An oracle separating conjectures about incompleteness in the finite domain

Dose, Titus

doi:10.1016/j.tcs.2020.01.003

Cited by 3 publications

(7 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Our oracle supports this estimation, as it rules out relativizable proofs for "CON ⇒ NP ∩ coNP" and "TFNP ⇒ NP ∩ coNP." By Dose [Dos20c;Dos20a], the same holds for the converse implications. Overall, we recognize a strong independence between NP ∩ coNP and all remaining hypotheses: (i) There does not exist a relativizable proof for NP ∩ coNP, even if we simultaneously assume all remaining hypotheses.…”

Section: Introductionmentioning

confidence: 73%

“…(ii) There exists a relativizable proof for the implication NP ∩ coNP ⇒ CON ∨ SAT [KMT03]. But there does not exist a relativizable proof showing that NP ∩ coNP implies one of the remaining hypotheses [Dos20c;Dos20a].…”

Section: Introductionmentioning

confidence: 99%

“…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%

See 2 more Smart Citations

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

Ehrmanntraut¹,

Egidy²,

Glaßer³

2022

Preprint

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionmentioning

confidence: 73%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Ehrmanntraut¹,

Egidy²,

Glaßer³

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…In the specific case of the relation between the existence of a p-optimal propositional proof system and the existence of a complete problem for TFNP, it shows that these statements are independent in the relativized worlds (Corollary 5.3), hence it answers Pudlák and Krajíček's question in [19,25]. After this paper was posted as a preprint [17], Dose in [12], constructed an oracle such that relative to it:…”

Section: And Relative To Wmentioning

confidence: 85%

“…As we mentioned in the Introduction, the result of Dose in [12] and Theorem 5.2 are incomparable. It is proved in [14] that TFNP = FP is equivalent to the statement:…”

Section: Theorem 51 There Exists An Oraclementioning

confidence: 94%

New Relations and Separations of Conjectures About Incompleteness in the Finite Domain

Khaniki¹

2021

J. symb. log.

View full text Add to dashboard Cite

In [20] Krajíček and Pudlák discovered connections between problems in computational complexity and the lengths of first-order proofs of finite consistency statements. Later Pudlák [25] studied more statements that connect provability with computational complexity and conjectured that they are true. All these conjectures are at least as strong as [23–25].One of the problems concerning these conjectures is to find out how tightly they are connected with statements about computational complexity classes. Results of this kind had been proved in [20, 22].In this paper, we generalize and strengthen these results. Another question that we address concerns the dependence between these conjectures. We construct two oracles that enable us to answer questions about relativized separations asked in [19, 25] (i.e., for the pairs of conjectures mentioned in the questions, we construct oracles such that one conjecture from the pair is true in the relativized world and the other is false and vice versa). We also show several new connections between the studied conjectures. In particular, we show that the relation between the finite reflection principle and proof systems for existentially quantified Boolean formulas is similar to the one for finite consistency statements and proof systems for non-quantified propositional tautologies.

show abstract

Further oracles separating conjectures about incompleteness in the finite domain

Dose

2020

Theoretical Computer Science

View full text Add to dashboard Cite

An oracle separating conjectures about incompleteness in the finite domain

Cited by 3 publications

References 11 publications

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

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

New Relations and Separations of Conjectures About Incompleteness in the Finite Domain

Further oracles separating conjectures about incompleteness in the finite domain

Contact Info

Product

Resources

About