Scalable Floating-Point Satisfiability via Staged Optimization

2026-01-08

arXiv.org

AI Summary

This work addresses the challenge of balancing scalability and precision in floating-point satisfiability (FP-SAT) by proposing a three-stage progressive optimization framework. It first employs orthogonal projection to rapidly identify feasible regions and achieve partial monotonic descent, thereby avoiding flat plateaus. Subsequently, it refines the solution using ULP² optimization to approach bit-level accuracy, followed by a final refinement on an n-ULP lattice, where a zero-valued objective function certifies a valid solution. The approach requires neither heavyweight bit-level reasoning nor specialized abstractions, relying solely on black-box evaluations while synergistically integrating SMT solving with numerical optimization. Evaluated on benchmarks including SMT-COMP'25, the method achieves a 99.4% recall rate with zero false positives and demonstrates a 5–10× speedup over conventional SMT and numerical solvers.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions

Abstract

This work introduces StageSAT, a new approach to solving floating-point satisfiability that bridges SMT solving with numerical optimization. StageSAT reframes a floating-point formula as a series of optimization problems in three stages of increasing precision. It begins with a fast, projection-aided descent objective to guide the search toward a feasible region, proceeding to bit-level accuracy with ULP$^2$ optimization and a final $n$-ULP lattice refinement. By construction, the final stage uses a representing function that is zero if and only if a candidate satisfies all constraints. Thus, when optimization drives the objective to zero, the resulting assignment is a valid solution, providing a built-in guarantee of soundness. To improve search, StageSAT introduces a partial monotone descent property on linear constraints via orthogonal projection, preventing the optimizer from stalling on flat or misleading landscapes. Critically, this solver requires no heavy bit-level reasoning or specialized abstractions; it treats complex arithmetic as a black-box, using runtime evaluations to navigate the input space. We implement StageSAT and evaluate it on extensive benchmarks, including SMT-COMP'25 suites and difficult cases from prior work. StageSAT proved more scalable and accurate than state-of-the-art optimization-based alternatives. It solved strictly more formulas than any competing solver under the same time budget, finding most satisfiable instances without producing spurious models. This amounts to 99.4% recall on satisfiable cases with 0% false SAT, exceeding the reliability of prior optimization-based solvers. StageSAT also delivered significant speedups (often 5--10$\times$) over traditional bit-precise SMT and numeric solvers. These results demonstrate that staged optimization significantly improves performance and correctness of floating-point satisfiability solving.

Problem

• floating-point satisfiability
• SMT solving
• numerical optimization
• scalability
• soundness

Innovation

• floating-point satisfiability
• staged optimization
• ULP-based refinement
• orthogonal projection
• soundness guarantee