M. Navara

We characterize T-measures on weakly generated tribes, where T is a strict tri- angular norm and we give a Liapunoff Theorem for these measures. This gen- eralizes previous results obtained for monotonic T-measures or for Frank tri- angular norms.

We characterize T-measures on weakly generated tribes, where T is a strict tri- angular norm and we give a Liapunoff Theorem for these measures. This gen- eralizes previous results obtained for monotonic T-measures or for Frank tri- angular norms.

ABSTRACT As a natural generalization of a measure space, Butnariu and Klement introduced T-tribes of fuzzy sets with T-measures. They made the first steps towards a characterization of monotonic real-valued T-measures for a Frank triangular norm T. Later on, Mesiar and the authors of this paper found independently two generalizations, one for vector-valued T-measures with respect to Frank t-norms (in particular for nonmonotonic ones) [3], the other for monotonic real-valued T-measures with respect to general strict t-norms [15]. Here we present a common generalization – a characterization of nonmonotonic T-measures with respect to an arbitrary strict t-norm. Moreover, we prove this for vector-valued T-measures. Using this characterization, we generalize Ljapunov Theorem to this context.

... Note that Mamdani's approach is not fully coherent with the paradigm of approximate reasoning. ... to rather restrictive conditions, especially for the standard choice of the t-norm and the shapes of the membership functions of the ... the conditions Int1)–Int3) for a fuzzy interpolation. ...

ABSTRACT Zadeh's Fuzzy Logic was introduced as a mathematical tool for deduction working with different degrees of truth, instead of the two truth degrees of Classical Logic. The next step taken by Pavelka was to admit assumptions and theorems which are also satisfied only to some truth degree, as well as deduction rules transferring these degrees to the deduced formulas. In the semantics these possibly, but not necessarily, correspond to the degrees of satisfaction of formulas. A significant consequent progress towards this direction was achieved by Hajek's work on Rational Pavelka Logic; it forms formulas with graded truth as couples of a formula and a rational truth value. The success of the Rational Pavelka Logic follows from the properties of the standard MV-algebra, i.e. the real unit interval with the Łukasiewicz operations, which was used as its semantical interpretation. The main mathematical result is that the Rational Pavelka Logic is sound and complete in the sense that the syntactical provability degree and the semantical degree of satisfaction coincide. In this paper an alternative to Pavelka's approach is investigated. Instead of the standard MV-algebra, the truth values are taken from a perfect MV-algebra, in particular, Chang's MV-algebra. This is the opposite extreme among MV-algebras, it admits only infinitesimals besides the two classical truth degrees. We argue that such an example also has a motivation in deduction imitating human reasoning. As Chang's MV-algebra is not complete, the provability degree need not exist. Nevertheless, as the main result the following weak completeness theorem is proved: If the provability degree of a formula exists, it coincides with the degree of satisfaction. This result is obtained by adding to Pavelka's original approach a new rule of inference, Hájek's book-keeping axioms and an axiom valid in Chang's MV-algebra. An example that clarifies the differences between Classical Logic, Rational Pavelka Logic and Perfect Pavelka Logic is given, and some new fuzzy rules of inference are introduced.

ABSTRACT As a natural generalization of a measure space, Butnariu and Klement introduced T-tribes of fuzzy sets with T-measures. They made the first steps towards a characterization of monotonic real-valued T-measures for a Frank triangular norm T. Later on, Mesiar and the authors of this paper found independently two generalizations, one for vector-valued T-measures with respect to Frank t-norms (in particular for nonmonotonic ones) [3], the other for monotonic real-valued T-measures with respect to general strict t-norms [15]. Here we present a common generalization – a characterization of nonmonotonic T-measures with respect to an arbitrary strict t-norm. Moreover, we prove this for vector-valued T-measures. Using this characterization, we generalize Ljapunov Theorem to this context.

This paper surveys the present state of knowledge on propositional fuzzy logics extending SBL with an additional involutive negation. The involutive negation is added as a new propositional connective in order to improve the expressive power of the standard mathematical fuzzy logics based on continuous triangular norms.

... algorithm for a conditionally independent statistical model and two hidden states (a contribution to MI Schlesinger's proof) Vojtech Franc, Václav Hlavác, Mirko Navara {xfrancv,hlavac,navara}@cmp.felk.cvut.cz CTU–CMP–2002–10 October 3, 2002 ...