McKinley, Richard
(2013).
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Canonical proof nets for classical logic.
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Annals of pure and applied logic, 164(6), pp. 702-732.
Elsevier
10.1016/j.apal.2012.05.007

Text
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Proof nets provide abstract counterparts to sequent proofs modulo rule permutations; the idea being that if two proofs have the same underlying proof-net, they are in essence the same proof. Providing a convincing proof-net counterpart to proofs in the classical sequent calculus is thus an important step in understanding classical sequent calculus proofs. By convincing, we mean that (a) there should be a canonical function from sequent proofs to proof nets, (b) it should be possible to check the correctness of a net in polynomial time, (c) every correct net should be obtainable from a sequent calculus proof, and (d) there should be a cut-elimination procedure which preserves correctness. Previous attempts to give proof-net-like objects for propositional classical logic have failed at least one of the above conditions. In Richard McKinley (2010) [22], the author presented a calculus of proof nets (expansion nets) satisfying (a) and (b); the paper defined a sequent calculus corresponding to expansion nets but gave no explicit demonstration of (c). That sequent calculus, called LK∗ in this paper, is a novel one-sided sequent calculus with both additively and multiplicatively formulated disjunction rules. In this paper (a self-contained extended version of Richard McKinley (2010) [22]), we give a full proof of (c) for expansion nets with respect to LK∗, and in addition give a cut-elimination procedure internal to expansion nets – this makes expansion nets the first notion of proof-net for classical logic satisfying all four criteria.

## Item Type: |
Journal Article (Original Article) |
---|---|

## Division/Institute: |
08 Faculty of Science > Institute of Computer Science (INF) > Logic and Theory Group (LTG) 08 Faculty of Science > Institute of Computer Science (INF) |

## UniBE Contributor: |
Mc Kinley, Richard |

## Subjects: |
000 Computer science, knowledge & systems 500 Science > 510 Mathematics |

## ISSN: |
0168-0072 |

## Publisher: |
Elsevier |

## Submitter: |
Florian Ranzi |

## Date Deposited: |
16 Apr 2014 11:28 |

## Last Modified: |
16 Jan 2015 11:12 |

## Publisher DOI: |
10.1016/j.apal.2012.05.007 |

## Uncontrolled Keywords: |
Proof nets, Identity of proofs, Classical logic, Cut elimination |

## BORIS DOI: |
10.7892/boris.45208 |

## URI: |
https://boris.unibe.ch/id/eprint/45208 |