Bit-Precise CHC Satisfiability Using Theory-Modular Reasoning

2026-08-05

AI Summary

This work addresses the satisfiability problem for Constrained Horn Clauses (CHCs) over the theory of fixed-size bit-vectors (𝒯_B), a key challenge in bit-precise program verification. The paper introduces Mosaic, a novel framework that enables modular cooperative reasoning between bit-vector and integer arithmetic theories for the first time. By decomposing CHC problems into theory-specific fragments and facilitating cross-theory translation and information exchange, Mosaic overcomes the scalability limitations of existing CHC solvers in bit-level verification tasks. Implemented on top of Z3 and Spacer, Mosaic demonstrates substantial performance improvements over native Spacer on benchmarks involving bit-vector operations, thereby validating its effectiveness and superiority.

Abstract

Deciding satisfiability of Constrained Horn Clauses (CHCs) modulo the theory of fixed-size bit-vectors ($\mathcal{T}_B$) is fundamental to bit-precise program verification. However, state-of-the-art CHC-solvers often struggle with $\mathcal{T}_B$, limiting scalability in bit-precise reasoning. We present Mosaic, a theory-modular framework for deciding satisfiability of CHCs modulo $\mathcal{T}_B$ by combining reasoning in $\mathcal{T}_B$ and the theory of Integer Arithmetic ($\mathcal{T}_I$). Given a CHC set modulo $\mathcal{T}_B$, Mosaic partitions it into two fragments interpreted over $\mathcal{T}_B$ and $\mathcal{T}_I$. Moreover, it implements an algorithm that reasons about the fragments in a modular fashion, exchanges information between them via sound translations across theories, and determines satisfiability w.r.t. the original CHC set. We implemented a prototype of Mosaic using Z3 and Spacer and evaluated it on bit-manipulating benchmarks. Our evaluation shows that Mosaic significantly outperforms Spacer on these benchmarks.

Problem

• Constrained Horn Clauses
• bit-precise verification
• fixed-size bit-vectors
• satisfiability
• program verification

Innovation

• Constrained Horn Clauses
• bit-precise reasoning
• theory-modular reasoning
• fixed-size bit-vectors
• satisfiability