The limits of tractability in Resolution-based propositional proof systems

Dantchev, Stefan; Martin, Barnaby

Abstract

We study classes of propositional contradictions based on the Least Number Principle (LNP) in the refutation system of Resolution and its generalisations with bounded conjunction, Res(k). We prove that any first-order sentence with no finite models that admits a Σ1 interpretation of the LNP, relativised to a set that is quantifier-free definable, generates a sequence of propositional contradictions that have polynomially-sized refutations in the system Res(k), for some k. When one considers the LNP with total order we demonstrate that a Π1 interpretation of this is sufficient to generate such a propositional sequence with polynomially-sized refutations in the system Res(k). On the other hand, we prove that a very simple first-order sentence that admits a Π1 interpretation of the LNP (with partial and not total order) requires exponentially-sized refutations in both Resolution and Res(k), for all constant k.

Citation

Dantchev, S., & Martin, B. (2012). The limits of tractability in Resolution-based propositional proof systems. Annals of Pure and Applied Logic, 163(3), 656-668. https://doi.org/10.1016/j.apal.2011.11.001

• Journal Article Type: Article
• Online Publication Date: Dec 3, 2011
• Publication Date: Jun 1, 2012
• Deposit Date: Dec 19, 2011
• Publicly Available Date: Sep 12, 2017
• Journal: Annals of Pure and Applied Logic
• Print ISSN: 0168-0072
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 163
• Issue: 3
• Pages: 656-668
• DOI: https://doi.org/10.1016/j.apal.2011.11.001
• Public URL: https://durham-repository.worktribe.com/output/1523879