encode constraints as smt

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 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.

Frequent SMT solver invocations in symbolic/concolic execution impose a severe performance bottleneck. Method: This paper proposes a generalized reuse technique targeting unsatisfiable (UNSAT) cores, moving beyond conventional caching that only supports reuse of syntactically equivalent or structurally similar formulas. We systematically investigate *semantic-preserving mappings of UNSAT cores under arbitrary variable substitutions*, leveraging UNSAT core extraction, variable substitution modeling, subformula isomorphism checking, and cache index optimization to enable cross-formula-structure reuse. Contribution/Results: Evaluated on standard benchmarks, our approach achieves a 74% UNSAT core reuse rate—33 percentage points higher than Utopia—significantly reducing solver invocations and substantially decreasing execution time in complex scenarios. By enabling reuse across semantically equivalent but syntactically divergent formulas, our method overcomes the semantic limitations inherent in prior caching strategies.

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 critical challenge of reliably integrating automated reasoning tools—such as theorem provers, SAT/SMT solvers, and termination analyzers—with proof assistants to build highly trustworthy systems. It presents a systematic survey and comparative analysis of two principal technical approaches: certification and formal verification. The study examines core methodologies including logical encoding, result replay and checking, and integration mechanisms within proof assistants. By elucidating the respective strengths and limitations of these methods and illustrating them through multiple successful case studies, the paper offers clear methodological guidance for constructing high-assurance automated reasoning systems, thereby substantially enhancing the verifiability and trustworthiness of their outputs.

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 efficiently leveraging information from SMT solvers in interactive theorem provers without relying on their full proofs. We propose a lightweight proof reconstruction approach based on “hints”: by extracting derived facts generated during SMT solving, we guide Lean’s built-in automation tactics. This method offers a rich yet loosely coupled middle ground between depending entirely on SMT proofs and using only the original premise set. Integrating cvc5 with Lean, we implement the QuerySMT tactic, which significantly outperforms existing SMT integration tools on standard Lean benchmarks, thereby enhancing the automation capabilities of interactive proof development.

This work addresses the challenge in SMT solving where advanced tasks such as MaxSMT and unsatisfiable core extraction require complete sets of theory lemmas, which traditional lazy lemma generation cannot provide. Existing eager encoding approaches suffer from limited theoretical coverage, lack of support for theory combinations, and redundant lemma production. To overcome these limitations, the paper proposes a theory-agnostic lemma enumeration method built upon an enhanced AllSMT framework. By integrating divide-and-conquer strategies, projection-based enumeration, and theory-driven formula partitioning, the approach enables efficient, parallelizable, and complete lemma set generation. It supports arbitrary theory combinations and demonstrates significantly improved efficiency and scalability, substantially outperforming classical eager encoding techniques on complex SMT instances.