Machines that Can Output Empty Words

Glaßer, Christian; Travers, Stephen

doi:10.1007/s00224-007-9087-5

Cited by 2 publications

(2 citation statements)

References 19 publications

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“…For instance, for all sets A, B ∈ 1NP, it holds that A∪ B = 0 A ∪ 1B ∈ 1NP, implying that 1NP is closed under∪. Contrary to that, there exists an oracle relative to which 1NP∨ 1NP = 1NP [13].…”

Section: The Symmetric Difference Of Sets a And B Is Defined Asmentioning

confidence: 76%

The complexity of unions of disjoint sets

Glaßer

Selman

Travers

et al. 2008

Journal of Computer and System Sciences

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This paper is motivated by the open question whether the union of two disjoint NPcomplete sets always is NP-complete. We discover that such unions retain much of the complexity of their single components. More precisely, they are complete with respect to more general reducibilities. Moreover, we approach the main question in a more general way: We analyze the scope of the complexity of unions of m-equivalent disjoint sets. Under the hypothesis that NE = coNE, we construct degrees in NP where our main question has a positive answer, i.e., these degrees are closed under unions of disjoint sets.

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Section: The Symmetric Difference Of Sets a And B Is Defined Asmentioning

confidence: 76%

The complexity of unions of disjoint sets

Glaßer

Selman

Travers

et al. 2008

Journal of Computer and System Sciences

Self Cite

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“…It is known that AUPH ⊆ UPH ⊆ U PH ⊆ UAP [LR94,CGRS04], where UAP (unambiguous alternating polynomial-time) is the analog of UP for alternating polynomialtime Turing machines. These hierarchies received renewed interests in some recent papers (see, for instance, [ACRW04,CGRS04, ST,GT05]). Spakowski and Tripathi [ST], developing on circuit complexity-theoretic proof techniques of Sheu and Long [SL96], and of Ko [Ko89], obtained results on the relativized structure of these hierarchies.…”

Section: Introductionmentioning

confidence: 99%

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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Machines that Can Output Empty Words

Cited by 2 publications

References 19 publications

The complexity of unions of disjoint sets

The complexity of unions of disjoint sets

Hierarchical Unambiguity

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