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arxiv: 2606.00697 · v1 · pith:URE6IDXDnew · submitted 2026-05-30 · 💻 cs.LO

MCSAT Modulo Transcendental Arithmetics

Jorge Gallego-Hern\'andez , Enrico Lipparini , Alessio Mansutti This is my paper

Pith reviewed 2026-06-28 18:04 UTC · model grok-4.3

classification 💻 cs.LO
keywords MCSATtranscendental arithmeticnon-linear real arithmeticSMT solvingsine functionexponential functionYices2
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The pith

An MCSAT extension solves quantifier-free real arithmetic formulas that include sine and exponential functions.

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

The paper develops a procedure for deciding formulas that combine non-linear real arithmetic with transcendental functions, which lie outside the decidable fragment. It extends the MCSAT calculus by first abstracting the input to plain non-linear real arithmetic, then letting MCSAT together with an NRA plugin construct a partial model incrementally. A dedicated Transcendental Real Arithmetic plugin sits between MCSAT and the NRA component, using real-analysis checks to verify consistency of the partial model with the transcendental constraints and to add necessary refinements when inconsistency is detected. The authors implemented the approach inside Yices2 for sine and exponential and report that the resulting prototype solves more SAT and UNSAT instances than existing solvers.

Core claim

The framework abstracts the input formula to NRA, lets MCSAT and an NRA plugin incrementally build a partial model of the abstracted formula, and relies on a Transcendental Real Arithmetic plugin to ensure consistency of that partial model with the original transcendental constraints and to refine the NRA abstraction accordingly.

What carries the argument

The Transcendental Real Arithmetic plugin, which acts as intermediary between MCSAT and the NRA plugin by checking consistency via real analysis and returning sound refinements to the abstracted formula.

If this is right

  • The same MCSAT-plus-plugin architecture can be instantiated inside an SMT solver for the sine and exponential functions.
  • The resulting solver decides both satisfiable and unsatisfiable instances of the extended theory.
  • The abstraction-refinement loop terminates with a correct answer on the tested benchmarks.
  • Soundness of the overall procedure rests on the real-analysis checks performed by the transcendental plugin.

Where Pith is reading between the lines

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

  • The intermediary-plugin pattern could be reused for other transcendental or special functions once suitable real-analysis oracles are available.
  • The same abstraction strategy might be combined with decision procedures for other decidable fragments to enlarge the solvable class of mixed arithmetic formulas.
  • Empirical gains on both SAT and UNSAT instances suggest that hybrid algebraic and analytic checks can reduce the practical impact of undecidability in this domain.

Load-bearing premise

The Transcendental Real Arithmetic plugin can reliably detect inconsistencies and produce sound refinements to the NRA abstraction using only methods from real analysis, without introducing incompleteness or false positives.

What would settle it

A concrete quantifier-free formula containing sine or exponential for which the procedure returns SAT but no model exists, or returns UNSAT when a model does exist, would show the claim is false.

Figures

Figures reproduced from arXiv: 2606.00697 by Alessio Mansutti, Enrico Lipparini, Jorge Gallego-Hern\'andez.

Figure 1
Figure 1. Figure 1: The MCSAT framework. formulas of the form τ < 0 or τ = 0. The δ-weakening of φ, denoted φ δ , is the formula obtained from φ by replacing the literals in each clause following the rules: τ < 0 → τ < δ, ¬(τ < 0) → ¬(τ < −δ), τ = 0 → −δ ≤ τ ≤ δ, and ¬(τ = 0) → ⊤. A formula φ is said to be δ-satisfiable if and only if its δ-weakening φ δ is satisfiable. Definition 3 (δ-satisfiability procedure). A δ-satisfiab… view at source ↗
Figure 2
Figure 2. Figure 2: Schematic view of our procedure. flow reaches the Decide step. Suppose MCSAT performs a Boolean decision, adding (y = −1) 7→ ⊤ to the trail. The algorithm now performs the Consistency check, which fails as the literals x 2 = y and y = −1 caused the feasibility set of x to become empty. The plugin may learn the (tautological) clause (x 2 ̸= y ∨ y ≥ 0), and return the decision (y = −1) 7→ ⊤ as the cause of t… view at source ↗
Figure 3
Figure 3. Figure 3: Cactus plots of the results of the benchmark. [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
read the original abstract

