Promise problems and access to unambiguous computation

Abstract

This paper studies the power of three types of access to unambiguous computation: nonadaptive access, fault-tolerant access, and guarded access. (1) Though for NP it is known that nonadaptive access has exponentially terse adaptive simulations, we show that UP has no such relativizable simulations: there are worlds in which (k+1)-truth-table access to UP is not subsumed by k-Turing access to UP. (2) Though fault-tolerant access (i.e., “1-helping” access) to NP is known to be no more powerful than NP itself, we give both structural and relativized evidence that fault tolerant access to UP suffices to recognize even sets beyond UP. Furthermore, we completely characterize, in terms of locally positive reductions, the sets that fault-tolerantly reduce to UP. (3) In guarded access, Grollmann and Selman's natural notion of access to unambiguous computation, a deterministic polynomial-time Turing machine asks questions to a nondeterministic polynomial-time Turing machine in such a way that the nondeterministic machine never accepts ambiguously. In contrast to guarded access, the standard notion of access to unambiguous computation is that of access to a set that is uniformly unambiguous—even for queries that it never will be asked by its questioner, it must be unambiguous. We show that these notions, though the same for nonadaptive reductions, differ for Turing and strong nondeterministic reductions.

The full version of this paper is available as an April 1992 University of Rochester Department of Computer Science technical report, with the same title as this paper. As space requires the omission here of all proofs, and some of the discussion, we urge the interested reader to consult that full technical report version of this paper.

Research supported in part by the National Science Foundation under research grant CCR-9057486 and by a grant from MITL.

Research supported in part by the National Science Foundation under research grant CCR-8957604. Research done in part during a visit to Princeton University supported by DIMACS (Center for Discrete Mathematics and Theoretical Computer Science), a National Science Foundation Science and Technology Center—NSF-STC88-09648.

Research supported in part by the National Science Foundation under grant CCR-8957604 and the Slovak Academy of Sciences under the grant “Complexity of Sequential and Parallel Computations.”