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Proof Complexity of Non-classical Logics

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Lectures on Logic and Computation (ESSLLI 2011, ESSLLI 2010)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7388))

Abstract

Proof complexity is an interdisciplinary area of research utilising techniques from logic, complexity, and combinatorics towards the main aim of understanding the complexity of theorem proving procedures. Traditionally, propositional proofs have been the main object of investigation in proof complexity. Due their richer expressivity and numerous applications within computer science, also non-classical logics have been intensively studied from a proof complexity perspective in the last decade, and a number of impressive results have been obtained.

In these notes we give an introduction to this recent field of proof complexity of non-classical logics. We cover results from proof complexity of modal, intuitionistic, and non-monotonic logics. Some of the results are surveyed, but in addition we provide full details of a recent exponential lower bound for modal logics due to Hrubeš [60] and explain the complexity of several sequent calculi for default logic [16,13]. To make the text self-contained, we also include necessary background information on classical proof systems and non-classical logics.

Part of these notes are based on the survey [12] and the research paper [13]. This paper was produced while the first author was visiting Sapienza University of Rome under support of grant N. 20517 by the John Templeton Foundation. The work of the second author was supported by the DFG-funded Research Centre on Spatial Cognition (SFB/TR 8), project I1-[OntoSpace].

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  1. Institut für Theoretische Informatik, Leibniz-Universität Hannover, Germany

    Olaf Beyersdorff

  2. Research Center on Spatial Cognition (SFB/TR 8), Universität Bremen, Germany

    Oliver Kutz

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  1. Olaf Beyersdorff
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  1. Department of Computing, Imperial College London, 180 Queen’s Gate, SW7 2AZ, London, UK

    Nick Bezhanishvili

  2. Department of Informatics and MAthematical Modelling, Technical University of Denmark, Asmussens Alle, Bld. 305, 2800 Kgs., Lyngby,, Denmark

    Valentin Goranko

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Beyersdorff, O., Kutz, O. (2012). Proof Complexity of Non-classical Logics. In: Bezhanishvili, N., Goranko, V. (eds) Lectures on Logic and Computation. ESSLLI ESSLLI 2011 2010. Lecture Notes in Computer Science, vol 7388. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31485-8_1

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