Showing papers by "Juris Hartmanis published in 1986"

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 UPA has no complete languages and other relativizations for which PB ≠ UPB ≠ NPB and UPB 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 BPPA 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.

On sparse oracles separating feasible complexity classes

Abstract: This note clarifies which oracles separate NP from P and which do not. In essence, we are changing our research paradigm from the study of which problems can be relativized in two conflicting ways to the study and characterization of the class of oracles achieving a specified relativization. Results of this type have the potential to yield deeper insights into the nature of relativization problems and focus our attention on new and interesting classes of languages.