Extending CDCL to disjunctions of parity equations

TL;DR

This paper introduces CDCL(⊕), a generalization of CDCL for XNF formulas, connecting it with Res(⊕) proofs, and demonstrates its effectiveness on benchmarks involving linear clauses and Tseitin formulas.

Contribution

It develops CDCL(⊕), a novel solver framework that extends CDCL to handle disjunctions of parity equations, with theoretical proof of its simulation capabilities and practical implementation.

Findings

Xorcle outperforms Kissat and CryptoMiniSAT on XNF benchmarks.

Xorcle's runtime scales nearly polynomially on Tseitin formulas.

Theoretical connection established between CDCL(⊕) and Res(⊕) proof systems.

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.