Unambiguous Computation: Boolean Hierarchies and Sparse Turing-Complete Sets

This paper studies, for UP, two topics that have been intensely studied for NP: Boolean hierarchies and the consequences of the existence of sparse Turing-complete sets. Unfortunately, as is often the case, the results for NP draw on special properties of NP that do not seem to carry over straightforwardly to UP. For example, it is known for NP (and more generally for any class containing \Sigma^* and \emptyset and closed under union and intersection) that the symmetric difference hierarchy, the Boolean hierarchy, and the Boolean closure all are equal. We prove that closure under union is not needed for this claim: For any class \cal K that contains \Sigma^* and \emptyset and is closed under intersection (e.g., UP, US, and FewP), the symmetric difference hierarchy over \cal K, the Boolean hierarchy over \cal K, and the Boolean closure of \cal K all are equal. On the other hand, we show that two hierarchies---the Hausdorff hierarchy and the nested difference hierarchy---which in the NP case are equal to the Boolean closure fail to be equal for the UP case in some relativized worlds. Regarding sparse Turing-complete sets for UP, we prove that if UP has sparse Turing-complete sets, then the levels of the unambiguous polynomial hierarchy are simpler than one would otherwise expect: they collapse one level in terms of their location in the promise unambiguous polynomial hierarchy. We obtain related results under the weaker assumption that UP has sparse Turing-hard sets.

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