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 Resolution.
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References
Dantchev, S.: Rank complexity gap for Lovász-Schrijver and Sherali-Adams proof systems. In: Proceedings of the 39th Annual ACM Symposium on Theory of Computing, San Diego, California, USA, June 11-13, pp. 311–317. ACM, New York (2007)
Dantchev, S., Riis, S.: On relativisation and complexity gap for resolution-based proof systems. In: Baaz, M., Makowsky, J.A. (eds.) CSL 2003. LNCS, vol. 2803, pp. 142–154. Springer, Heidelberg (2003)
Krajíĉek, J.: Bounded Arithmetic, Propositional Logic, and Complexity Theory. Cambridge University Press, Cambridge (1995)
Krajíĉek, J.: On the weak pigeonhole principle. Fundamenta Mathematica 170, 123–140 (2001)
Pudlák, P.: Proofs as games. American Mathematical Monthly (June-July 2000)
Riis, S.: A complexity gap for tree-resolution. Computational Complexity 10 (2001)
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Dantchev, S., Martin, B. (2010). The Limits of Tractability in Resolution-Based Propositional Proof Systems. In: Ferreira, F., Löwe, B., Mayordomo, E., Mendes Gomes, L. (eds) Programs, Proofs, Processes. CiE 2010. Lecture Notes in Computer Science, vol 6158. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13962-8_11
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DOI: https://doi.org/10.1007/978-3-642-13962-8_11
Publisher Name: Springer, Berlin, Heidelberg
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