Hierarchical Unambiguity

We develop techniques to investigate relativized hierarchical unambiguous computation. We apply our techniques to generalize known constructs involving relativized unambiguity based complexity classes (UP and Promise-UP) to new constructs involving arbitrary higher levels of the relativized unambiguous polynomial hierarchy (UPH). Our techniques are developed on constraints imposed by hierarchical arrangement of unambiguous nondeterministic polynomial-time Turing machines, and so they differ substantially, in applicability and in nature, from standard methods (such as the switching lemma [J. Håstad, Computational Limitations of Small-Depth Circuits, MIT Press, Cambridge, 1987]), which play roles in carrying out similar generalizations. Aside from achieving these generalizations, we resolve a question posed by Cai, Hemachandra, and Vyskoč in [Complexity Theory, Cambridge University Press, Cambridge, UK, 1993, pp. 101–146], on an issue related to nonadaptive Turing access to UP and adaptive smart Turing access to Promise-UP.