smt solving

This study systematically evaluates the solving performance of SAT and SMT solvers on ultra-large-scale Sudoku puzzles (25×25), focusing on solution efficiency and success rates across varying difficulty levels. Method: We construct a unified benchmarking framework integrating state-of-the-art SMT solvers (Z3, CVC5) and DPLL-based SAT solvers, coupled with a novel constraint-based Sudoku generator that supports controllable difficulty and structural diversity, alongside quantitative difficulty metrics. Contribution/Results: First, this work presents the first systematic empirical evaluation of SMT solvers on 25×25 Sudoku. Second, we propose a scalable, configurable Sudoku generator enabling precise difficulty tuning and enhanced puzzle diversity. Third, experimental results demonstrate that SMT solvers significantly outperform SAT solvers in both solving speed and robustness—particularly on harder instances—thereby empirically validating the benefits of theory-aware reasoning and high-level encodings for large-scale constraint satisfaction problems.

This work presents the first systematic solution to the problems of model counting and sampling in the theory of bit-vectors. By leveraging bit-blasting to translate bit-vector formulas into conjunctive normal form (CNF), the approach integrates modern CNF counters and samplers to uniformly support a wide range of modes, including exact and approximate counting, projected and unprojected counting, as well as near-uniform and uniform sampling. The resulting tool, csb, addresses a critical gap in efficient counting and sampling for bit-vector constraints and demonstrates substantial performance advantages over existing methods in empirical evaluations, highlighting its practicality and effectiveness.

This work addresses the limitations of MCSat in solving complex SMT problems involving nonlinear integer and real arithmetic by reformulating it as a theory-agnostic proof system. By formally capturing key implementation mechanisms from the Yices2 solver, the authors derive a unified and general framework of MCSat inference rules, which they instantiate across multiple theories—including propositional logic, nonlinear real arithmetic, and uninterpreted functions. This approach not only integrates core design choices of modern SMT solvers but also establishes the first unified MCSat calculus supporting multiple theories, substantially enhancing its expressiveness and applicability. The effectiveness of the proposed framework is demonstrated through representative examples.

SMT solvers exhibit poor efficiency on quantified formulas arising from real-world applications—especially when formulas are easily encodable yet computationally expensive to solve. This paper introduces a novel quantification mechanism based on set-bounded quantifiers, where variable domains are restricted to finite sets, and integrates quantifier elimination with filtering operators from finite relational theory. Our contributions are threefold: (1) We define a decidable fragment of constraints wherein bounded quantification is realized via constrained set derivation; (2) we identify the fundamental cause of undecidability in unrestricted filtering operations; and (3) we establish a formal framework unifying quantifier-free logic with filtering operators. Experiments demonstrate that our approach significantly outperforms state-of-the-art quantification techniques on the satisfiable SLEEC benchmark, while matching the performance of the specialized solver LEGOS on unsatisfiable benchmarks.

This work addresses the low computational efficiency of stable model solving for Answer Set Programming Modulo Theories (ASPMT) under real-number constraints. We propose the first semantics-preserving, compact fragment-to-SMT automatic compilation method. Our approach leverages *gringo* for partial grounding and uniformly encodes the remaining logical variables and real arithmetic constraints into SMT-LIB format, enabling stable model computation via general-purpose SMT solvers such as Z3. The key innovation lies in defining translation rules grounded in the functional stable model semantics, thereby establishing the first formal semantic bridge between ASP and SMT—supporting nonmonotonic reasoning and modeling of continuous change. Experimental evaluation demonstrates substantial improvements in expressiveness and solving efficiency for real-arithmetic reasoning and dynamic system modeling. The method provides a scalable, automated foundation for formal verification of physical processes.

This work addresses the challenge of solving complex string constraints that are poorly supported in current SMT solvers by proposing a novel string equation solving method. Building upon the classical Nielsen transformation, the approach uniquely integrates string exponentiation with generalized Parikh images and introduces an equality decomposition technique, thereby substantially extending the scope of traditional methods. Experimental results demonstrate that the proposed method effectively solves intricate string equations, including those involving rare SMT string inputs, significantly enhancing the capability to handle expressive string constraints.

This work addresses the automation bottleneck in automated theorem proving for first-order classical logic by proposing a novel approach that integrates the strengths of ordered saturation and subgoal reduction. The method directly encodes the connection calculus as a SAT problem and, for the first time, incorporates symmetry-breaking techniques to devise three innovative SAT encoding schemes. These encodings transcend the conventional paradigm of reducing first-order logic to propositional logic through simplification. The resulting solver, upCoP, demonstrates the effectiveness and practicality of the proposed approach in empirical evaluations, achieving a significant improvement in proof search efficiency.

This work addresses the inefficiency of directly handling quantified formulas in SMT solving and the absence of a general-purpose automated grounder supporting aggregation operations, which often compels users to resort to error-prone manual grounding. To overcome these limitations, the paper introduces relational algebra into the SMT grounding process for the first time, establishing a theoretical grounding framework that supports first-order logic with aggregation and enables finite equivalent transformations of certain quantified formulas over infinite domains. Building on an embedded SQLite engine for efficient relational algebra evaluation, the authors develop xmt-lib, an SMT-LIB-compliant grounder. Empirical evaluation on public benchmarks demonstrates that xmt-lib substantially enhances the performance of the Z3 solver, outperforming purely declarative approaches.

This work addresses the challenge of establishing quantifier-free equivalence between bit-vector and finite-field arithmetic in zero-knowledge proof circuit verification. We introduce BitModEq, a novel tactic implemented within the Lean theorem prover, which integrates range lemmas, case analysis, and bit-blasting to enable the first verified, automatic translation and equivalence proof from finite-field operations to bit-vector representations in Lean. Evaluated on standard ZKP arithmetization benchmarks, our approach solves 19% more instances than state-of-the-art SMT solvers, effectively overcoming their scalability limitations in handling conversion operators and inequalities.

This work investigates whether large language models can autonomously construct software systems with formal reasoning capabilities, specifically by generating from scratch a complete DPLL(T)-style SMT solver. The resulting solver supports quantifier-free uninterpreted functions (QF_UF), incorporates preprocessing and the Nieuwenhuis–Oliveras congruence closure algorithm, and automatically produces formal proofs in Lean for unsatisfiable instances. To the best of our knowledge, this is the first SMT solver fully generated by a large language model without any human-written code that is capable of emitting machine-checkable proofs. Experimental evaluation on standard SMT-LIB benchmarks demonstrates competitive performance, substantially advancing the frontier of AI-driven autonomous construction of sophisticated formal reasoning tools.