LLM-Guided Quantified SMT Solving over Uninterpreted Functions

2026-01-08

arXiv.org

AI Summary

This work addresses the fundamental challenge posed by quantified formulas containing uninterpreted functions (UFs) over nonlinear real arithmetic in SMT solving, where conventional instantiation techniques suffer from explosive search spaces due to a lack of semantic insight. We propose AquaForte, a novel framework that integrates the mathematical intuition of large language models into SMT solving for the first time. By decomposing constraints and generating semantically plausible candidate function definitions through structured prompting, AquaForte guides the solver via adaptive instantiation and an iterative clause exclusion mechanism. The approach preserves completeness while significantly accelerating the solving of satisfiable instances. Evaluated on SMT-COMP benchmarks, AquaForte successfully solves multiple problems that timed out for both Z3 and CVC5, demonstrating particularly strong performance on satisfiable formulas.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Search and Retrieval-Augmented AI: Large language models for search
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Quantified formulas with Uninterpreted Functions (UFs) over non-linear real arithmetic pose fundamental challenges for Satisfiability Modulo Theories (SMT) solving. Traditional quantifier instantiation methods struggle because they lack semantic understanding of UF constraints, forcing them to search through unbounded solution spaces with limited guidance. We present AquaForte, a framework that leverages Large Language Models to provide semantic guidance for UF instantiation by generating instantiated candidates for function definitions that satisfy the constraints, thereby significantly reducing the search space and complexity for solvers. Our approach preprocesses formulas through constraint separation, uses structured prompts to extract mathematical reasoning from LLMs, and integrates the results with traditional SMT algorithms through adaptive instantiation. AquaForte maintains soundness through systematic validation: LLM-guided instantiations yielding SAT solve the original problem, while UNSAT results generate exclusion clauses for iterative refinement. Completeness is preserved by fallback to traditional solvers augmented with learned constraints. Experimental evaluation on SMT-COMP benchmarks demonstrates that AquaForte solves numerous instances where state-of-the-art solvers like Z3 and CVC5 timeout, with particular effectiveness on satisfiable formulas. Our work shows that LLMs can provide valuable mathematical intuition for symbolic reasoning, establishing a new paradigm for SMT constraint solving.

Problem

• Quantified SMT
• Uninterpreted Functions
• Non-linear Real Arithmetic
• Satisfiability Modulo Theories
• Semantic Guidance

Innovation

• LLM-guided reasoning
• quantified SMT
• uninterpreted functions
• adaptive instantiation
• semantic guidance