The authors construct an oracle Φ that satisfies the following properties:
NP^Φ = PSPACE^Φ (property P1). This is achieved by encoding the PSPACE-complete problem QBFΦ into the oracle Φ.
UP^Φ has no many-one complete sets (property P2). This is achieved by carefully constructing the oracle Φ in a way that for each NP^Φ-machine Ni, either:
The authors show that their oracle construction can be performed as stated and that the desired properties P1 and P2 hold relative to the constructed oracle Φ.
The oracle provides new separations of classical conjectures on optimal proof systems and complete sets in promise classes. It answers several questions by Pudlák, e.g., the implications UP ⇒ CONN and SAT ⇒ TFNP are false relative to the 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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