A Cut-Free Sequent Calculus for the Analysis of Finite-Trace Properties in Concurrent Systems

2025-12-02

AI Summary

This paper addresses the formal verification of prefix-closed properties over finite traces in concurrent systems. To enable compositional reasoning, it introduces LMC—the first cut-free sequent calculus framework for such properties. Methodologically, LMC rests on three key innovations: (1) closure ℓ-monoids as the minimal algebraic structure for modeling finite-trace properties; (2) a division-free modal logic based on distributive residuated lattices, uniformly capturing both prefix closure and its residuals; and (3) a Gentzen-style syntactic formulation achieved by integrating Belnap’s structural operators with Moortgat’s modal rules. LMC is proven sound and complete with respect to closure ℓ-monoids and satisfies cut elimination. Consequently, it provides a novel, semantically grounded yet proof-theoretically efficient tool for compositional verification of prefix-closed finite-trace properties.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Machine Learning: Large Multimodal Models (LMMs)

Application Category

• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

We address the problem of identifying a proof-theoretic framework that enables a compositional analysis of finite-trace properties in concurrent systems, with a particular focus on those specified via prefix-closure. To this end, we investigate the interaction of a prefix-closure operator and its residual (with respect to set-theoretic inclusion) with language intersection, union, and concatenation, and introduce the variety of closure $ell$-monoids as a minimal algebraic abstraction of finite-trace properties to be conveniently described within an analytic proof system. Closure $ell$-monoids are division-free reducts of distributive residuated lattices equipped with a forward diamond/backward box residuated pair of unary modal operators, where the diamond is a topological closure operator satisfying $Diamond(x cdot y) leq Diamond x cdot Diamond y$. As a logical counterpart to these structures, we present $mathsf{LMC}$, a Gentzen-style system based on the division-free fragment of the Distributive Full Lambek Calculus. In $mathsf{LMC}$, structural terms are built from formulas using Belnap-style structural operators for monoid multiplication, meet, and diamond. The rules for the modalities and the structural diamond are taken from Moortgat's system $mathsf{NL}(Diamond)$. We show that the calculus is sound and complete with respect to the variety of closure $ell$-monoids and that it admits cut elimination.

Problem

• Develops a cut-free sequent calculus for analyzing finite-trace properties in concurrent systems.
• Introduces closure ℓ-monoids as an algebraic abstraction for prefix-closure properties.
• Ensures soundness, completeness, and cut elimination for the logical system LMC.

Innovation

• Introduces closure ℓ-monoids as algebraic abstraction for finite-trace properties
• Presents LMC, a Gentzen-style cut-free sequent calculus for analysis
• Uses structural operators and modal rules from NL(◇) for compositionality