Extending CDCL to disjunctions of parity equations

2026-05-14

AI Summary

Traditional CDCL solvers are limited by the Resolution proof system and struggle to efficiently handle problems involving parity (XOR) constraints. This work proposes CDCL(⊕), an extension of conflict-driven clause learning to XNF formulas that supports branching, propagation, and learning over disjunctions of XOR equations. We establish a bidirectional simulation between CDCL(⊕) and the Res(⊕) proof system, introduce novel inference rules such as 1-UIP learning tailored for linear clauses, and incorporate XOR-aware branching, linear-algebraic unit propagation, linear-clause conflict analysis, and an extended LRUP(⊕) proof logging technique. The prototype solver Xorcle significantly outperforms Kissat and CryptoMiniSAT on native XNF benchmarks and demonstrates near-polynomial scalability on CNF-encoded Tseitin formulas.

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.

Problem

• CDCL
• parity equations
• linear clauses
• Resolution
• XNF

Innovation

• CDCL(⊕)
• Res(⊕)
• XNF
• parity reasoning
• linear clauses