Promise Problems and Access to Unambiguous Computation

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) 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.

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