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 machines. Often, counting classes, probabilistic classes, and intersection classes 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 P ≠ UP if and only if there exists a set S in P of Boolean formulas with at most one satisfying assignment such that S ∩ SAT is not in P. 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. If UP has complete languages then there exists a set R in P of Boolean formulas with at most one satisfying assignment so that SAT ∩ R is complete for UP. Finally, we indicate the wide applicability of our techniques to counting and probabilistic classes by using them to examine the probabilistic class BPP. There is a relativized world 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 ∈ P and MAJORITY = {f | f is true for at least half of all assignments} is the canonical PP-complete set.

This research was supported by NSF Research Grant DCR-8301766. The second author was supported by a Hertz Foundation Fellowship. Part of this research was done during the “Complexity Year” at the Mathematical Sciences Research Institute in Berkeley.