Inductive Satisfiability Certification for Universal Quantifiers and Uninterpreted Function Symbols

2026-02-18

AI Summary

This work addresses the limitations of existing SMT solvers in deciding the satisfiability of linear integer arithmetic formulas involving universal quantifiers and uninterpreted function symbols, where reliance on explicit small models often leads to failure. The paper introduces a novel satisfiability proof method based on inductive reasoning, marking the first integration of induction into satisfiability certification for this class of formulas without constructing explicit models. By synergistically combining inductive inference, linear integer arithmetic, and SMT techniques, the approach substantially broadens the scope of tractable instances. It successfully verifies several satisfiable formulas that state-of-the-art SMT solvers cannot decide, thereby advancing the capability of automated reasoning for complex quantified logical formulas.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs

Abstract

The combination of uninterpreted function symbols and universal quantification occurs in many applications of automated reasoning, for example, due to their ability to reason about arrays. Yet the satisfiability of such formulas is, in general, undecidable. In practice, SMT solvers are often successful in the unsatisfiable case, using heuristics. However, in the satisfiable case, they rely on explicit model construction, which fails for formulas whose smallest model is not small enough. We introduce an alternative approach that certifies satisfiability using induction arguments, and apply it to the case of linear integer arithmetic. The resulting algorithm is able to prove satisfiability of formulas that are out of reach for current SMT solvers.

Problem

• satisfiability
• universal quantifiers
• uninterpreted function symbols
• inductive certification
• SMT solvers

Innovation

• inductive satisfiability certification
• universal quantifiers
• uninterpreted function symbols
• SMT solvers
• linear integer arithmetic