New relations and separations of conjectures about incompleteness in the fnite domain
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Our main results are in the following three sections: 1. We prove new relations between proof complexity conjectures that are discussed in pu18. 2. We investigate the existence of p-optimal proof systems for TAUT, assuming the collapse of C and N C (the nondeterministic version of C) for some new classes C and also prove new conditional independence results for strong theories, assuming nonexistence of p-optimal proof systems. 3. We construct two new oracles V and W. These two oracles imply several new separations of proof complexity conjectures in relativized worlds. Among them, we prove that existence of a p-optimal proof system for TAUT and existence of a complete problem for TFNP are independent of each other in relativized worlds which was not known before.
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