Approximating Infinite Solution Sets by Discretization of the Scales of Truth Degrees

The present paper discusses the problem of approximating possibly infinite sets of solutions by finite sets of solutions via discretization of scales of truth degrees. Infinite sets of solutions we have in mind in this paper typically appear in constraint-based problems such as "find all collections in a given finite universe satisfying constraint C". In crisp setting, i.e. when collections are conceived as crisp sets, the set of all such collections is finite and often computationally tractable. In fuzzy setting, i.e. when collections are conceived as fuzzy sets, the set of all such collections may be infinite and, ipso facto, computationally intractable when one uses the unit interval [0,1] as the scale of membership degrees. A natural solution to this problem is to uses, instead of [0,1], a finite subset K of [0, 1] which approximates [0,1] to a satisfactory degree. This idea is pursued in the present paper. To be sufficiently specific, we illustrate the idea on a particular method, namely, on formal concept analysis. We present several results including estimation of degrees of similarity of the finitary approximation to the possibly infinite original case by means of the degree of approximation of if of [0, 1].