Modal Logics for Local and Global Similarity Relations
Ana Deneva, Dimiter Vakarelov · Fundamenta Informaticae · 1997
In the everyday language we very often say that some object x is similar to some other object y. If we want some real AI systems to deal with this notion we have to make it exact. The aim of this paper is to analyze several kinds of similarity relations between objects and to introduce modal logics for reasoning about similarity. The paper is a continuation of [3], where a characterization of the so called there positive and negative similarity relations is given. Very often, when we say that two objects are similar, we mean that they have a common property. For instance, we say that the son is similar to his father, because both of them have blue eyes. In [3] this kind of similarity is called positive similarity but in the present paper we will use the name of “local positive similarity”, because we want to introduce also the notion of global similarity. In the above sense two objects x and y are in the relation of local positive similarity if there exists a property A from a given set of properties Pr, such that A is possessed both by x and y . But we can say also that the son is similar to his father, because they both are not smokers. We will call this kind of similarity “local negative similarity”. In this sense two objects x and y are in the relation of local negative similarity if there exists a property A from a given set of properties Pr, such that A is possessed neither by x nor by y. The simplest way to formalize these two kinds of similarity relations is to use the notion of Property system, given in [3]. Namely, S = (Ob, Pr, f) is called a property system (P-system) if Ob is a non empty set, whose elements are called objects, Pr is a non empty set, whose elements are called properties, and f is a function, called information function, assigning to each object x a set f(x) ⊆ Pr, called information of x. The elements of f(x) are called properties of x and if A ∈ f(x) , we say that x possesses the property A. Now the formal definitions of the above similarity relations are the following: Local positive similarity — xΣ0y iff (∃A ∈ Pr) (A ∈ f(x) and A ∈ f(y)) iff f(x) ∩ f(y) 6= ∅,