Disjoint NP-Pairs

Glaßer, Christian; Selman, Alan L.; Sengupta, Samik; Zhang, Liyu

doi:10.1137/s0097539703425848

Cited by 34 publications

(8 citation statements)

References 17 publications

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“…If P is a length-optimal proof system, then its canonical pair is a complete disjoint NP pair with respect to polynomial reductions (i.e., every disjoint NP pair is reducible to it). 16 Therefore the following conjecture is a strengthening of Conjecture CON N .…”

Section: Disjoint Np Pairsmentioning

confidence: 90%

“…Glaßer et al [16] constructed an oracle relative to which there is no complete disjoint NP-pair. Other than that, we have little supporting evidence.…”

Section: Disjoint Np Pairsmentioning

confidence: 99%

“…A combinatorial characterization of the canonical pair has only been found for the resolution proof system. In [16] they also constructed an oracle relative to which there exists a complete disjoint NP-pair, but no length-optimal proof system exists, i.e., Conjecture DisjNP fails, but Conjecture CON N holds true.…”

Section: Disjoint Np Pairsmentioning

confidence: 99%

“…Construct oracles that show that relativized conjectures are different or show that they are equivalent for pairs of conjectures presented in this article. Apparently the only separation that is known is a separation of Conjectures CON and DisjNP, see [16]. 4.…”

Section: Proof Let Sat(x Y) Be a δ Bmentioning

confidence: 99%

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Incompleteness in the Finite Domain

Pudlák¹

2017

Bull. symb. log

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Motivated by the problem of finding finite versions of classical incompleteness theorems, we present some conjectures that go beyond NP = coNP. These conjectures formally connect computational complexity with the difficulty of proving some sentences, which means that high computational complexity of a problem associated with a sentence implies that the sentence is not provable in a weak theory, or requires a long proof. Another reason for putting forward these conjectures is that some results in proof complexity seem to be special cases of such general statements and we want to formalize and fully understand these statements. In this paper we review some conjectures that we have presented earlier [26,32,33,34], introduce new conjectures, systematize them and prove new connections between them and some other statements studied before.

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Section: Disjoint Np Pairsmentioning

confidence: 90%

“…Glaßer et al [16] constructed an oracle relative to which there is no complete disjoint NP-pair. Other than that, we have little supporting evidence.…”

Section: Disjoint Np Pairsmentioning

confidence: 99%

Section: Disjoint Np Pairsmentioning

confidence: 99%

Section: Proof Let Sat(x Y) Be a δ Bmentioning

confidence: 99%

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Incompleteness in the Finite Domain

Pudlák¹

2017

Bull. symb. log

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“…Pudlák [Pud17] states as open problem to construct oracles that show that the relativized conjectures are different or show that they are equivalent. Such oracles have been constructed by Verbitskii [Ver91], Glaßer et al [Gla+04], Khaniki [Kha19], Dose [Dos20c;Dos20a;Dos20b], and Dose and Glaßer [DG20]. The restriction to relativizable proofs arises from the following idea: We consider the mentioned hypotheses as conjectures, hence we expect that they are equivalent.…”

Section: Introductionmentioning

confidence: 99%

Oracle with $\mathrm{P=NP\cap coNP}$, but no Many-One Completeness in UP, DisjNP, and DisjCoNP

Ehrmanntraut¹,

Egidy²,

Glaßer³

2022

Preprint

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We construct an oracle relative to which P = NP ∩ coNP, but there are no many-one complete sets in UP, no many-one complete disjoint NP-pairs, and no many-one complete disjoint coNP-pairs. This contributes to a research program initiated by Pudlák [Pud17], which studies incompleteness in the finite domain and which mentions the construction of such oracles as open problem. The oracle shows that NP ∩ coNP is indispensable in the list of hypotheses studied by Pudlák. Hence one should consider stronger hypotheses, in order to find a universal one.

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Do there exist complete sets for promise classes?

Beyersdorff

Sadowski

2011

Mathematical Logic Quarterly

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In this paper we investigate the following two questions: Q1: Do there exist optimal proof systems for a given language ? Q2: Do there exist complete problems for a given promise class C?For concrete languages (such as TAUT or SAT) and concrete promise classes C (such as NP ∩ coNP, UP, BPP, disjoint NP-pairs etc.), these questions have been intensively studied during the last years, and a number of characterizations have been obtained. Here we provide new characterizations for Q1 and Q2 that apply to almost all promise classes C and languages , thus creating a unifying framework for the study of these practically relevant questions.While questions Q1 and Q2 are left open by our results, we show that they receive affirmative answers when a small amount of advice is available in the underlying machine model. For promise classes with promise condition in coNP, the advice can replaced by a tally NP-oracle.

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Disjoint NP-Pairs

Cited by 34 publications

References 17 publications

Incompleteness in the Finite Domain

Incompleteness in the Finite Domain

Oracle with $\mathrm{P=NP\cap coNP}$, but no Many-One Completeness in UP, DisjNP, and DisjCoNP

Do there exist complete sets for promise classes?

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