Internet-Publication. — 16 p. English. (OCR-слой).
[Soft Computing Laboratory, Dept. Mathematics and Informatics, University of Salerno, Italy].
AbstractIn this paper we describe the relationship between many-valued logics (in particular Basic logic and ÃLukasiewicz logic) and semirings.
We will also give definitions of automata on BL-algebras and MV-algebras.
Introduction.Semirings are algebraic structures with two associative binary operations, where one distributes over the other, introduced by Vandiver [15] in 1934. In more recent times semirings have been deeply studied, especially in relation with applications ([8]). For example semirings have been used to model formal languages and automata theory (see [6]), and semirings over real numbers ((max, +)-semirings) are the basis for the idempotent analysis [11].
In this paper, following the lines established in [5], we make further steps in establishing a relationship between semirings and many-valued logics.
Many-valued logic has been proposed to model phenomena in which uncertainty and vagueness are involved. One of the more general classes of many-valued logics is the BL-logic defined in [9] (see also [10]) as the logic of continuous t-norms. Special cases of BL-logics are ÃLukasiewicz, Godel and Product logic. In particular ÃLukasiewicz logic has been deeply investigated, together with its algebraic counterpart, MV-algebras, introduced by Chang in [1] to prove completeness theorem of ÃLukasiewicz logic. MV-algebras have nice algebraic properties and can be considered as intervals of lattice-ordered groups (see [2]).
Introduction.Preliminaries.Semirings and MV-algebras.Semiring connection between MV-algebras and l-groups.
ApplicationsBL-Automata.
Conclusions and Acknowledgments.
References.