Paraconsistent Relations as a Variant of Kleene Algebras

TL;DR

This paper introduces paraconsistent Kleene algebras with tests (PKAT), extending traditional algebraic frameworks to model programs with vague or contradictory outcomes, supported by new algebraic semantics.

Contribution

It proposes PKAT, a novel algebraic structure that captures vague and inconsistent computations, expanding the applicability of Kleene algebras.

Findings

Introduction of PKAT for vague and contradictory computations

Development of two algebraic semantics based on twisted structures

Potential applications in modeling inconsistent program behaviors

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.