Continuous Optimization for Satisfiability Modulo Theories on Linear Real Arithmetic

2026-03-24

AI Summary

This work proposes FourierSMT, a novel framework addressing the limited parallelism and poor scalability of traditional SMT solvers on large-scale linear real arithmetic problems. By reformulating the SMT problem as a continuous optimization task, FourierSMT introduces the extended Walsh–Fourier expansion (xWFE) to the mixed Boolean–real domain for the first time. It leverages extended binary decision diagrams (xBDDs) to reduce computational complexity and establishes an equivalence between solution completeness and the expected value of xWFE through randomized rounding and probabilistic sampling of circuit outputs. Evaluated on scheduling and placement benchmarks involving up to 10,000 variables and 700,000 constraints, FourierSMT achieves an 8× speedup over state-of-the-art SMT solvers, demonstrating significantly improved scalability and solving efficiency.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Mixed Discrete/Continuous Search
• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning

Application Category

• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

Efficient solutions for satisfiability modulo theories (SMT) are integral in industrial applications such as hardware verification and design automation. Existing approaches are predominantly based on conflict-driven clause learning, which is structurally difficult to parallelize and therefore scales poorly. In this work, we introduce FourierSMT as a scalable and highly parallelizable continuous-variable optimization framework for SMT. We generalize the Walsh-Fourier expansion (WFE), called extended WFE (xWFE), from the Boolean domain to a mixed Boolean-real domain, which allows the use of gradient methods for SMT. This addresses the challenge of finding satisfying variable assignments to high-arity constraints by local updates of discrete variables. To reduce the evaluation complexity of xWFE, we present the extended binary decision diagram (xBDD) and map the constraints from xWFE to xBDDs. We then show that sampling the circuit-output probability (COP) of xBDDs under randomized rounding is equivalent to the expectation value of the xWFEs. This allows for efficient computation of the constraints. We show that the reduced problem is guaranteed to converge and preserves satisfiability, ensuring the soundness of the solutions. The framework is benchmarked for large-scale scheduling and placement problems with up to 10,000 variables and 700,000 constraints, achieving 8-fold speedups compared to state-of-the-art SMT solvers. These results pave the way for GPU-based optimization of SMTs with continuous systems.

Problem

• Satisfiability Modulo Theories
• Linear Real Arithmetic
• Scalability
• Parallelization
• High-arity Constraints

Innovation

• FourierSMT
• extended Walsh-Fourier expansion
• xBDD
• continuous optimization
• SMT