Continuous Optimization for Satisfiability Modulo Theories on Linear Real Arithmetic

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.

💡 Research Summary

This paper introduces FourierSMT, a novel continuous‑optimization framework for solving Satisfiability Modulo Theories (SMT) problems that involve linear real arithmetic (LRA). Traditional SMT solvers are built on conflict‑driven clause learning (CDCL) and struggle to scale on modern multi‑core or GPU architectures because their search is fundamentally sequential. FourierSMT overcomes this limitation by converting each SMT constraint into a piecewise‑multilinear polynomial via an extended Walsh‑Fourier expansion (xWFE) and then evaluating these polynomials efficiently using an extended binary decision diagram (xBDD).

The authors first generalize the classic Walsh‑Fourier transform, which is defined only on Boolean hypercubes, to a mixed Boolean‑real domain. For each atomic linear inequality α_i they introduce a Boolean indicator δ_i(y) that evaluates to +1 or –1 depending on whether the inequality holds. A constraint c containing n Boolean variables and k real atoms can then be expressed uniquely as

 f_c(x, y) = Σ_{S⊆