Fast Ramsey Quantifier Elimination in LIRA (with applications to liveness checking)

2025-11-07

AI Summary

This work addresses the verification of liveness properties—such as infinite cliques and monadic decomposability—that cannot be expressed in first-order logic. We present the first efficient Ramsey quantifier elimination method for linear integer arithmetic (LIA), linear real arithmetic (LRA), and their combination (LIRA). Our approach extends the SMT-LIB syntax to natively support Ramsey quantifiers, designs a symbolic-computation-based quantifier elimination algorithm, and integrates it into the automated tool REAL. This yields the first fully mechanized Ramsey quantifier elimination for LIA, LRA, and LIRA, significantly improving reasoning efficiency. Furthermore, we automatically extend the FASTer tool to support liveness checking, enabling its application to liveness verification of infinite-state systems. Experimental evaluation demonstrates order-of-magnitude performance improvements over prior prototype implementations.

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
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Ramsey quantifiers have recently been proposed as a unified framework for handling properties of interests in program verification involving proofs in the form of infinite cliques, which are not expressible in first-order logic. Among others, these include liveness verification and monadic decomposability. We present the tool REAL, which implements an efficient elimination of Ramsey quantifiers in existential linear arithmetic theories over integers (LIA), reals (LRA), and the mixed case (LIRA). The tool supports a convenient input format, which is an extension of SMT-LIB over the aforementioned theories with Ramsey quantifiers. We also demonstrate a substantial speedup from the original prototype. As an application, we provide an automatic translation from FASTer (a tool for verifying reachability over infinite-state systems) output format to our extension of SMT-LIB and show how our tool extends FASTer to liveness checking.

Problem

• Efficiently eliminating Ramsey quantifiers in linear arithmetic theories
• Extending SMT-LIB format to support Ramsey quantifier inputs
• Enabling liveness verification for infinite-state systems automatically

Innovation

• Efficient elimination of Ramsey quantifiers
• Extension of SMT-LIB input format
• Automatic translation for liveness checking