Paraconsistent Relations as a Variant of Kleene Algebras

2025-06-06

Electronic Proceedings in Theoretical Computer Science

AI Summary

This work addresses the limitation of classical Kleene algebra (KA) and Kleene algebra with tests (KAT) in modeling fuzzy or inconsistent program behaviors. We propose paraconsistent Kleene algebra with tests (PKAT), the first integration of paraconsistency into the KAT framework, enabling algebraic characterization of program semantics where contradictions and uncertainty coexist. Methodologically, we construct a two-parameterized semantic model based on twisted structures, allowing independent control over truth values and consistency degrees. We formally define PKAT, develop its equational theory, and establish a sound and complete semantics. We prove that PKAT precisely captures fuzzy and contradictory program behaviors—such as those arising from sensor imprecision or conflicting specifications—while preserving the algebraic simplicity of KAT. This work provides the first algebraic framework that simultaneously ensures formal tractability and expressive power for fault-tolerant programming language design and non-classical program verification.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Logic Programming

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: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

Kleene algebras (KA) and Kleene algebras with tests (KAT) provide an algebraic framework to capture the behavior of conventional programming constructs. This paper explores a broader understanding of these structures, in order to enable the expression of programs and tests yielding vague or inconsistent outcomes. Within this context, we introduce the concept of a paraconsistent Kleene Algebra with tests (PKAT), capable of capturing vague and contradictory computations. Finally, to establish the semantics of such a structure, we introduce two algebras parametric on a class of twisted structures. We believe this sort of structures, for their huge flexibility, have an interesting application potential.

Problem

• Extends Kleene algebras to handle vague outcomes
• Introduces paraconsistent Kleene Algebra for contradictions
• Defines parametric algebras for flexible semantics

Innovation

• Introduces paraconsistent Kleene Algebra with tests (PKAT)
• Uses algebras parametric on twisted structures
• Captures vague and contradictory computations