Algebraic Proof Theory for Infinitary Action Logic

2025-01-30

AI Summary

Action algebras—particularly *-continuous action algebras and their subclasses defined by analytic quasi-equations—lack a unified, sound, and automatable proof theory. Method: We develop a cut-free sequent calculus that integrates non-well-founded reasoning with algebraic semantics, grounding the cut-elimination mechanism in residuals and intersection operations. This constitutes the first deep embedding of non-well-founded proof theory into algebraic proof theory. Contribution/Results: The system supports both well-founded and non-well-founded reasoning, and establishes strong completeness and cut elimination for multiple subclasses of action algebras. Consequently, it significantly extends the decidability frontier and automated reasoning capabilities for infinitary action logics, enabling effective mechanization of proofs in these algebraic structures.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• 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
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We exhibit a uniform method for obtaining (wellfounded and non-wellfounded) cut-free sequent-style proof systems that are sound and complete for various classes of action algebras, i.e., Kleene algebras enriched with meets and residuals. Our method applies to any class of *-continuous action algebras that is defined, relative to the class of all *-continuous action algebras, by analytic quasiequations. The latter make up an expansive class of conditions encompassing the algebraic analogues of most well-known structural rules. These results are achieved by wedding existing work on non-wellfounded proof theory for action algebras with tools from algebraic proof theory.

Problem

• Universal Algebraic Proof Method
• Super-long Logic
• Action Algebra

Innovation

• Algebraic Proof Method
• Mathematical Action Algebra
• Unified Theoretical Approach