FINITE DERIVATIVE CLOSURE OF PARTIAL FUNCTIONS AND A METHOD OF INFERENCE

Abstract

We formulate a concept called a finite derivative closure of partial functions which is an abstraction of recursive structures in finite tree automata (transducers). Then we apply this concept to inference of functions from partial functions. 1. Preliminary Remarks and Notations We can say that a partial function gives informations of a function partially. If a function co is decided by a recursive structure S and if a partial function co which is a restriction of cp contains the same structure S, then we can obtain co by naturally extending CO using the structure S. So we are interested in the problem to extract recursive structures from partial functions. In this paper, we formulate a finite derivative closure as such a recursive structure. Under some restrictions, we discuss the methods for extracting such structures from given partial functions with finite domains. We now give some notations needed in this paper. Let A and B be sets and cc be a partial function from A to B. D(cc) denotes the domain of co, that is, D(cp)= { x E A ; (3 y e: B)[cp(x)= y1 } . For a subset X of A, we set co(X )= { y E A ; (2 x E Xf D(co)) [S0(x)=y]}. For an element y of B and a subset Y of B, we set c9-1(Y)={x A; co(x)=y} and co1(Y)=U5EYCp-1(y). cc is said to be one-to-one (or 1-1) if co(x)=co(y) implies x=y for each x, y in D(cp). co is said to be onto if cc(A)=B. Let co and 0 be partial functions. If D(co) D(cb) and cb(x) is equal to co(x) for each x in D(cp), then we say that co is a restriction of 0 or 0 is an extension of co and we denote the situation by cc 0. For a nonempty set W and a nonnegative integer in, 17m(W) denotes the family of all partial functions from W to Wm. It should be noted that, in case of m=0, W° is a set containing only one element. Hence a partial function from W to W° is identified with a subset of W. For an f in 77m(W), in is denoted by d(f) and called the degree of f. We set 77(W)=U7°77m(W). For an f in 17(W), the notation 1=(f„ ••• , fm) means that (1) d(f)=7,1, (2) for each i=1, • • • , m, f z is in 171(W) and D(fz)=D(f ), and * Faculty of Economics , Niigata University, Igarashi, Niigata, Japan.