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Relational semantics for Kleene logic and action logic. (English) Zbl 1099.03014
Recall that (1) Kleene algebras in the sense of D. Kozen provide solutions of the finite axiomatization problem for the algebra of regular sets by finitely many equations as does (2) action logic. In this interesting paper, the authors treat them as non-classsical logics with the usual Hilbert-style axiomatization and semantics. Several soundness theorems and completeness theorems are proved mostly in great detail. In their discussions of semantics based on a four-valued approach, the authors imagine a “dual” to C. E. Shannon’s concept of information, which can be used to clarify the role of the logics and algebras. Finally, in the authors’ star semantics, the reviewer wonders: can partial generalized Galois logics be extended with an operation, if the new operation is (1) circularly definable or (2) is the closure of a circularly definable operation?

##### MSC:
 03B47 Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics) 68Q70 Algebraic theory of languages and automata 03B60 Other nonclassical logic 03B45 Modal logic (including the logic of norms) 03D05 Automata and formal grammars in connection with logical questions 03C13 Model theory of finite structures
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