An Overview Of Some Semantic And Syntactic Complexity Classes

Cox, James L.; Pay, Tayfun

doi:10.48550/arxiv.1806.03501

Cited by 3 publications

(4 citation statements)

References 22 publications

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“…On the other hand, this equivalence for LWPP, in terms of descriptive richness, between one target value and a polynomial number of values, may not hold even for quite similar counting-class situations. In particular, in this section we prove that in some relativized worlds, for the analog (which we will call LWPP + ) of LWPP defined in terms of #P rather than GapP functions (following Cox and Pay [5], who recently introduced the #P-based analogue of WPP), having even two target values for acceptance (the class Two-LWPP + ) yields a richer class of languages than having one value. So it is not the case that single targets and lists of targets inherently function identically as to descriptive richness for counting classes.…”

mentioning

confidence: 81%

The Robustness of LWPP and WPP, with an Application to Graph Reconstruction

Hemaspaandra

Spakowski

et al. 2020

comput. complex.

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We show that the counting class LWPP [8] remains unchanged even if one allows a polynomial number of gap values rather than one. On the other hand, we show that it is impossible to improve this from polynomially many gap values to a superpolynomial number of gap values by relativizable proof techniques.The first of these results implies that the Legitimate Deck Problem (from the study of graph reconstruction) is in LWPP (and thus low for PP, i.e., PP Legitimate Deck = PP) if the weakened version of the Reconstruction Conjecture holds in which the number of nonisomorphic preimages is assumed merely to be polynomially bounded. This strengthens the 1992 result of Köbler, Schöning, and Torán [15] that the Legitimate Deck Problem is in LWPP if the Reconstruction Conjecture holds, and provides strengthened evidence that the Legitimate Deck Problem is not NP-hard.We additionally show on the one hand that our main LWPP robustness result also holds for WPP, and also holds even when one allows both the rejection-and acceptance-gap-value targets to simultaneously be polynomial-sized lists; yet on the other hand, we show that for the #P-based analog of LWPP the behavior much differs in that, in some relativized worlds, even two target values already yield a richer class than one value does.

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mentioning

confidence: 81%

The Robustness of LWPP and WPP, with an Application to Graph Reconstruction

Hemaspaandra

Spakowski

et al. 2020

comput. complex.

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“…(It is known that FewP is contained in the class known as SPP and is indeed so-called SPP-low [30,17,18], however that does not make our containments in restricted counting classes uninteresting, as it seems unlikely that SPP is contained in any restricted counting class, since SPP's "no" case involves potentially exponential numbers of accepting paths, not zero such paths.) The interesting, recent paper of Cox and Pay [16] draws on the result of Borchert, Hemaspaandra, and Rothe [7] that appears as our Theorem 4.1 to establish that FewP ⊆ RC {2 t −1 | t∈N + } (note that the right-hand side is the restricted counting class defined by the Mersenne numbers), a result that itself implies FewP ⊆ RC {1,3,5,...} . "RC" (restricted counting) classes [7] are central to this paper.…”

Section: Related Workmentioning

confidence: 99%

“…This led us to revisit the issue of identifying the sets S ⊆ N + that satisfy FewP ⊆ RC S , studied previously by, for example, Borchert, Hemaspaandra, and Rothe [7] and Cox and Pay [16]. In particular, Borchert, Hemaspaandra, and Rothe showed, by the iterative constant-setting technique, the following theorem.…”

Section: Gaps Ambiguity and Iterative Constant-settingmentioning

confidence: 99%

Gaps, Ambiguity, and Establishing Complexity-Class Containments via Iterative Constant-Setting

Hemaspaandra¹,

Juvekar²,

Nadjimzadah³

et al. 2021

Preprint

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Cai and Hemachandra used iterative constant-setting to prove that Few ⊆ ⊕P (and thus that FewP ⊆ ⊕P). In this paper, we note that there is a tension between the nondeterministic ambiguity of the class one is seeking to capture, and the density (or, to be more precise, the needed "nongappy"ness) of the easy-to-find "targets" used in iterative constant-setting. In particular, we show that even less restrictive gap-size upper bounds regarding the targets allow one to capture ambiguity-limited classes. Through a flexible, metatheorem-based approach, we do so for a wide range of classes including the logarithmic-ambiguity version of Valiant's unambiguous nondeterminism class UP. Our work lowers the bar for what advances regarding the existence of infinite, P-printable sets of primes would suffice to show that restricted counting classes based on the primes have the power to accept superconstant-ambiguity analogues of UP. As an application of our work, we prove that the Lenstra-Pomerance-Wagstaff Conjecture implies that all O(log log n)-ambiguity NP sets are in the restricted counting class RC PRIMES .

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“…The analog of WPP defined using #P functions rather than GapP functions already exists in the literature, namely it is the class known as F = P that was recently introduced by Cox and Pay [CP18]. Similarly, we here define, and denote as LWPP + , the analog of LWPP except defined using #P functions rather than GapP functions.…”

Section: Lwpp +mentioning

confidence: 99%

The Robustness of LWPP and WPP, with an Application to Graph Reconstruction

Hemaspaandra

Spakowski

et al. 2017

Preprint

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We show that the counting class LWPP [FFK94] remains unchanged even if one allows a polynomial number of gap values rather than one. On the other hand, we show that it is impossible to improve this from polynomially many gap values to a superpolynomial number of gap values by relativizable proof techniques.The first of these results implies that the Legitimate Deck Problem (from the study of graph reconstruction) is in LWPP (and thus low for PP, i.e., PP Legitimate Deck = PP) if the weakened version of the Reconstruction Conjecture holds in which the number of nonisomorphic preimages is assumed merely to be polynomially bounded. This strengthens the 1992 result of Köbler, Schöning, and Torán [KST92] that the Legitimate Deck Problem is in LWPP if the Reconstruction Conjecture holds, and provides strengthened evidence that the Legitimate Deck Problem is not NP-hard.We additionally show on the one hand that our main LWPP robustness result also holds for WPP, and also holds even when one allows both the rejection-and acceptance-gap-value targets to simultaneously be polynomial-sized lists; yet on the other hand, we show that for the #P-based analog of LWPP the behavior much differs in that, in some relativized worlds, even two target values already yield a richer class than one value does. Despite that nonrobustness result for a #P-based class, we show that the #P-based "exact counting" class C = P remains unchanged even if one allows a polynomial number of target values for the number of accepting paths of the machine.

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An Overview Of Some Semantic And Syntactic Complexity Classes

Cited by 3 publications

References 22 publications

The Robustness of LWPP and WPP, with an Application to Graph Reconstruction

The Robustness of LWPP and WPP, with an Application to Graph Reconstruction

Gaps, Ambiguity, and Establishing Complexity-Class Containments via Iterative Constant-Setting

The Robustness of LWPP and WPP, with an Application to Graph Reconstruction

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