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arxiv: 2605.15002 · v1 · pith:2FIT4BPBnew · submitted 2026-05-14 · 💻 cs.LO · cs.CC

Extending CDCL to disjunctions of parity equations

Paul Beame , Glenn Sun This is my paper

Pith reviewed 2026-06-30 19:24 UTC · model grok-4.3

classification 💻 cs.LO cs.CC
keywords CDCLRes(⊕)XNFparity equationsclause learningproof simulationSAT solverslinear clauses
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The pith

Extending CDCL to disjunctions of parity equations allows polynomial simulation of Res(⊕)

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

This paper shows that CDCL can be generalized to XNF formulas made of disjunctions of parity equations while maintaining a close connection to the Res(⊕) proof system. Specifically, CDCL(⊕) produces Res(⊕) proofs and, with nondeterministic decisions and restarts, can polynomially simulate any Res(⊕) proof. This relationship mirrors the one between standard CDCL and Resolution, suggesting that practical solvers can leverage stronger proof techniques for parity constraints. The authors introduce new inference rules to translate common CDCL mechanisms like 1-UIP learning into this setting and demonstrate an implementation that performs well on relevant problems.

Core claim

CDCL(⊕) not only produces Res(⊕) proofs, but also polynomially simulates Res(⊕) given nondeterministic decisions and restarts. The connection is established using a new set of inference rules for Res(⊕) that translate Resolution-based subroutines such as 1-UIP clause learning to the setting of linear clauses. This enables branching on arbitrary parity equations, linear-algebraic reasoning in unit propagation, and learning linear clauses through conflict analysis.

What carries the argument

New inference rules for Res(⊕) that enable translation of 1-UIP clause learning and other CDCL subroutines to disjunctions of parity equations

If this is right

  • CDCL(⊕) can produce proofs whose size is polynomial in the size of the shortest Res(⊕) proof.
  • XNF formulas that have short Res(⊕) proofs but long Resolution proofs become tractable for CDCL-based solvers.
  • An implementation called Xorcle with adapted heuristics and LRUP(⊕) logging can solve native XNF benchmarks faster than existing solvers.
  • On CNF Tseitin formulas, the method scales nearly polynomially without preprocessing.

Where Pith is reading between the lines

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

  • Direct integration of parity reasoning into CDCL could replace separate parity modules in solvers like CryptoMiniSAT.
  • The use of nondeterministic decisions points to opportunities for developing better branching heuristics on parity equations.
  • Similar extensions might be possible for other proof systems beyond Resolution and Res(⊕).

Load-bearing premise

The new inference rules for Res(⊕) correctly allow the translation of CDCL subroutines to linear clauses while preserving the soundness of the proof system.

What would settle it

An XNF formula that admits a short Res(⊕) proof but on which CDCL(⊕) requires superpolynomial time even when allowed nondeterministic decisions and restarts.

read the original abstract

Because CDCL produces proofs in the Resolution proof system, problems provably hard for Resolution are also provably hard for CDCL. Exponentially shorter proofs can sometimes be found using stronger proof systems such as $\text{Res}(\oplus)$, a generalization of Resolution to XNF formulas, whose constraints are disjunctions of parity equations ("linear clauses") such as $(x \oplus y) \lor \lnot (y \oplus z)$. While some modern solvers like CryptoMiniSAT reason on Boolean clauses with separate parity equations, reasoning about more general linear clauses is less explored. We present $\text{CDCL}(\oplus)$, a generalization of CDCL to XNF formulas, and prove a bidirectional connection with $\text{Res}(\oplus)$: $\text{CDCL}(\oplus)$ not only produces $\text{Res}(\oplus)$ proofs, but also polynomially simulates $\text{Res}(\oplus)$ given nondeterministic decisions and restarts, mirroring the classical relationship between CDCL and Resolution. Our key technical tool is a new set of inference rules for $\text{Res}(\oplus)$ that helps us translate Resolution-based subroutines such as 1-UIP clause learning. Altogether, $\text{CDCL}(\oplus)$'s parity reasoning includes branching on arbitrary parity equations, linear-algebraic reasoning during unit propagation, and learning linear clauses through conflict analysis. We provide a proof-of-concept implementation of $\text{CDCL}(\oplus)$ called Xorcle, which includes adaptations of existing CDCL heuristics to XNF formulas and an extension of LRUP proof logging that we call $\text{LRUP}(\oplus)$. On a selected suite of benchmarks focusing on native XNF formulas, Xorcle outperforms existing solvers such as Kissat and CryptoMiniSAT. Additionally, on Tseitin formulas written in CNF, even without preprocessing, Xorcle's running time appears to scale nearly polynomially.

