Characterizing X-separability and one-side invertibility in lambda - beta - Omega -calculus

Given a finite set T identical to (T/sub 1/, . . . ,T/sub t/) of terms of the lambda - beta -K-calculus and a set X/sub T/ identical to (x/sub 1/, . . ., x/sub n/) of free variables (occurring in the elements of T), X/sub T/-separability is the problem of deciding whether there exists a simultaneous substitution for the elements of X/sub T/ transforming T into the set Z identical to (Z/sub 1/, . . . Z/sub t/) of arbitrary terms. The X/sub T/-separability problem is proved to be solvable for any approximation T/sup Hash / of the set T by terms in lambda - beta - Omega -normal form. Since the characterization is constructive, if the terms T/sup , Hash //sub i/ identical to lambda x/sub 1/ . . . x/sub n/. T/sup Hash //sub i/ (i=1, . . ., t) are closed then the sequence T/sup Hash //sub 1/, . . ., T/sup Hash //sub t/ induces a family of mappings (from n to t dimensions) whose surjectivity and right-invertibility becomes decidable. The left-invertibility of this family is proved to be decidable too.<>