Promise problems and access to unambiguous computation

Cai, Jin-Yi; Hemachandra, Lane A.; Vyskoc, Jozef

doi:10.1007/3-540-55808-x_14

Cited by 15 publications

(3 citation statements)

References 22 publications

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“…Subsection 4.2 studies the issue of simulating nonadaptive access to UΣ p h , the h'th level of the unambiguous polynomial hierarchy, by adaptive access to UΣ p h . Theorem 4.7 of this subsection generalizes a result of Cai, Hemachandra, and Vyskoč [CHV92] from the case of h = 1 to the case of any arbitrary h ≥ 1. Lemma 3.1 is used as a key tool for proving Theorem 4.7.…”

Section: Resultssupporting

confidence: 56%

“…However, for the case of unambiguous nondeterministic computation such a relationship between nonadaptive access and adaptive access is not known. Cai, Hemachandra, and Vyskoč [CHV92] showed that even proving the superiority of adaptive Turing access over nonadaptive Turing access with one single query more might be nontrivial for unambiguous nondeterministic computation: Theorem 4.6 ([CHV92]) For any total, polynomial-time computable and polynomially bounded function k(•), there exists an oracle A such that…”

Section: Simulating Nonadaptive Access By Adaptive Access (Non-promis...mentioning

confidence: 99%

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Hierarchical Unambiguity

Spakowski¹,

Tripathi²

2009

SIAM J. Comput.

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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 [Hås87]), which play roles in carrying out similar generalizations.Aside from achieving these generalizations, we resolve a question posed by Cai, Hemachandra, and Vyskoč [CHV93] on an issue related to nonadaptive Turing access to UP and adaptive smart Turing access to Promise-UP. * A preliminary version of this paper was presented at the MFCS '06 conference. † Supported in part by the DFG under grants RO 1202/9-1 and RO 1202/9-3. hierarchy: They proved that there is a relativized world where Σ p 2 = Π p 2 . However, Baker and Selman [BS79] noted that their proof techniques do not apply at higher levels of the polynomial hierarchy because of certain constraints in their counting argument. Thus, it required the development of entirely different proof techniques for separating all the levels of the relativized polynomial hierarchy. The landmark paper by Furst, Saxe, and Sipser [FSS84] established the connection between the relativization of the polynomial hierarchy and lower bounds for small depth circuits computing certain functions. Techniques for proving such lower bounds were developed in a series of papers [FSS84, Sip83,Yao85,Hås87], which were motivated by questions about the relativized structure of the polynomial hierarchy. Yao [Yao85] finally succeeded in separating the levels of the relativized polynomial hierarchy by applying these new techniques. Håstad [Hås87] gave the most refined presentation of these techniques via the switching lemma. Even to date, Håstad's switching lemma [Hås87] is used as an essential tool to separate relativized hierarchies, composed of classes stacked one on top of another. (See, for instance,[Hås87,Ko89,BU98,ST] where the switching lemma is used as a strong tool for proving the feasibility of oracle constructions.)A major contribution of our paper lies in demonstrating that known oracle constructions involving the initial levels of the unambiguous polynomial hierarchy (UPH) and the promise unambiguous polynomial hierarchy (U PH), i.e. UP and P Promise-UP s , respectively, can be extended to oracle constructions involving arbitrary higher levels of UPH by application only of pure counting arguments. In fact, it seems implausible to achieve these extensions by well-known techniques from circuit complexity (e.g., the switching lemma [Hås87] and the polynomial method surveyed in [Bei93,Reg97]).

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Section: Resultssupporting

confidence: 56%

Section: Simulating Nonadaptive Access By Adaptive Access (Non-promis...mentioning

confidence: 99%

Hierarchical Unambiguity

Spakowski¹,

Tripathi²

2009

SIAM J. Comput.

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“…However, the class P USAT s of languages with smart polynomialtime Turing reductions to USAT is apparently larger: it is closed under complements, contains FewP [CHV93], 2 and contains the graph-isomorphism problem:…”

Section: Corollary 1 Uap Is Closed Under All Boolean Operationsmentioning

confidence: 99%

A Protocol for Serializing Unique Strategies

Crasmaru

Glaßer

Regan

et al. 2004

Lecture Notes in Computer Science

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Abstract. We devise an efficient protocol by which a series of twoperson games Gi with unique winning strategies can be combined into a single game G with unique winning strategy, even when the result of G is a non-monotone function of the results of the Gi that is unknown to the players. In computational complexity terms, we show that the class UAP of Niedermeier and Rossmanith [NR98] of languages accepted by unambiguous polynomial-time alternating TMs is self-low, i.e., UAP UAP = UAP. It follows that UAP contains the Graph Isomorphism problem, nominally improving the problem's classification into SPP by Arvind and Kurur [AK02] since UAP is a subclass of SPP [NR98]. We give some other applications, oracle separations, and results on problems related to unique-alternation formulas.

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Hierarchical Unambiguity

Spakowski

Tripathi

2006

Lecture Notes in Computer Science

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Promise problems and access to unambiguous computation

Cited by 15 publications

References 22 publications

Hierarchical Unambiguity

Hierarchical Unambiguity

A Protocol for Serializing Unique Strategies

Hierarchical Unambiguity

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