Hierarchical Unambiguity

TL;DR

This paper develops new techniques to analyze relativized hierarchical unambiguous computation, generalizing known classes to higher levels of the unambiguous polynomial hierarchy and resolving a related open question.

Contribution

It introduces novel methods based on hierarchical constraints to extend unambiguous complexity classes and addresses an open problem on Turing access to UP classes.

Findings

Generalized relativized unambiguous classes to higher levels

Developed techniques distinct from standard methods like the switching lemma

Resolved an open question on Turing access to UP classes

Abstract

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 \mathcal{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 [Hastad, Computational Limitations of Small-Depth Circuits, MIT Press, 1987]), which play roles in carrying out similar generalizations. Aside from achieving these generalizations, we resolve a question posed by Cai, Hemachandra, and Vyskoc [J. Cai, L. Hemachandra, and J. Vyskoc, Promises and fault-tolerant database access, In K. Ambos-Spies, S. Homer, and U. Schoening, editors, Complexity Theory, pages 101-146. Cambridge University Press, 1993] on an issue related to nonadaptive Turing access to UP and adaptive smart Turing access to \mathcal{UP}.