We construct an oracle relative to which NP = PSPACE, but UP has no many-one complete sets. This combines the properties of an oracle by Hartmanis and Hemachandra [HH88] and one by Ogiwara and Hemachandra [OH93]. The oracle provides new separations of classical conjectures on optimal proof systems and complete sets in promise classes. This answers several questions by Pudl\'ak [Pud17], e.g., the implications UP CONN and SAT TFNP are false relative to our oracle. Moreover, the oracle demonstrates that, in principle, it is possible that TFNP-complete problems exist, while at the same time SAT has no p-optimal proof systems.
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Dingel et al. (2024) studied this question.
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