New relations and separations of conjectures about incompleteness in the finite domain

0561027 - MÚ 2023 RIV US eng J - Journal Article
Khaniki, Erfan
New relations and separations of conjectures about incompleteness in the finite domain.
Journal of Symbolic Logic. Roč. 87, č. 3 (2022), s. 912-937. ISSN 0022-4812. E-ISSN 1943-5886
EU Projects: European Commission(XE) 339691 - FEALORA
Institutional support: RVO:67985840
Keywords : disjoint NE-sets * finite consistency * oracles * propositional proof systems
OECD category: Pure mathematics
Impact factor: 0.6, year: 2022
Method of publishing: Limited access
https://doi.org/10.1017/jsl.2021.99

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.
Permanent Link: https://hdl.handle.net/11104/0333783