Mathematical Foundations of Computer Science 2006
Volume 4162 of the series Lecture Notes in Computer Science pp 3957
From Deduction Graphs to Proof Nets: Boxes and Sharing in the Graphical Presentation of Deductions
 Herman GeuversAffiliated withInstitute for Computing and Information Sciences, Radboud University Nijmegen
 , Iris LoebAffiliated withInstitute for Computing and Information Sciences, Radboud University Nijmegen
Abstract
Deduction graphs [3] provide a formalism for natural deduction, where the deductions have the structure of acyclic directed graphs with boxes. The boxes are used to restrict the scope of local assumptions. Proof nets for multiplicative exponential linear logic (MELL) are also graphs with boxes, but in MELL the boxes have the purpose of controlling the modal operator !. In this paper we study the apparent correspondences between deduction graphs and proof nets, both by looking at the structure of the proofs themselves and at the process of cutelimination defined on them. We give two translations from deduction graphs for minimal proposition logic to proof nets: a direct one, and a mapping via socalled context nets. Context nets are closer to natural deduction than proof nets, as they have both premises (on top of the net) and conclusions (at the bottom). Although the two translations give basically the same results, the translation via context nets provides a more abstract view and has the advantage that it follows the same inductive construction as the deduction graphs. The translations behave nicely with respect to cutelimination.
 Title
 From Deduction Graphs to Proof Nets: Boxes and Sharing in the Graphical Presentation of Deductions
 Book Title
 Mathematical Foundations of Computer Science 2006
 Book Subtitle
 31st International Symposium, MFCS 2006, Stará Lesná, Slovakia, August 28September 1, 2006. Proceedings
 Pages
 pp 3957
 Copyright
 2006
 DOI
 10.1007/11821069_4
 Print ISBN
 9783540377917
 Online ISBN
 9783540377931
 Series Title
 Lecture Notes in Computer Science
 Series Volume
 4162
 Series ISSN
 03029743
 Publisher
 Springer Berlin Heidelberg
 Copyright Holder
 SpringerVerlag Berlin Heidelberg
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 Editors

 Rastislav Královič ^{(16)}
 Paweł Urzyczyn ^{(17)}
 Editor Affiliations

 16. Department of Computer Science, Comenius University, Mlynská dolina
 17. Institute of Informatics, University of Warsaw
 Authors

 Herman Geuvers ^{(18)}
 Iris Loeb ^{(18)}
 Author Affiliations

 18. Institute for Computing and Information Sciences, Radboud University Nijmegen, 6500 GL, Nijmegen, The Netherlands
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