Abstract
Jeřábek showed in 2008 that cuts in propositional-logic deep-inference proofs can be eliminated in quasipolynomial time. The proof is an indirect one relying on a result of Atserias, Galesi and Pudlák about monotone sequent calculus and a correspondence between this system and cut-free deep-inference proofs. In this paper we give a direct proof of Jeřábek’s result: we give a quasipolynomial-time cut-elimination procedure in propositional-logic deep inference. The main new ingredient is the use of a computational trace of deep-inference proofs called atomic flows, which are both very simple (they trace only structural rules and forget logical rules) and strong enough to faithfully represent the cut-elimination procedure.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Atserias, A., Galesi, N., Pudlák, P.: Monotone simulations of non-monotone proofs. Journal of Computer and System Sciences 65(4), 626–638 (2002)
Brünnler, K.: Deep Inference and Symmetry in Classical Proofs. Logos Verlag, Berlin (2004), http://www.iam.unibe.ch/~kai/Papers/phd.pdf
Brünnler, K.: Deep inference and its normal form of derivations. In: Beckmann, A., Berger, U., Löwe, B., Tucker, J.V. (eds.) CiE 2006. LNCS, vol. 3988, pp. 65–74. Springer, Heidelberg (2006), http://www.iam.unibe.ch/~kai/Papers/n.pdf
Brünnler, K., Tiu, A.F.: A local system for classical logic. In: Nieuwenhuis, R., Voronkov, A. (eds.) LPAR 2001. LNCS (LNAI), vol. 2250, pp. 347–361. Springer, Heidelberg (2001), http://www.iam.unibe.ch/~kai/Papers/lcl-lpar.pdf
Bruscoli, P., Guglielmi, A.: On the proof complexity of deep inference. ACM Transactions on Computational Logic 10(2), 1–34 (2009), Article 14, http://cs.bath.ac.uk/ag/p/PrComplDI.pdf
Buss, S.R.: The undecidability of k-provability. Annals of Pure and Applied Logic 53(1), 75–102 (1991)
Guglielmi, A.: Deep inference and the calculus of structures, http://alessio.guglielmi.name/res/cos
Guglielmi, A.: A system of interaction and structure. ACM Transactions on Computational Logic 8(1), 1–64 (2007), http://cs.bath.ac.uk/ag/p/SystIntStr.pdf
Guglielmi, A., Gundersen, T.: Normalisation control in deep inference via atomic flows. Logical Methods in Computer Science 4(1:9), 1–36 (2008), http://arxiv.org/pdf/0709.1205
Jeřábek, E.: Proof complexity of the cut-free calculus of structures. Journal of Logic and Computation (2008) (in press), http://www.math.cas.cz/~jerabek/papers/cos.pdf
Wegener, I.: The Complexity of Boolean Functions. John Wiley & Sons Ltd and B. G. Teubner, Stuttgart (1987)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bruscoli, P., Guglielmi, A., Gundersen, T., Parigot, M. (2010). A Quasipolynomial Cut-Elimination Procedure in Deep Inference via Atomic Flows and Threshold Formulae. In: Clarke, E.M., Voronkov, A. (eds) Logic for Programming, Artificial Intelligence, and Reasoning. LPAR 2010. Lecture Notes in Computer Science(), vol 6355. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17511-4_9
Download citation
DOI: https://doi.org/10.1007/978-3-642-17511-4_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-17510-7
Online ISBN: 978-3-642-17511-4
eBook Packages: Computer ScienceComputer Science (R0)