Many-valued logics were first considered by J. Łukasiewicz in 1920. MV-algebras were introduced by C.C.Chang in 1958 to prove the completeness theorem for infinite-valued Łukasiewicz logic. In the last 25 years the importance of MV-algebras and Łukasiewicz logic has been increasing, for three main reasons: (i) the discovery of a categorical equivalence between MV-algebras and lattice-ordered Abelian groups with an Archimedean unit; (ii) the deep relations between MV-algebras and polyhedral and toric geometry: suffice to say that the strong Oda conjecture is equivalent to the joint refinability of MV-algebraic bases; (iii) the applications of many-valued logic to the treatment of uncertain information, e.g. in the Rényi-Ulam game of Twenty Questions with errors, i.e. Berlekamp’s theory of feedback error-correcting coding; remarkably enough, the tautology problem of infinite-valued logic is coNP-complete, precisely as its two-valued counterpart.
The overall aim of this project is the application of techniques from algebraic topology, polyhedral geometry and functional analysis to the study of the fine structure of MV-algebras and the deductive-algorithmic theory of Łukasiewicz logic. The minimization problem for finitely axiomatized theories and the characterization of projective MV-algebras and its application to unification theory in Łukasiewicz logic, are just two challenging problems, with ramifications to various mathematical areas. These problems will be investigated by refined techniques arising from MV-algebraic representation theory. The basic methodology has been introduced in the applicant’s joint papers with the researcher in charge, published (or to appear) in Communications in Contemporary Mathematics, Forum Mathematicum, Algebra Universalis, Journal of Algebra.
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