First-order SMTL Logic and quasi-witnessed models
Un apropament matemàtic al problema de vaguetat
Algebraic study of axiomatic extensions of triangular norm based fuzzy logics
Generalized continuous and left-continuous t-norms arising from algebraic semantics for fuzzy logics
Publication Type:
Journal ArticleSource:
Information Sciences, Elsevier, Volume 180, Issue 8, p.1354–1372 (2010)Abstract:
This paper focuses on the issue of how generalizations of continuous and left- continuous t-norms over linearly ordered sets should be from a logical point of view. Taking into account recent results in the scope of algebraic semantics for fuzzy logics over chains with a monoidal residuated operation, we advocate linearly ordered BL-algebras and MTL-algebras as adequate generalizations of continuous and left-continuous t-norms respectively. In both cases, the underlying basic structure is that of linearly ordered residuated lattices. Although the residuation property is equivalent to left-continuity in t-norms, continuous t-norms have received much more attention due to their simpler structure. We review their complete description in terms of ordinal sums and discuss the problem of describing the structure of their generalization to BL-chains. In particular we show the good behavior of BL-algebras over a finite or complete chain, and discuss the partial knowledge of rational BL-chains. Then we move to the general non-continuous case corresponding to left-continuous t-norms and MTL-chains. The unsolved problem of describing the structure of left-continuous t-norms is presented together with a fistful of construction-decomposition techniques that apply to some distinguished families of t-norms and, finally, we discuss the situation in the general study of MTL-chains as a natural generalization of left-continuous t-norms.
Secure and Optimal Base Contraction in Graded {\L}ukasiewicz Logics
Publication Type:
Conference PaperSource:
Artificial Intelligence Research and Development. Proceedings of the 12th International Conference of the Catalan Association fo Artificial Intelligence, IOS Press, Volume 202, Cardona, Catalonia, Spain, p.265-274 (2009)ISBN:
978-1-60750-061-2Keywords:
Base Contraction; T-norm fuzzy logic; Partial meetAbstract:
The operation of base contraction was characterized using remainder sets for several logics. The case of {\L}ukasiewicz logics with truth-constants requires to switch from remainders to maximal consistent subsets. We characterize first contraction operators that establish a security-threshold, and use these to define optimal operators, which are provably sound w.r.t. axioms. Finally, these are shown to collapse to the former case for any finite base.
On Triangular Norm based Fuzzy Description Logics
Publication Type:
Conference PaperSource:
IFSA-EUSFLAT, Calouste Gulbekian Foundation, 20-24 July 2009. Lisbon, Portugal, p.891-896 (2009)ISBN:
978-989-95079-6-8Keywords:
Description Logics; Fuzzy Description Logics; t-norm based fuzzy logics; Truth-constants; Involutive negationAbstract:
Description Logics (DLs) are knowledge representation languages useful to represent concepts and roles. Fuzzy Description Logics (FDLs) incorporate both vague concepts and vague roles
modeling them as fuzzy sets and fuzzy relations respectively.
In the present paper, following ideas from H\'ajek, we propose the use of t-norm based (fuzzy) logics with truth constants in the language as logics underlying the fuzzy description language. We introduce the
languages ALC_L*(Al[S]) and ALC_L*_(Al[S]) as an adequate syntactical counterpart of some semantic calculi given in different works dealing with FDLs.
