Valuations in Nilpotent Minimum Logic

The Euler characteristic can be defined as a special kind of valuation on finite distributive lattices. This work begins with some brief consideration on the role of the Euler characteristic on NM algebras, the algebraic counterpart of Nilpotent Minimum logic. Then, we introduce a new valuation, a modified version of the Euler characteristic we call idempotent Euler characteristic. We show that the new valuation encodes information about the formul{\ae} in NM propositional logic

A temporal semantics for Nilpotent Minimum logic

In [Ban97] a connection among rough sets (in particular, pre-rough algebras) and three-valued {\L}ukasiewicz logic {\L}3 is pointed out. In this paper we present a temporal like semantics for Nilpotent Minimum logic NM ([Fod95, EG01]), in which the logic of every instant is given by {\L}3: a completeness theorem will be shown. This is the prosecution of the work initiated in [AGM08] and [ABM09], in which the authors construct a temporal semantics for the many-valued logics of G\"odel ([G\"od32], [Dum59]) and Basic Logic ([H\'aj98]).Comment: 19 pages, 2 table

First-order Nilpotent Minimum Logics: first steps

Following the lines of the analysis done in [BPZ07, BCF07] for first-order G\"odel logics, we present an analogous investigation for Nilpotent Minimum logic NM. We study decidability and reciprocal inclusion of various sets of first-order tautologies of some subalgebras of the standard Nilpotent Minimum algebra. We establish a connection between the validity in an NM-chain of certain first-order formulas and its order type. Furthermore, we analyze axiomatizability, undecidability and the monadic fragments.Comment: In this version of the paper the presentation has been improved. The introduction section has been rewritten, and many modifications have been done to improve the readability; moreover, numerous references have been added. Concerning the technical side, some proofs has been shortened or made more clear, but the mathematical content is substantially the same of the previous versio

Spectral duality for finitely generated nilpotent minimum algebras, with applications

We establish a categorical duality for the finitely generated Lindenbaum-Tarski algebras of prepositional nilpotent minimum logic. The latter's conjunction is semantically interpreted by a left-continuous (but not continuous) triangular norm; implication is obtained through residuation. Our duality allows one to transfer to nilpotent minimum, logic several known results about inutitionistic logic with the prelinearity axiom (also called Gödel-Dummett logic), mutatis mutandis. We give several such applications.Fil: Aguzzoli, Stefano. Università degli Studi di Milano; ItaliaFil: Busaniche, Manuela. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Matemática Aplicada del Litoral. Universidad Nacional del Litoral. Instituto de Matemática Aplicada del Litoral; ArgentinaFil: Marra, Vincenzo. Università degli Studi di Milano; Itali

DUALITIES AND REPRESENTATIONS FOR MANY-VALUED LOGICS IN THE HIERARCHY OF WEAK NILPOTENT MINIMUM.

In this thesis we study particular subclasses of WNM algebras. The variety of WNM algebras forms the algebraic semantics of the WNM logic, a propositional many-valued logic that generalizes some well-known case in the setting of triangular norms logics. WNM logic lies in the hierarchy of schematic extensions of MTL, which is proven to be the logic of all left-continuous triangular norms and their residua. In this work, I have extensively studied two extensions of WNM logic, namely RDP logic and NMG logic, from the point of view of algebraic and categorical logic. We develop spectral dualities between the varieties of algebras corresponding to RDP logic and NMG logic, and suitable defined combinatorial categories. Categorical dualities allow to give algorithmic construction of products in the dual categories obtaining computable descriptions of coproducts (which are notoriously hard to compute working only in the algebraic side) for the corresponding finite algebras. As a byproduct, representation theorems for finite algebras and free finitely generated algebras in the considered varieties are obtained. This latter characterization is especially useful to provide explicit construction of a number of objects relevant from the point of view of the logical interpretation of the varieties of algebras: normal forms, strongest deductive interpolants and most general unifiers