smt encoding

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.

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 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 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.

MILP solvers’ outputs lack trustworthy verification in critical applications such as hardware verification, compiler optimization, and machine-assisted theorem proving. Method: This paper proposes the first formal verification framework for VIPR 1.0—a general-purpose certificate format—by fully encoding its inference rule system into unambiguous, SMT-expressible first-order logic formulas and constructing a solver-agnostic verifier compliant with the SMT-LIB standard to ensure algorithmic verifiability. Contribution/Results: The framework eliminates ambiguities inherent in the original VIPR specification and enables rigorous, implementation-independent validation of MILP certificates. Experimental evaluation on public benchmark suites confirms the verifier’s correctness and practical feasibility, demonstrating substantial improvements in both the rigor and generality of MILP certificate verification.

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.

This work addresses the challenge of efficiently integrating constraints and logical reasoning in complex combinatorial search by proposing a scalable translation-based Constraint Answer Set Programming (CASP) framework. The approach seamlessly combines Answer Set Programming with Satisfiability Modulo Theories (SMT), leveraging mature SMT solvers—such as CVC5, Yices, and Z3—to enable efficient reasoning over mixed integer and real-valued constraints. The framework introduces a more expressive input language that supports weak constraint optimization and provides a unified architecture for incorporating novel constraint types. Experimental results demonstrate that the system substantially outperforms state-of-the-art solvers, including CLINGCON, CLINGO[DL], and CLINGO[LP], achieving significant improvements in both expressiveness and solving efficiency on standard benchmarks.

This work addresses the lack of formal guarantees regarding semantic preservation during problem reformulation and solver correctness in constraint programming. It presents the first end-to-end verified framework implemented in the Lean theorem prover, enabling formal proofs of parameterized equivalence, equisatisfiability, and symmetry-breaking correctness for entire families of problems. The approach combines general, parameterized proofs with instance-level certificate checking, thereby eliminating the need to trust external solvers. Verified certificates are produced via backend transformations, and a single high-level proof suffices for arbitrarily large instances. This methodology achieves dramatic search-space reductions—up to a factor of twenty million—and enables full verification of the largest instances in just a few minutes.

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.