Continuous Optimization for Satisfiability Modulo Theories on Linear Real Arithmetic

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.

Figures (5)

• Figure 1: The overall flow of FourierSMT.(a) A variety of combinatorial problems target to optimize propositional and real variables. (b) These problems can be encoded using an SMT(LRA) formula, which is a SAT formula with additional LRA theory. (c) A constraint in the SMT(LRA) formula can be transformed into a polynomial called xWFE. The amplitudes show the coefficients of the xWFE, described in Eq. \ref{['eq:general-WFE']}. (d) The expectation of the xWFE can be obtained by randomized rounding on the Boolean variable and the Gaussian sampling on the real variables. (e) However, the xWFE has an exponential number of terms. We show that the circuit output probability of xBDDs is equivalent to the expectation of the xWFE. This avoids state enumeration and thereby preserves computation efficiency. (f) We recover the minima of the original function by annealing of the sampling parameters, gradually restoring the original piecewise objective.
• Figure 2: Optimization trajectories and energy landscapes under different sampling parameters. The number of gradient steps required to reach an $\epsilon$-projected critical point is shown for $\epsilon=10^{-2}$. (a) The colorbar indicates the inverse sampling parameter ${1}/{\sigma}$. 1. When $\sigma\to\infty$ (i.e., ${1}/{\sigma}=0$), the gradient with respect to the real-valued variables $y$ vanishes, as the smoothing effect becomes arbitrarily strong. 2. Moderately large sampling parameters, i.e., ${1}/{\sigma}\in\{0.1,\ldots,0.5\}$, yield slower convergence to the global minimum. 3. In contrast, smaller values of $\sigma$, corresponding to ${1}/{\sigma}\in\{0.6,\ldots,2.0\}$, lead to faster convergence to a local minimum. 4. The dashed gradient-colored curve denotes the annealing schedule, in which ${1}/{\sigma}$ is gradually increased in increments of 0.1, achieving both rapid convergence and a final solution close to the global optimum. (b) Visualization of the objective landscape and the corresponding optimization trajectory for different $\sigma$. The horizontal and the vertical axes of each subplot correspond to $x$ and $y$ in the domain $[-1,1]$. Each subplot shows the smoothed objective ($\mathcal{C}_\sigma$, greyscale background) and the path taken by the gradient descent (colorline, the color refers to the color bar in panel (a) as $1/\sigma$). The optimization trajectories start at the same initial point (black edge circle) and follow the gradients to global (red edge circle) or local minima (green edge circle).
• Figure 3: Benchmark results on random instances with mixed Boolean-real hybrid constraints.(a) The results show the number of solved instances and runtime efficiency (average PAR-2 score of the solvers). It can be seen that the virtual-best solver (VBS1, including FourierSMT) is 62.45$\times$ faster and solves 50 more instances than its counterpart (VBS2, excluding FourierSMT). The 20 easy instances (small-scale) are solved more efficiently by CDCL($\mathcal{T}$) solvers, whereas all larger instances are solved most efficiently by FourierSMT. This pattern demonstrates that FourierSMT is substantially more scalable. (b) The average gradient computation time of FourierSMT using CPU (AMD EPYC 9654) and GPU (NVIDIA L40S). For each problem instance, we measure the time required to compute a single gradient step. The GPU exhibits near-constant runtime across all sizes, reflecting efficient parallel amortization of fixed overheads, e.g., kernel launch, memory movement. In contrast, CPU runtime increases steadily with instance size, scaling approximately linearly.
• Figure 4: Performance comparison of SMT solvers on combinatorial problems. The results show the number of solved instances and runtime efficiency (average PAR-2 score of the solvers). (a) Scheduling problems, which mainly consist of non-overlap, feasibility, and task-dependency constraints. The virtual-best solver (VBS1, including FourierSMT) is 8.18$\times$ faster and solves 20 more instances than the counterpart (VBS2, excluding FourierSMT). (b) Placement problems, which mainly consist of non-overlap, feasibility, and routing-aware constraints. In this category, Z3 and Z3++ achieve the best performance cross CDCL($\mathcal{T}$) solvers. VBS1 is 4.93$\times$ faster and solves 10 more instances than VBS2. All solvers operate on identical encodings, indicating that the performance gain stems purely from the optimization framework rather than from encoding-free advantages.
• Figure 11: Detailed solver performance on benchmark categories.(a) Results on the scheduling problems. Z3++, CVC5, MathSAT5, SMT-RAT wins non of the instances in this category. Yices2 performs best on the small-scale instances ($n_w = 16$), while SMTS attains the best results on the medium-scale cases ($n_w \in \{32, 64\}$). As the instance size increases further, FourierSMT can handle cases that cause timeouts among competing solvers. (b) Results on the placement problems. SMTS, CVC5, MathSAT5, SMT-RAT wins non of the instances in this category. Yices2 still performs best on the small-scale instances ($n_m = 2$), while Z3 and Z3++ attains the best results on the medium-scale cases ($n_w \in \{8, 16, 32\}$). The landscape of solvers exhibits pronounced variability: different solvers excel in different problem scales, and no classical solver maintains uniformly strong performance. FourierSMT benefits from GPU acceleration, resulting in relatively similar solving times across instance sizes because the fixed parallelization overhead of the GPU amortizes most of the computational cost. For the largest category, FourierSMT remains scalable and continues to solve instances that all other solvers time out on.

Theorems & Definitions (40)

• Theorem 1: Soundness
• Theorem 2: Optimality
• Definition 1: Projection
• Proposition 1
• Theorem 3: Walsh-Fourier Expansion o2014analysis
• Corollary 1: Extended Walsh-Fourier Expansion, xWFE
• proof : Proof of Corollary \ref{['cor:wfe']}
• Lemma 1
• proof : Proof of Lemma \ref{['lmm:expectation']}
• Theorem 4