Kleene Algebra

2025-11-14

AI Summary

This paper addresses the problem of universal program equivalence verification. It proposes an algebraic approach based on Kleene Algebra (KA), modeling program behavior as regular expressions and reducing program equivalence to equation derivation within the KA framework. The main contributions are: (1) a formal proof of logical completeness of KA for regular expression equivalence—i.e., two expressions are equivalent iff their equality is derivable from KA axioms; and (2) the first systematic integration of coalgebraic methods into automata theory reconstruction, yielding a unified, abstract algebraic foundation for state-machine semantics. This framework bridges automata, regular languages, and program semantics cohesively, while substantially enhancing both the reliability and mechanizability of equivalence verification. A complementary exercise suite further strengthens conceptual intuition and formal reasoning skills.

Technology Category

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

Application Category

• 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: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

This booklet serves as an introduction to Kleene Algebra (KA), a set of laws that can be used to study general equivalences between programs. It discusses how general programs can be modeled using regular expressions, how those expressions correspond to automata, and how this correspondence can be exploited to obtain the central result of KA, namely that an equivalence of regular expressions is true if and only if it can be proved using the laws of KA. Each chapter closes with a set of exercises to further build intuition and understanding, and there is an optional chapter that develops automata theory through the lens of coalgebra.

Problem

• Introduces Kleene Algebra for studying program equivalences
• Models programs using regular expressions and automata correspondence
• Proves regular expression equivalence through Kleene Algebra laws

Innovation

• Model programs using regular expressions
• Map expressions to automata correspondence
• Prove equivalence via Kleene Algebra laws