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 / 0 minor

Summary. The paper introduces CDCL(⊕), a generalization of CDCL to XNF formulas whose constraints are disjunctions of parity equations. It claims to establish a bidirectional connection with the Res(⊕) proof system: CDCL(⊕) produces Res(⊕) proofs and, given nondeterministic decisions and restarts, polynomially simulates Res(⊕) by means of a new set of inference rules that translate standard CDCL subroutines such as 1-UIP clause learning. The paper also presents a proof-of-concept implementation called Xorcle, including adapted heuristics and an extension of LRUP proof logging called LRUP(⊕), together with experimental results on selected XNF benchmarks and Tseitin formulas.

Significance. If the simulation result holds, the work provides a direct analogue of the classical CDCL–Resolution relationship for a stronger proof system that incorporates linear-algebraic reasoning over GF(2). This connection would clarify the proof-theoretic power of parity-aware solvers and could guide the design of stronger conflict-analysis procedures. The implementation supplies initial practical evidence that native handling of linear clauses can be competitive on formulas that are native to the XNF setting.

major comments (2)
  1. [Abstract and simulation section] Abstract and the section describing the simulation: the central claim that CDCL(⊕) polynomially simulates Res(⊕) rests on a new set of inference rules for Res(⊕) that translate 1-UIP learning. The manuscript asserts the existence of such rules and the simulation but supplies neither the explicit rules nor the detailed derivation, leaving the soundness and completeness of the translation unverified.
  2. [Experimental evaluation] Experimental evaluation section: the reported near-polynomial scaling on Tseitin formulas and outperformance on the selected XNF suite are presented without a description of the experimental methodology, benchmark selection procedure, or statistical analysis. Post-hoc benchmark selection further weakens the evidential value of the performance claims.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major point below and agree that revisions are needed to provide missing details on the simulation and experimental methodology.

read point-by-point responses

  1. Referee: [Abstract and simulation section] Abstract and the section describing the simulation: the central claim that CDCL(⊕) polynomially simulates Res(⊕) rests on a new set of inference rules for Res(⊕) that translate 1-UIP learning. The manuscript asserts the existence of such rules and the simulation but supplies neither the explicit rules nor the detailed derivation, leaving the soundness and completeness of the translation unverified.

    Authors: We agree that the manuscript does not supply the explicit inference rules or a detailed derivation of the polynomial simulation. The revised version will include the full set of new inference rules for Res(⊕) that translate standard CDCL subroutines such as 1-UIP clause learning, along with a complete proof establishing the polynomial simulation under nondeterministic decisions and restarts. This will explicitly verify soundness and completeness of the translation. revision: yes

  2. Referee: [Experimental evaluation] Experimental evaluation section: the reported near-polynomial scaling on Tseitin formulas and outperformance on the selected XNF suite are presented without a description of the experimental methodology, benchmark selection procedure, or statistical analysis. Post-hoc benchmark selection further weakens the evidential value of the performance claims.

    Authors: We agree that the experimental section requires additional detail. The revision will expand this section to describe the full experimental methodology, hardware/software environment, benchmark selection criteria (guided by the focus on native XNF formulas to evaluate parity-aware reasoning), and statistical analysis including medians and multiple runs. We will also clarify the rationale for the selected suite while acknowledging limitations of the evaluation. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central claim is a direct proof that CDCL(⊕) produces Res(⊕) proofs and polynomially simulates Res(⊕) under nondeterministic decisions and restarts. This is established via an explicit new set of inference rules for Res(⊕) that translate standard CDCL subroutines such as 1-UIP clause learning. No step reduces by construction to fitted parameters, self-definitions, or load-bearing self-citations; the simulation is presented as independently derived from the new rules and the classical CDCL-Resolution relationship. The implementation (Xorcle) and benchmarks are separate from the proof and do not affect the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard properties of linear algebra over GF(2) and resolution; the main technical addition is constructed inference rules rather than new postulated entities or fitted parameters.

axioms (2)
  • standard math Linear algebra over GF(2) supports unit propagation and conflict analysis on parity equations
    Invoked explicitly for linear-algebraic reasoning during unit propagation.
  • domain assumption Standard CDCL subroutines such as 1-UIP can be translated to XNF via new inference rules
    Central premise for the simulation proof and clause learning adaptation.

pith-pipeline@v0.9.1-grok · 5876 in / 1581 out tokens · 47068 ms · 2026-06-30T19:24:50.337369+00:00 · methodology

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Reference graph

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