We propose a framework for solving quantifier-free formulas from (undecidable) extensions of non-linear real arithmetic (NRA) with transcendental functions, such as exponential and trigonometric ones. The framework extends the Model Constructive Satisfiability calculus (MCSAT), and leverages procedures for NRA and methods from real analysis. At its core, our procedure abstracts the input formula to NRA, and lets MCSAT and an NRA plugin incrementally build a partial model of the abstracted formula. A Transcendental Real Arithmetic plugin, acting as an intermediary between MCSAT and the NRA plugin, ensures the consistency of the partial model and is responsible for refining the abstracted formula. We implemented our procedure in the Yices2 SMT solver for the sine and exponential functions, and conducted an extensive empirical evaluation that shows that our prototype outperforms state-of-the-art solvers on both SAT and UNSAT instances.

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 proposes a framework extending the Model Constructive Satisfiability (MCSAT) calculus to quantifier-free formulas in non-linear real arithmetic (NRA) augmented with transcendental functions such as sine and exponential. The input is abstracted to NRA; MCSAT together with an NRA plugin incrementally constructs a partial model; a Transcendental Real Arithmetic plugin, positioned between MCSAT and the NRA plugin, checks consistency of the partial model via real-analysis methods and produces refinements. The procedure is implemented in Yices2 for sine and exponential; an extensive empirical evaluation is reported to show that the prototype outperforms state-of-the-art solvers on both SAT and UNSAT instances.

Significance. If the soundness of the transcendental plugin is established and the performance claims hold under reproducible conditions, the work would constitute a practical advance for SMT solving over an undecidable fragment. The combination of MCSAT with NRA abstraction and real-analysis refinements, together with the reported implementation and empirical comparison, would be a concrete contribution to the handling of transcendental arithmetic in verification tools.

major comments (2)
  1. [Abstract] Abstract and framework description: the central soundness claim rests on the Transcendental Real Arithmetic plugin reliably detecting inconsistencies and producing sound refinements using only real-analysis methods, yet no derivation, pseudocode, or consistency argument for this plugin is supplied; without these details the soundness of the overall procedure cannot be verified from the text.
  2. [Implementation and evaluation] Implementation and evaluation sections: the claim that the prototype outperforms state-of-the-art solvers on both SAT and UNSAT instances is load-bearing for the paper's main result, but the manuscript provides no concrete description of the benchmarks, the precise comparison baseline, or how the transcendental consistency checks avoid false positives or incompleteness.
minor comments (2)
  1. Notation for the abstraction and refinement steps is introduced without a clear running example that would illustrate how the NRA abstraction and the transcendental plugin interact on a concrete formula.
  2. The paper positions the approach as necessarily incomplete (as expected for an undecidable fragment) but does not discuss any completeness guarantees on restricted subclasses that might be of independent interest.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The two major comments identify important gaps in the presentation of the Transcendental Real Arithmetic plugin and the experimental evaluation. We address each point below and will incorporate the necessary additions in a revised manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract and framework description: the central soundness claim rests on the Transcendental Real Arithmetic plugin reliably detecting inconsistencies and producing sound refinements using only real-analysis methods, yet no derivation, pseudocode, or consistency argument for this plugin is supplied; without these details the soundness of the overall procedure cannot be verified from the text.

    Authors: We agree that the current manuscript does not supply a derivation, pseudocode, or explicit consistency argument for the Transcendental Real Arithmetic plugin. In the revised version we will insert a dedicated subsection (placed after the framework overview) that contains (i) high-level pseudocode for the plugin’s main procedures, (ii) a description of the real-analysis techniques employed for exp and sin, and (iii) a concise soundness argument showing that detected inconsistencies are genuine and that the produced refinements preserve satisfiability of the original formula. This addition will make the overall soundness claim verifiable directly from the text. revision: yes

  2. Referee: [Implementation and evaluation] Implementation and evaluation sections: the claim that the prototype outperforms state-of-the-art solvers on both SAT and UNSAT instances is load-bearing for the paper's main result, but the manuscript provides no concrete description of the benchmarks, the precise comparison baseline, or how the transcendental consistency checks avoid false positives or incompleteness.

