on 25-07-2014
Benjamin Bedregal, UFRN, Brasil
July 25, 2014, Friday, 14h.
Abstract: There exist infinitely many ways to extend the classical propositional connectives to the set [0,1], preserving their behaviors in the extremes 0 and 1 exactly as in the classical logic. However, it is a consensus that this issue is not sufficient, and, therefore, these extensions must also preserve some minimal logical properties of the classical connectives. The notions of t-norms (conjunction), t-conorms (disjunction), fuzzy negations and fuzzy implications taking these considerations into account. In previous works, the author, joint with other colleagues, generalizes these notions to the set 𝕌={[a,b]|0≤a≤b≤1} , providing canonical constructions to obtain, for example, interval-valued t-norms that are the best interval representations of t-norms. In this talk, we will make a revision of fuzzy negations, interval-vaçlued fuzzy negations and provide generalizations, in a natural way, of several notions related with fuzzy negations, such as the ones of equilibrium point and negation-preserving automorphism. We show that the main properties of these notions are preserved in those generalizations.
Interval Mathematics. Its applications in Fuzzy Logic and Its Limitations.
Regivan Santiago, UFRN, Brasil
July 25, 2014, Friday, 15h15.
Abstract: During the 50's, Moore and Sunaga proposed an arithmetic for closed intervals, [a,b]={x∈R:a≤x≤b}, in order to provide an object able to capture the numerical errors during computations. The idea is that if we want to compute a value, f(x), for a function f:ℝ→ℝ, the user provide an interval [a,b] such that x∈[a,b] and the composition of interval arithmetical operations, a function F:Iℝ→Iℝ, provides the value F([a,b])=[c,d], such that f(x)∈F([a,b]). This property is known as interval correctness. In 2006, Santiago and Bedregal investigated it from a topological viewpoint. The result was a method called interval representation, which is a way to define interval functions from real functions. This method has been efficient in applications and in the definition of Fuzzy connectives. The concept of interval and its arithmetic is the basis for the most important definition in Fuzzy theory, however some problems arise because the algebraic structure of intervals. In this talk we discuss some connections between Interval Mathematics and Fuzzy Logic.
In 1979 Richard Statman proved, using proof-theory, that the purely implicational fragment of Intuitionistic Logic is PSPACE-complete. He showed a polynomially bounded translation from full Intuitionistic Propositional Logic into its implicational fragment. By the PSPACE-completeness of S4, proved by Ladner, and the Godel translation from S4 into Intuitionistic Logic, the PSPACE-completeness of purely implicational fragment of Intuitionistic Logic is drawn. The sub-formula principle for a deductive system for a logic L states that whenever {γ1,…,γk}⊢Lα there is a proof in which each formula occurrence is either a sub-formula of α or of some of γi. In this work we extend Statman's result and show that any propositional (possibly modal) structural logic satisfying a particular statement of the sub-formula principle is PSPACE-complete. As a consequence, EXPTIME-complete propositional logics, such as PDL and the common-knowledge epistemic logic with at least 2 agents satisfy this particular sub-formula principle, if and only if, PSPACE=EXPTIME.
Room: 4.35, Mathematics
Support: SQIG/Instituto de Telecomunicações with support from FCT and FEDER namely by the FCT project PEst-OE/EEI/LA0008/2013
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