New Relations and Separations of Conjectures About Incompleteness in the Finite Domain

Abstract

In [20] Krajíček and Pudlák discovered connections between problems in computational complexity and the lengths of first-order proofs of finite consistency statements. Later Pudlák [25] studied more statements that connect provability with computational complexity and conjectured that they are true. All these conjectures are at least as strong as [23–25].One of the problems concerning these conjectures is to find out how tightly they are connected with statements about computational complexity classes. Results of this kind had been proved in [20, 22].In this paper, we generalize and strengthen these results. Another question that we address concerns the dependence between these conjectures. We construct two oracles that enable us to answer questions about relativized separations asked in [19, 25] (i.e., for the pairs of conjectures mentioned in the questions, we construct oracles such that one conjecture from the pair is true in the relativized world and the other is false and vice versa). We also show several new connections between the studied conjectures. In particular, we show that the relation between the finite reflection principle and proof systems for existentially quantified Boolean formulas is similar to the one for finite consistency statements and proof systems for non-quantified propositional tautologies.

P-Optimal Proof Systems for Each NP-Complete Set but no Complete Disjoint NP-Pairs Relative to an Oracle

“…Regarding Pudlák's conjectures, however, our oracle O extends Khaniki's result as relative to O we have the even stronger result that there is no relativizable proof for the implication DisjNP ⇒ SAT. Since due to the oracle V by Khaniki [Kha19] none of the implications DisjCoNP ⇒ DisjNP, TFNP ⇒ DisjNP, and SAT ⇒ DisjNP can be proven relativizably, our oracle shows that DisjNP is independent of each of the conjectures DisjCoNP, TFNP, and SAT in relativized worlds, i.e., none of the six possible implications has a relativizable proof.…”

“…Let us now focus on the properties 1 and 2 of the oracle. Regarding these, our oracle has similar properties as the aforementioned oracle W by Khaniki [Kha19]: both oracles show that there is no relativizable proof for the implication CON ⇒ SAT. Relative to Khaniki's oracle W it even holds that each total polynomial search problem has a polynomial time solution, which implies not only ¬SAT but also that all optimal proof systems for SAT are P-optimal [KM00].…”

“…Pudlák [Pud17] also defines the conjecture RFN 1 and lists it between CON ∨ SAT and P = NP, i.e., CON ∨ SAT ⇒ RFN 1 ⇒ P = NP. Khaniki [Kha19] even shows CON ∨ SAT ⇔ RFN 1 , which is why we omit RFN 1 in the figure. For a definition of RFN 1 we refer to [Pud17].…”

“…Khaniki [Kha19] partially answers this question: besides showing two of the conjectures to be equivalent he presents two oracles V and W showing that SAT and CON (as well as TFNP and CON) are independent in relativized worlds which means that none of the two possible implications between the two conjectures has a relativizable proof. To be more precise, relative to V, there exist P-optimal propositional proof systems but no many-one complete disjoint coNP-pairs, where -as mentioned above-the latter implies TFNP and SAT.…”