    Authors: The evaluation section currently reports aggregate results but, as the referee observes, lacks sufficient concrete detail. We will expand it to include: an explicit list of benchmark families together with their provenance and sizes; the precise versions, configurations, and command-line options of all competing solvers; and a paragraph explaining that the transcendental consistency checks are designed to be sound (hence no false positives) by invoking only decidable real-analysis procedures, while noting the inherent incompleteness stemming from the undecidability of the theory. We will also attach a supplementary archive containing the full benchmark set and raw solver output to support reproducibility. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper presents an algorithmic framework extending MCSAT with an NRA plugin and a new Transcendental Real Arithmetic intermediary that uses real-analysis methods for consistency and refinement. No equations, fitted parameters, predictions, or self-citations are described that reduce any central claim to its own inputs by construction. The implementation in Yices2 and empirical evaluation are presented as independent validation of the new procedure on an undecidable fragment. The derivation chain is self-contained as a constructive combination of existing calculi and analysis techniques.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The framework rests on standard properties of real numbers and transcendental functions from real analysis; the main addition is the algorithmic plugin rather than new mathematical entities or fitted constants.

axioms (1)
  • domain assumption Methods from real analysis suffice to decide consistency of partial models involving exponential and trigonometric functions over the reals
    Invoked when the Transcendental Real Arithmetic plugin refines the NRA abstraction.
invented entities (1)
  • Transcendental Real Arithmetic plugin no independent evidence
    purpose: Intermediary ensuring consistency between MCSAT partial models and the NRA abstraction for transcendental functions
    New component introduced to bridge MCSAT and the NRA plugin; no independent evidence of correctness supplied in the abstract.

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Forward citations

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Reference graph

Works this paper leans on

14 extracted references · 12 canonical work pages · cited by 1 Pith paper

  1. [1]

    MCSAT Modulo Transcendental Arith- metics

    doi:10.1007/s10817-009-9149-2. [art26] Artifact for “MCSAT Modulo Transcendental Arith- metics”. https://cloud.software.imdea.org/index.php/s/ PgqwRYckKrLXT4q,

    work page doi:10.1007/s10817-009-9149-2

  2. [2]

    Specification-based monitoring of cyber-physical systems: a survey on theory, tools and applications

    [BDD+18] Ezio Bartocci, Jyotirmoy Deshmukh, Alexandre Donzé, Georgios Fainekos, Oded Maler, Dejan Ni ˇckovi´c, and Sriram Sankara- narayanan. Specification-based monitoring of cyber-physical systems: a survey on theory, tools and applications. In Lectures on Runtime Verification: Introductory and Advanced Topics, pages 135–175. Springer, 2018.doi:10.1007/...

    work page doi:10.1016/s1574-6526(06)80020-9 2018

  3. [3]

    [Bou24] Olivier Bournez

    doi:10.1007/978-3-030-79876-5_7. [Bou24] Olivier Bournez. Complexity over the reals,

    work page doi:10.1007/978-3-030-79876-5_7

  4. [4]

    Available at www.lix.polytechnique.fr/Labo/Olivier.Bournez/load/MPRI- PROJET/Cours-2024-COMPR.pdf, accessed June 2,

    2024

  5. [5]

    [CGI+18] Alessandro Cimatti, Alberto Griggio, Ahmed Irfan, Marco Roveri, and Roberto Sebastiani

    Second Edition.doi:10.1007/3-540-33099-2. [CGI+18] Alessandro Cimatti, Alberto Griggio, Ahmed Irfan, Marco Roveri, and Roberto Sebastiani. Incremental linearization for satisfiability and verification modulo nonlinear arithmetic and transcendental functions.ACM Trans. Comput. Log., 19(3):19:1– 19:52, 2018.doi:10.1145/3230639. [CGLS22] Alessandro Cimatti, ...

    work page doi:10.1007/3-540-33099-2 2018

  6. [6]

    doi:10.1007/978-3-031-19992-9_9

    Springer International Publishing. doi:10.1007/978-3-031-19992-9_9. [Col98] George E. Collins. Quantifier elimination for real closed fields by cylindrical algebraic decomposition. pages 85–121,

    work page doi:10.1007/978-3-031-19992-9_9

  7. [7]

    [CX23] Rizeng Chen and Bican Xia

    doi:10.1007/978-3-7091-9459-1_4. [CX23] Rizeng Chen and Bican Xia. Deciding first-order formulas involving univariate mixed trigonometric-polynomials. InISSAC, 2023.doi:10.1145/3597066.3597104. [DMJ13] Leonardo De Moura and Dejan Jovanovi ´c. A model-constructing satisfiability calculus. InVMCAI, 2013.doi:10.1007/ 978-3-642-35873-9_1. [Dut14] Bruno Dutert...

