Notre Dame Journal of Formal Logic

Relational Semantics for Kleene Logic and Action Logic

Katalin Bimbó and J.~Michael Dunn

Abstract

Kleene algebras and action logic were proposed to be solutions to the finite axiomatization problem of the algebra of regular sets (of strings). They are treated here as nonclassical logics—with Hilbert-style axiomatizations and semantics. We also provide intuitive accounts in terms of information states of the semantics which provide further insights into the formalisms. The three types of "Kripke-style'' semantics which we define develop insights from gaggle theory, and from our four-valued and generalized Kripke semantics for the minimal substructural logic. Soundness and completeness are proven each time.

Article information

Source
Notre Dame J. Formal Logic, Volume 46, Number 4 (2005), 461-490.

Dates
First available in Project Euclid: 12 December 2005

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1134397663

Digital Object Identifier
doi:10.1305/ndjfl/1134397663

Mathematical Reviews number (MathSciNet)
MR2183055

Zentralblatt MATH identifier
1099.03014

Subjects
Primary: 03B47: Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics) {For proof-theoretic aspects see 03F52}
Secondary: 03B45: Modal logic (including the logic of norms) {For knowledge and belief, see 03B42; for temporal logic, see 03B44; for provability logic, see also 03F45} 68Q70: Algebraic theory of languages and automata [See also 18B20, 20M35] 03D05: Automata and formal grammars in connection with logical questions [See also 68Q45, 68Q70, 68R15]

Keywords
gaggle theory Routley-Meyer semantics Kripke semantics nonclassical logics modal logics join semi-lattice residuation reflexive transitive closure Kleene star regular languages

Citation

Bimbó, Katalin; Dunn, J.~Michael. Relational Semantics for Kleene Logic and Action Logic. Notre Dame J. Formal Logic 46 (2005), no. 4, 461--490. doi:10.1305/ndjfl/1134397663. https://projecteuclid.org/euclid.ndjfl/1134397663


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