Alternatives to floating-point satisfiability

Floating-point satisfiability (FPSAT) is the problem of determining if a given set of constraints involving floating-point variables and operations is satisfiable. It is crucial for verifying the correctness of numerical algorithms and software, especially in safety-critical domains where floating-point inaccuracies can lead to significant errors.

At a glance

Executive summary

Floating-point satisfiability (FPSAT) is the problem of determining if there exists an assignment of floating-point numbers that satisfies a given set of constraints. It is a specialized area within constraint satisfaction and verification, particularly relevant for analyzing programs that heavily rely on floating-point arithmetic.

TL;DR

If you need to verify properties of floating-point computations use FPSAT; if you need to find optimal solutions to numerical problems use numerical optimization; if you need to solve general logical formulas with integer/real variables use SMT solving; if you need to minimize floating-point errors use ULP optimization.

Key points

Consider FPSAT when your primary concern is the exact satisfaction of constraints involving floating-point numbers, not just finding approximate solutions.
If the problem involves finding the best possible values for variables under numerical constraints, numerical optimization is likely more appropriate.
For general logical reasoning that may include floating-point variables but also complex logical structures, SMT solvers offer broader capabilities.
If the goal is to reduce the impact of rounding errors in floating-point computations, ULP optimization is the direct approach.
Evaluate whether your problem can be precisely formulated with floating-point arithmetic as the core constraint type, which is the domain of FPSAT.

Our Take

### Our Take The landscape of computational problem-solving has evolved significantly, particularly in the realms of floating-point satisfiability (FPS) and Satisfiability Modulo Theories (SMT) solving, alongside numerical and ULP (Unbounded Linear Programming) optimization. Each of these methodologies serves distinct purposes, yet they intersect in their applications and underlying principles. FPS focuses on determining the satisfiability of floating-point expressions, which is crucial for verifying software and hardware systems that rely on numerical computations. Recent studies, such as those by D'Antoni et al. (2018), highlight the challenges of floating-point arithmetic, particularly in terms of precision and rounding errors. FPS provides a robust framework to address these issues, ensuring that solutions are not only feasible but also reliable in real-world applications. On the other hand, SMT solving extends beyond mere satisfiability to encompass a broader range of theories, including arithmetic, arrays, and bit-vectors. As demonstrated in the work of Barret et al. (2011), SMT solvers can efficiently handle complex constraints, making them suitable for various applications in formal verification and automated reasoning. However, they may struggle with the intricacies of floating-point arithmetic, which can lead to incomplete or inaccurate results. Numerical optimization and ULP optimization, as explored by Bertsimas and Tsitsiklis (1997), provide powerful tools for finding optimal solutions in continuous spaces. While these methods excel in scenarios requiring optimality, they may not inherently address the satisfiability aspect of problems, which can be critical in verification tasks. In conclusion, while FPS and SMT solving share common ground in addressing numerical challenges, they cater to different needs within the computational spectrum. Numerical and ULP optimization add another layer, focusing on optimal solutions rather than satisfiability. This nuanced interplay suggests that a hybrid approach, leveraging the strengths of each methodology, may yield the most effective results in complex problem-solving scenarios.

Alternative	Difference	Papers (with floating-point satisfiability)	Avg viability
SMT solving	—	1	—
numerical optimization	—	1	—
ULP optimization	—	1	—