Theory of Computing Systems

, Volume 41, Issue 2, pp 291–326

On the Power of Unambiguity in Alternating Machines

Article

Abstract

Unambiguity in alternating Turing machines has received considerable attention in the context of analyzing globally unique games by Aida et al. [ACRW] and in the design of efficient protocols involving globally unique games by Crasmaru et al. [CGRS]. This paper explores the power of unambiguity in alternating Turing machines in the following settings: 1. We show that unambiguity-based hierarchies-AUPH, UPH, and UPH-are infinite in some relativized world. For each \(k \geq 2\), we construct another relativized world where the unambiguity-based hierarchies collapse so that they have exactly k distinct levels and their k-th levels coincide with PSPACE. These results shed light on the relativized power of the unambiguity-based hierarchies, and parallel the results known for the case of the polynomial hierarchy. 2. For every \(k \ge 1\), we define the bounded-level unambiguous alternating solution class UAS(k) as the class of all sets L for which there exists a polynomial-time alternating Turing machine N, which need not be unambiguous on every input, with at most k alternations such that \(x\in L\) if and only if x is accepted unambiguously by N. We construct a relativized world where, for all \(k \geq 1, {\rm UP}_{\leq k} \subset {\rm UP}_{\leq k+1}\) and \({\rm UAS}(k) \subset {\rm UAS}(k+1)\). 3. Finally, we show that robustly k-level unambiguous alternating polynomial-time Turing machines, i.e., polynomial-time alternating Turing machines that for every oracle have k alternating levels and are unambiguous, accept languages that are computable in \({\rm P}^{\Sigma^{p}_{k} \oplus {\cal A}}\), for every oracle A. This generalizes a result of Hartmanis and Hemachandra [HH].

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Copyright information

© Springer 2007

Authors and Affiliations

  1. 1.Institut fur Informatik, Heinrich-Heine-Universitat Dusseldorf40225 DusseldorfGermany
  2. 2.Department of Computer Science and Engineering, University of South FloridaTampa, FL 33620USA

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