On the Efficient Generation of Language Instances

Abstract

Polynomial-time Turing machines that output instances of a given language are considered, where the instances are required to have a certain length specified by the input. Two types of generating machines are investigated. The first, called a constructor, is deterministic and outputs one string in the language having the specified input length, if such a string exists. A generator is nondeterministic and may output different strings in the language using different computations on the same input; it is required, however, that for any string in the language satisfying the input constraint, there be some computation of the generator on this input that produces the string. Although most P and NP languages examined appear to have such polynomial-time constructors and generators, it is shown that the question of whether all NP languages have such machines is related to other open questions in complexity theory and that even under the assumption that P is not equal to NP, the question cannot be resolved using te...