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Logic Seminar: Models for the Gödel-Kripke logic based on rough sets

Author: Lona | Image: Lona

Speaker: Laura Gamboa Guzman (Iowa State University)

Abstract: The purpose of this talk is to introduce the basic aspects of the rough set theory, its relation with basic modal logic, and how to obtain a 3-valued semantics for the Gödel-Kripke logic based on the algebra of rough sets. Computer scientist Zdzislaw Pawlak introduced the rough set theory in the 1980s in the context of information tables/systems, which are a way to assign information or attributes to each element from a fixed universe to reason formally about the lower and upper approximations that one can obtain from the information available of crisp sets (i.e., conventional sets). On the other hand, what we refer to as the Gödel-Kripke logic is a modal logic that extends Gödel’s propositional logic, which is an intermediate logic between intuitionistic logic and classical propositional logic, with two modal connectives for necessity and possibility. We took as inspiration the article by X. Caicedo and R.O. Rodriguez, A Gödel Modal Logic, where a fuzzy version of Kripke semantics for modal logic is presented, in which both the accessibility relation and the truth values of propositional letters at each world take values in the standard Gödel algebra [0,1]. An interpretation of the basic modal language is then proposed in terms of rough sets instead of numerical values, taking advantage of the similarity between the corresponding algebraic structures since the classical interpretation of this logic is given in terms of operations in a Heyting algebra. Finally, I will present some results that compare both interpretations for the propositional fragment and a single-modality fragment of this logic. This work was mainly developed as my undergrad thesis and I am still working on some related questions with Ph.D. Maricarmen Martinez from the Universidad de los Andes.