A modal account of similarity-based reasoning

One of the goals of a variety of approximate reasoning models is to cope with inference patterns more flexible than those of classical reasoning. Among them, similarity-based reasoning aims at modeling notions of resemblance or proximity among propositions and consequence relations which make sense in such a setting. One way of proceeding is to. equip the set of interpretations or possible worlds with a similarity relation S, that is, a reflexive, symmetric, and t-norm-transitive fuzzy relation. We explore a modal approach to similarity-based reasoning, and we define three multimodal systems with similarity-based Kripke model semantics. A similarity-based Kripke model is a structure , in which W is the set of possible worlds, [[ [[ represents an assignment of possible worlds to atomic formulas, and S is a similarity function S: W × W ~ G, where G is a subset of the unit interval [0, 1] such that O, 1 e G. We provide soundness and completeness results for these systems with respect to some classes of the above structures. © 1997 Elsevier Science Inc. 1. I N T R O D U C T I O N One of the goals of a variety of approximate reasoning models is to cope with inference patterns more flexible than those of classical reasoning. For Address correspondence to Institut d'Investigaci6 en Intel . lig~ncia Artificial (IIIA-CSIC), Campus Universitat Aut6noma de Barcelona, 08193 Cerdanyola, Catalonia, Span. E-mail: {est eva,pere,godo~icardo}@iiia, csic. es. Received May 1 1996; accepted October 1 1996. International Journal of Approximate Reasoning 1997; 16:235-260 © 1997 Elsevier Science Inc. 0888-613X/97/$17.00 655 Avenue of the Americas, New York, NY 10010 PII S0888-613X(96)00126-0 236 Francesc Esteva et al. instance, a pattern like this one, if A approximately entails B, and we observe A', then it is plausible, to some extent, to conclude B whenever A' is close enough to A, would be a generalization of the well-known modus ponens rule. This kind of patterns have been the focus of a huge amount of research in the field of fuzzy logic, where, in general, the statement "A approximately entails B" has been modeled as a fuzzy rule whereas A, B, and A' are modeled as fuzzy facts (see for instance [18]). However, terms like "approximately" or "close," although fuzzy, denote notions of resemblance or proximity among propositions which need not be fuzzy. One way of proceeding is to equip the set of interpretations or possible worlds, f~, with a similarity relation S, that is, a reflexive, symmetric, and t-norm-transitive fuzzy relation [17]. This kind of approach was started by Ruspini [16] by proposing a similarity-based semantics for fuzzy logic, trying to capture inference patterns like the so-called generalized modus ponens. Given a similarity relation S on the set 12 of interpretations of a boolean language L, Ruspini proposes two measures, the implication and consistency measures, to account for the degree with which a proposition B is an approximate consequence from, or is consistent with, another proposition A, respectively. Namely, Is(BIA) = inf sup S(Wl,W2), Wll~a w21~B Cs(BIA) = sup sup S(W1,W2). wll~a w2~B Based on such measures, Ruspini provides then a formalization of the generalized modus ponens in fuzzy logic. This framework has been recently extended in [8] and [6] and compared with the possibilistic approach in [5] and [7]. See also [12] for another approach to similarity-based reasoning. From a logical point of view, several formalisms can be envisaged to capture a notion of similarity-based reasoning system. In [9], Esteva, Garcia, and Godo describe some of the work done in this direction and point out some open problems. The basic reference is [3], where the authors consider different types of graded consequence relations which make sense in similarity-based reasoning. They also turn their attention to a nonstandard fuzzy logic approach, and the frameworks of sphere semantics and multimodal logics are examined at the semantic level, as Farifias and Herzig did for possibility theory [10]. Links among all approaches were also provided. Similarity-Based Reasoning 2.37 Usually, the notion of a similarity relation, relative to a t-norm ®, on a set W is defined [17] as a function S : W × W ~ [0, 1], verifying: 1. reflexivity 1 S(w,w' ) = 1 iff w = w'; 2. symmetry S(w, w') = S(w', w); 3. ®-transitivity S(w,w") > S(w,w') ® S(w',w"). In this paper we will consider a slightly more general notion of similarity relation. We shall consider a (G, ® )-similarity relation on a set W as a function