Giangiacomo Gerla

Fuzzy logic, continuity and effectiveness

Abstract. It is shown the complete equivalence between the theory of continuous (enumeration) fuzzy closure operators and the theory of (effective) fuzzy deduction systems in Hilbert style. Moreover, it is proven that any truth-functional semantics whose connectives are interpreted in [0,1] by continuous functions is axiomatizable by a fuzzy deduction system (but not by an effective fuzzy deduction system, in general).

Similarity-based unification

Unification plays a central rule in Logic Programming. We ”soften” the unification process by admitting that two first order expressions can be ”similar” up to a certain degree and not necessarly identical. An extension of the classical unification theory is proposed accordingly. Indeed, in our approach, inspirated by the unification algorithm of Martelli-Montanari, the systems of equations go through a series of ”sound” transformations until a solvable form is found yielding a substitution that is proved to be a most general extended unifier for the given system of equations.

Closure systems and L-subalgebras

Abstract Let X be a set and l a class of L-subsets of X. l is a “closure system” if it is closed with respect to greatest lower bounds. If ƒ is an L-subset, then the g.l.b. \ tf of { g ϵ l|g⩾ƒ } is said to be generated by ƒ . We examine closure systems of L-subalgebras and propose formulas for the L-subalgebra of a required type generated by a given L-subset.

Graded consequence relations and fuzzy closure operator

ABSTRACT In this work the connections between the fuzzy closure operators and the graded consequence relations are examined Namely, as it is well known, in the crisp case there is a complete equivalence between the notion of closure operator and the one of consequence relation. We extend this result by proving that the graded consequence relations are related to a particular class of fuzzy closure operators, namely the class of fuzzy closure operators that can be obtained by a chain of classical closure operators.

Fuzzy subgroups and similarities

Abstract Given a set S, we show that there is a strict relation between the notion of similarity on S and the one of fuzzy subgroup of transformations in S . Such a relation enables us to extablish a connection between fuzzy subgroups and distances.

An Extension Principle for Fuzzy Logics

Let S be a set, P(S) the class of all subsets of S and F(S) the class of all fuzzy subsets of S. In this paper an “extension principle” for closure operators and, in particular, for deduction systems is proposed and examined. Namely we propose a way to extend any closure operator J defined in P(S) into a fuzzy closure operator J* defined in F(S). This enables us to give the notion of canonical extension of a deduction system and to give interesting examples of fuzzy logics. In particular, the canonical extension of the classical propositional calculus is defined and it is showed its connection with possibility and necessity measures. Also, the canonical extension of first order logic enables us to extend some basic notions of programming logic, namely to define the fuzzy Herbrand models of a fuzzy program. Finally, we show that the extension principle enables us to obtain fuzzy logics related to fuzzy subalgebra theory and graded consequence relation theory. Mathematics Subject Classification: 03B52.

Similarity in Logic Programming

By introducing a similarity relation R between constant and predicate symbols in the language of a logic program P, it is possible to perform approximate inferences. Indeed, it allows us to manage alternative instances of entities that can be considered“equal” with a given degree. We analyze the semantics of this approach exploiting an abstract interpretation technique. The abstract domain is obtained by considering suitable equivalence relations associated with the similarity R. The optimality of the abstract semantics is proved and the definition of fuzzy Herbrand model is also introduced.

Extension principles for fuzzy set theory

In several cases we show that it is possible to extend a notion in classical mathematics by identifying each fuzzy subset with the continuous chain of its closed cuts and by applying this notion to these cuts. In particular this idea is applied to extend functions from subsets into subsets (for instance, closure operators) and functions from sets into real numbers (for instance, measures).

Approximate Reasoning Based on Similarity

The connection between similarity logic and the theory of closure operators is examined. Indeed one proves that the consequence relation dened in (14) can be obtained by composing two closure operators and that the resulting operator is still a closure oper- ator. Also, we extend any similarity into a similarity which is compatible with the logical equivalence, and we prove that this gives the same consequence relation. Mathematics Subject Classication: 03B52, 94D05,68T27.

Logics with approximate premises

M. Ying proposed a propositional calculus in which the reasoning may be approximate by allowing the antecedent clause of a rule to match its premise only approximately. The aim of this note is to relate Yings proposal to an extension principle for closure operators proposed by the authors. In this way it is possible to show that, in a sense, Yings apparatus can be reduced to a fuzzy logic as defined by Pavelka [J. Pavelka, “On fuzzy logic I. Many valued rules of inference,” Zeitschrift fur Math. Logik und Grundlagen Math., 25, 45–52 (1979)].