Inducing Syntactic Cut-Elimination for Indexed Nested Sequents

Conference paper

DOI: 10.1007/978-3-319-40229-1_29

Part of the Lecture Notes in Computer Science book series (LNCS, volume 9706)
Cite this paper as:
Ramanayake R. (2016) Inducing Syntactic Cut-Elimination for Indexed Nested Sequents. In: Olivetti N., Tiwari A. (eds) Automated Reasoning. IJCAR 2016. Lecture Notes in Computer Science, vol 9706. Springer, Cham

Abstract

The key to the proof-theoretical study of a logic is a cutfree proof calculus. Unfortunately there are many logics of interest lacking suitable proof calculi. The proof formalism of nested sequents was recently generalised to indexed nested sequents in order to yield cutfree proof calculi for extensions of the modal logic K by Geach (Lemmon-Scott) axioms. The proofs of completeness and cut-elimination therein were semantical and intricate. Here we identify a subclass of the labelled sequent formalism and show that it corresponds to the indexed nested sequent formalism. This correspondence is then exploited to induce syntactic proofs for indexed nested sequents using the elegant existing proofs in the labelled sequent formalism. A larger goal of this work is to demonstrate how specialising existing proof-theoretical transformations (adapting these as required to remain within the subclass) is an alternative proof method which can alleviate the need for independent proofs from ‘scratch’ in each formalism. Moreover, such coercion can be used to induce new cutfree calculi. We demonstrate by presenting the first indexed nested sequent calculi for intermediate logics.

Keywords

Proof theory Cut-elimination Nested labelled sequents Modal logic 

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Technische Universität WienViennaAustria

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