Complexity classes without machines: On complete languages for UP

Abstract

This paper develops techniques for studying complexity classes that are not covered by known recursive enumerations of their machines. Counting classes, probabilistic classes, and intersection classes often lack such enumerations. Concentrating on the counting class UP, we show that there are relativizations for which UP A has no complete languages and other relativizations for which P B ≠ UP B ≠ NP B and UP B has complete languages. Among other results we show that 1. (1) UP has complete languages if and only if there exists a set R in P of Boolean formulas, each having at most one satisfying assignment so that SAT∩ R is complete for UP. 2. (2) P ≠ UP if and only if there exists a set S in P of Boolean formulas, each having at most one satisfying assignment, such that S ∩ SAT is not in P. 3. (3) P ≠ UP ∩ coUP if and only if there exists a set S in P of uniquely satisfiable Boolean formulas such that no polynomial-time machine can compute the solutions for the formulas in S. We suggest the wide applicability of our techniques to counting and probabilistic classes by using them to examine the probabilistic class BPP. There is a relativized word where BPP A has no complete languages. If BPP has complete languages, then it has a complete language of the form B ∩ Majority, where B ϵ P and Majority = {f¦f is true for at least half of all assignments} is the canonical PP-complete set.