# On the Power of Unambiguity in Alternating Machines

DOI: 10.1007/s00224-007-2014-y

- Cite this article as:
- Spakowski, H. & Tripathi, R. Theory Comput Syst (2007) 41: 291. doi:10.1007/s00224-007-2014-y

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## 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].