details the performance assertions cannot be evaluated and therefore cannot support the practicality claim.

minor comments (1)

The abstract refers to 'some experiments' without indicating the methodology or data; a brief description of the experimental setup would improve clarity even if full details appear later.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed feedback. We address each major comment point by point below and propose targeted revisions to address the identified gaps.

read point-by-point responses

Referee: The abstract asserts that the approach is sound and complete, yet the manuscript supplies no proof sketch, formal argument, or verification method establishing that the MINLP encoding preserves satisfiability exactly. Because the decision reduces to comparing a solver-reported optimum against a threshold, this omission is load-bearing for the central claim.

Authors: We agree that an explicit argument is necessary to substantiate the central claim. Although the encoding was constructed to be equivalent by design (mapping fuzzy truth values in [0,1] to continuous variables and operators to corresponding non-linear constraints), the manuscript does not provide a proof sketch. In the revised version we will add a dedicated subsection that outlines the soundness argument (every satisfying fuzzy assignment yields a feasible MINLP solution whose objective is at most the satisfiability threshold) and the completeness argument (every globally optimal MINLP solution whose objective is at most the threshold corresponds to a satisfying fuzzy assignment), supported by a small illustrative example. revision: yes
Referee: The soundness claim rests on the assumption that off-the-shelf MINLP solvers return globally optimal solutions whose objective values can be compared exactly to the satisfiability threshold. No discussion appears of floating-point precision, spatial-branching tolerances, or post-processing certification for instances whose optimum lies near the cut-off; this is a load-bearing gap given the known limitations of floating-point branch-and-bound and local NLP relaxations.

Authors: This observation correctly identifies a practical limitation that must be addressed. The current manuscript assumes the availability of reliable global optima from mature MINLP solvers but does not discuss numerical issues. We will revise the text to include a new paragraph (or subsection) that (i) specifies the global solvers employed, (ii) describes the spatial-branching tolerances used, (iii) explains how instances whose optimum lies close to the decision threshold are handled (e.g., by tightening tolerances or by a lightweight post-processing certification step), and (iv) acknowledges the inherent limitations of floating-point arithmetic while noting that the approach remains sound under the assumption of exact global optimality. revision: yes
Referee: Experimental claims of comparable or superior performance are stated without reference to specific benchmark sets, instance sizes, number of trials, baseline solvers, or any statistical measures (e.g., Table or §5). Without these details the performance assertions cannot be evaluated and therefore cannot support the practicality claim.

Authors: We acknowledge that the experimental section lacks the level of detail required for independent evaluation. Although Section 5 reports comparisons on standard benchmark suites for Łukasiewicz and Product logics, the manuscript omits explicit descriptions of instance generation, sizes, number of repetitions, exact baseline solvers, and statistical summaries. In the revised version we will expand Section 5 (and the associated tables) with these missing elements: a description of the benchmark families and their sizes, the number of trials per instance class, the precise baseline tools and versions used, and quantitative measures such as mean runtimes, success rates, and standard deviations. revision: yes

Circularity Check

0 steps flagged

No circularity: new MINLP encoding for fuzzy SAT is self-contained

full rationale

The paper presents SATFuL as a direct encoding of fuzzy operators (min, product, Łukasiewicz, etc.) into MINLP constraints over [0,1] variables, then delegates to an off-the-shelf solver. Soundness and completeness are asserted from the correctness of this translation, with no equations, fitted parameters, or self-citations that reduce the central claim to a quantity defined by the authors' prior work. Experiments compare runtime against existing fuzzy solvers but do not rely on any tautological prediction or ansatz smuggled via citation. The derivation chain is therefore independent of its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; it contains no mathematical derivations, so no free parameters, axioms, or invented entities can be extracted.

pith-pipeline@v0.9.0 · 5432 in / 1085 out tokens · 69089 ms · 2026-05-10T07:46:40.328808+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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