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Proof Nets for Herbrand’s Theorem

Proof Nets for Herbrand’s Theorem Proof Nets for Herbrand's Theorem ¨ RICHARD MCKINLEY, Universitat Bern This article explores Herbrand's theorem as the source of a natural notion of abstract proof object for classical logic, embodying the "essence" of a sequent calculus proof. We see how to view a calculus of abstract Herbrand proofs (Herbrand nets) as an analytic proof system with syntactic cut-elimination. Herbrand nets can also be seen as a natural generalization of Miller's expansion tree proofs to a setting including cut. We demonstrate sequentialization of Herbrand nets into a sequent calculus LKH ; each net corresponds to an equivalence class of LKH proofs under natural proof transformations. A surprising property of our cutreduction algorithm is that it is non-confluent despite not supporting the usual examples of non-confluent reduction in classical logic. Categories and Subject Descriptors: F.4.1 [Mathematical Logic and Formal Languages]: Mathematical Logic--Proof theory General Terms: Theory Additional Key Words and Phrases: Classical logic, cut elimination, Herbrand's theorem, proof nets ACM Reference Format: McKinley, R. 2013. Proof nets for Herbrand's theorem. ACM Trans. Comput. Logic 14, 1, Article 5 (February 2013), 31 pages. DOI:http://dx.doi.org/10.1145/2422085.2422090 1. INTRODUCTION ¨ This article is part of a program [Bellin et al. 2006; Fuhrmann and Pym http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png ACM Transactions on Computational Logic (TOCL) Association for Computing Machinery

Proof Nets for Herbrand’s Theorem

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References (47)

Publisher
Association for Computing Machinery
Copyright
Copyright © 2013 by ACM Inc.
ISSN
1529-3785
DOI
10.1145/2422085.2422090
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Abstract

Proof Nets for Herbrand's Theorem ¨ RICHARD MCKINLEY, Universitat Bern This article explores Herbrand's theorem as the source of a natural notion of abstract proof object for classical logic, embodying the "essence" of a sequent calculus proof. We see how to view a calculus of abstract Herbrand proofs (Herbrand nets) as an analytic proof system with syntactic cut-elimination. Herbrand nets can also be seen as a natural generalization of Miller's expansion tree proofs to a setting including cut. We demonstrate sequentialization of Herbrand nets into a sequent calculus LKH ; each net corresponds to an equivalence class of LKH proofs under natural proof transformations. A surprising property of our cutreduction algorithm is that it is non-confluent despite not supporting the usual examples of non-confluent reduction in classical logic. Categories and Subject Descriptors: F.4.1 [Mathematical Logic and Formal Languages]: Mathematical Logic--Proof theory General Terms: Theory Additional Key Words and Phrases: Classical logic, cut elimination, Herbrand's theorem, proof nets ACM Reference Format: McKinley, R. 2013. Proof nets for Herbrand's theorem. ACM Trans. Comput. Logic 14, 1, Article 5 (February 2013), 31 pages. DOI:http://dx.doi.org/10.1145/2422085.2422090 1. INTRODUCTION ¨ This article is part of a program [Bellin et al. 2006; Fuhrmann and Pym

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ACM Transactions on Computational Logic (TOCL) – Association for Computing Machinery

Published: Feb 1, 2013

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