Kleene Algebras and Logic: Boolean and Rough Set Representations, 3-valued, Rough and Perp Semantics

TL;DR

This paper develops a set-theoretic and modal semantics for Kleene algebras and their associated logic, demonstrating equivalence among various semantic frameworks including 3-valued, rough set, and perp semantics.

Contribution

It introduces a structural theorem linking Kleene algebra elements to set pairs and explores the semantics of Kleene negation as a set complement and modal operator.

Findings

Kleene algebra elements can be represented as ordered pairs of sets.

Kleene negation arises from set-theoretic complement.

All considered semantics for the logic are shown to be equivalent.

Abstract

A structural theorem for Kleene algebras is proved, showing that an element of a Kleene algebra can be looked upon as an ordered pair of sets. Further, we show that negation with the Kleene property (called the `Kleene negation') always arises from the set theoretic complement. The corresponding propositional logic is then studied through a 3-valued and rough set semantics. It is also established that Kleene negation can be considered as a modal operator, and enables giving a perp semantics to the logic. One concludes with the observation that all the semantics for this logic are equivalent.