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Publications and source records attributed to Titus Dose.
We build on a working program initiated by Pudl\'ak [Pud17] and construct an oracle relative to which each $\mathrm{coNP}$-complete set has $\mathrm{P}$-optimal proof systems and $\mathrm{NP}\cap\mathrm{coNP}$ does not have complete problems.
As one step in a working program initiated by Pudl\'ak [Pud17] we construct an oracle relative to which $\mathrm{P}\ne\mathrm{NP}$ and all non-empty sets in $\mathrm{NP}\cup\mathrm{coNP}$ have $\mathrm{P}$-optimal proof systems.
Pudl\'ak [Pud17] lists several major conjectures from the field of proof complexity and asks for oracles that separate corresponding relativized conjectures. Among these conjectures are: - $\mathsf{DisjNP}$: The class of all disjoint NP-pairs does not have many-one complete elements. - $\mathsf{SAT}$: NP does not contain many-one complete sets that have P-optimal proof systems. - $\mathsf{UP}$: UP does not have many-one complete problems. - $\mathsf{NP}\cap\mathsf{coNP}$: $\text{NP}\cap\text{coNP}$ does not have many-one complete problems. As one answer to this question, we construct an oracle relative to which $\mathsf{DisjNP}$, $\neg \mathsf{SAT}$, $\mathsf{UP}$, and $\mathsf{NP}\cap\mathsf{coNP}$ hold, i.e., there is no relativizable proof for the implication $\mathsf{DisjNP}\wedge \mathsf{UP}\wedge \mathsf{NP}\cap\mathsf{coNP}\Rightarrow\mathsf{SAT}$. In particular, regarding the conjectures by Pudl\'ak this extends a result by Khaniki [Kha19].
All versions of this paper contain errors. Therefore, the existence of an oracle relative to which (i) there exist complete disjoint coNP-pairs and (ii) there exist no complete total polynomial search problems must be considered as an open problem. In the following we refer to the version published in the proceedings of the 22nd International Symposium on Fundamentals of Computation Theory [Dos19] as this is the most recent version that has been published. The error is in the following sentence between the claims 4 and 5: Now let u' [symbol for strict extension] u be the minimal t'-valid oracle defined for all words of length q(n) (such an oracle exists according to Claim 4)." The problem is that here Claim 4 cannot be applied since for all \alpha, the function t' does not equal $t_\alpha$. References [Dos19] Titus Dose. Complete disjoint conp-pairs but no complete total polynomial search problems relative to an oracle. In Leszek Antoni Gasieniec, Jesper Jansson, and Christos Levcopoulos, editors, Fundamentals of Computation Theory - 22nd International Symposium, FCT 2019, Copenhagen, Denmark, August 12-14, 2019, Proceedings, volume 11651 of Lecture Notes in Computer Science, pages 153{167}. Springer, 2019.