Complete Disjoint coNP-Pairs but no Complete Total Polynomial Search Problems Relative to an Oracle

TL;DR

This paper discusses the open problem of the existence of an oracle where complete disjoint coNP-pairs exist but no complete total polynomial search problems, highlighting errors in previous versions and the complexity of resolving this issue.

Contribution

It clarifies the unresolved status of the problem and identifies errors in prior work, emphasizing the open nature of the question.

Findings

The existence of such an oracle remains an open problem.

Previous claims about the existence of complete disjoint coNP-pairs are invalidated.

The paper highlights the complexity of relating disjoint coNP-pairs and total polynomial search problems.

Abstract

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.