Cut Elimination for Shallow Modal Logics

  • Björn Lellmann
  • Dirk Pattinson
Conference paper

DOI: 10.1007/978-3-642-22119-4_17

Part of the Lecture Notes in Computer Science book series (LNCS, volume 6793)
Cite this paper as:
Lellmann B., Pattinson D. (2011) Cut Elimination for Shallow Modal Logics. In: Brünnler K., Metcalfe G. (eds) Automated Reasoning with Analytic Tableaux and Related Methods. TABLEAUX 2011. Lecture Notes in Computer Science, vol 6793. Springer, Berlin, Heidelberg

Abstract

Motivated by the fact that nearly all conditional logics are axiomatised by so-called shallow axioms (axioms with modal nesting depth ≤ 1) we investigate sequent calculi and cut elimination for modal logics of this type. We first provide a generic translation of shallow axioms to (one-sided, unlabelled) sequent rules. The resulting system is complete if we admit pseudo-analytic cut, i.e. cuts on modalised propositional combinations of subformulas, leading to a generic (but sub-optimal) decision procedure. In a next step, we show that, for finite sets of axioms, only a small number of cuts is needed between any two applications of modal rules. More precisely, completeness still holds if we restrict to cuts that form a tree of logarithmic height between any two modal rules. In other words, we obtain a small (Pspace-computable) representation of an extended rule set for which cut elimination holds. In particular, this entails Pspace decidability of the underlying logic if contraction is also admissible. This leads to (tight) Pspace bounds for various conditional logics.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Björn Lellmann
    • 1
  • Dirk Pattinson
    • 1
  1. 1.Department of ComputingImperial College LondonUK

Personalised recommendations