    work page doi:10.1007/978-3-7091-9459-1_4 2023

  8. [8]

    [FHT+07] Martin Fränzle, Christian Herde, Tino Teige, Stefan Ratschan, and Tobias Schubert

    Springer Berlin Hei- delberg.doi:10.1007/978-3-642-24364-6_9. [FHT+07] Martin Fränzle, Christian Herde, Tino Teige, Stefan Ratschan, and Tobias Schubert. Efficient solving of large non-linear arithmetic constraint systems with complex boolean structure. JSAT, 1:209–236, 2007.doi:10.3233/SAT190012. [GAC12] Sicun Gao, Jeremy Avigad, and Edmund M Clarke. Del...

    work page doi:10.1007/978-3-642-24364-6_9 2007

  9. [9]

    [GM25] Jorge Gallego-Hernández and Alessio Mansutti

    doi:10.1007/978-3-642-38574-2_14. [GM25] Jorge Gallego-Hernández and Alessio Mansutti. On the ex- istential theory of the reals enriched with integer powers of a computable number. InSTACS, 2025.doi:10.4230/ LIPICS.STACS.2025.37. [HIG25] Thomas Hader, Ahmed Irfan, and Graham-Lengrand, Stéphane. Decision heuristics in mcsat. In Ruzica Piskac and Zvonimir R...

    work page doi:10.1007/978-3-642-38574-2_14 2025

  10. [10]

    [JBDM13] Dejan Jovanovic, Clark Barrett, and Leonardo De Moura

    Springer Nature Switzerland. [JBDM13] Dejan Jovanovic, Clark Barrett, and Leonardo De Moura. The design and implementation of the model constructing satisfia- bility calculus. InFMCAD, 2013.doi:10.1109/FMCAD. 2013.7027033. [JDM13] Dejan Jovanovi ´c and Leonardo De Moura. Solving non-linear arithmetic.ACM Commun. Comput. Algebra, 46(3/4):104–105, 2013.doi:...

    work page doi:10.1109/fmcad 2013

  11. [11]

    [KRBT22] Gereon Kremer, Andrew Reynolds, Clark Barrett, and Cesare Tinelli

    doi:10.1145/2576802.2576828. [KRBT22] Gereon Kremer, Andrew Reynolds, Clark Barrett, and Cesare Tinelli. Cooperating techniques for solving nonlinear real arith- metic in the cvc5 smt solver (system description). InIJCAR, 2022.doi:10.1007/978-3-031-10769-6_7. [Lac03] Miklós Laczkovich. The removal ofπfrom some undecidable problems involving elementary fun...

    work page doi:10.1145/2576802.2576828 2022

  12. [12]

    [LR25] Enrico Lipparini and Stefan Ratschan

    Springer-Verlag.doi:10.1007/ 978-3-031-99984-0_6. [LR25] Enrico Lipparini and Stefan Ratschan. Satisfiability of non- linear transcendental arithmetic as a certificate search problem. J. Autom. Reason., 69(1), January 2025.doi:10.1007/ s10817-024-09716-3. [MW96] Angus Macintyre and Alex J. Wilkie. On the decidability of the real exponential field. InKreis...

    2025

  13. [13]

    1007/978-3-031-78750-8_13

    Springer Nature Switzerland.doi:10. 1007/978-3-031-78750-8_13. [Rat06] Stefan Ratschan. Efficient solving of quantified inequality constraints over the real numbers.ACM Trans. Comput. Log., 7(4):723–748, 2006.doi:10.1145/1183278.1183282. [Rat12] Stefan Ratschan. Applications of quantified constraint solving over the reals–bibliography.arXiv preprint arXiv...

    work page doi:10.1145/1183278.1183282 2006

  14. [14]

    APPENDIX ADDITIONAL MATERIAL FORSECTIONII-C Formulas:Throughout the paper, we assume that all formulas are quantifier-free and inconjunctive normal form (CNF)

    doi:10.1016/S0747-7171(10)80003-3. APPENDIX ADDITIONAL MATERIAL FORSECTIONII-C Formulas:Throughout the paper, we assume that all formulas are quantifier-free and inconjunctive normal form (CNF). We recall that aliteralis defined as an atomic formula or its negation. Aclauseis a disjunction of literals, and a CNF formula is a conjunction of clauses. The CD...

    work page doi:10.1016/s0747-7171(10)80003-3