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

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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, and that negation with the Kleene property (called the ‘Kleene negation’) is describable by the set-theoretic complement. The propositional logic \({\mathcal {L}}_{K}\) of Kleene algebras is shown to be sound and complete with respect to a 3-valued and a rough set semantics. It is also established that Kleene negation can be considered as a modal operator, due to a perp semantics of \({\mathcal {L}}_{K}\). Moreover, another representation of Kleene algebras is obtained in the class of complex algebras of compatibility frames. One concludes with the observation that \({\mathcal {L}}_{K}\) can be imparted semantics from different perspectives.

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  1. Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur, 208016, India

    Arun Kumar & Mohua Banerjee

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  1. Arun Kumar
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Correspondence to Arun Kumar.

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Kumar, A., Banerjee, M. Kleene Algebras and Logic: Boolean and Rough Set Representations, 3-Valued, Rough Set and Perp Semantics. Stud Logica 105, 439–469 (2017). https://doi.org/10.1007/s11225-016-9696-6

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