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Expansion trees with cut

Published online by Cambridge University Press:  08 October 2019

Federico Aschieri ,
Stefan Hetzl Open the ORCID record for Stefan Hetzl [Opens in a new window]  and
Daniel Weller
Federico Aschieri
Affiliation:
Institut für Logic and Computation, Technische Universität Wien, Austria
Stefan Hetzl*
Affiliation:
Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien
Daniel Weller
Affiliation:
Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien
*
*Corresponding author. Email: stefan.hetzl@tuwien.ac.at
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Abstract

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Herbrand’s theorem is one of the most fundamental insights in logic. From the syntactic point of view, it suggests a compact representation of proofs in classical first- and higher-order logics by recording the information of which instances have been chosen for which quantifiers. This compact representation is known in the literature as Miller’s expansion tree proof. It is inherently analytic and hence corresponds to a cut-free sequent calculus proof. Recently several extensions of such proof representations to proofs with cuts have been proposed. These extensions are based on graphical formalisms similar to proof nets and are limited to prenex formulas.

In this paper, we present a new syntactic approach that directly extends Miller’s expansion trees by cuts and also covers non-prenex formulas. We describe a cut-elimination procedure for our expansion trees with cut that is based on the natural reduction steps and shows that it is weakly normalizing.

Keywords

Proof theorycut-eliminationHerbrand’s theoremexpansion trees

Footnotes

†

Funded by FWF Lise Meitner Grant M 1930–N35 and START project Y544–N23

‡

Funded by the WWTF Vienna Research Group (VRG) 12-004.

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