Relativized limitations of left set technique and closure classes of sparse sets

Abstract

A number of theorems are proved by introducing the notion of k-families of sets of strings, and an algorithm which outputs the sets of certain k-families is given. The algorithm is used to disjunctively reduce the left set (or 1wdsr set) to a sparse set. The set output by the algorithm on an input corresponds to the set queried by the disjunctive reduction on the input.<>