From Deduction Graphs to Proof Nets: Boxes and Sharing in the Graphical Presentation of Deductions

  • Herman Geuvers
  • Iris Loeb
Conference paper

DOI: 10.1007/11821069_4

Part of the Lecture Notes in Computer Science book series (LNCS, volume 4162)
Cite this paper as:
Geuvers H., Loeb I. (2006) From Deduction Graphs to Proof Nets: Boxes and Sharing in the Graphical Presentation of Deductions. In: Královič R., Urzyczyn P. (eds) Mathematical Foundations of Computer Science 2006. MFCS 2006. Lecture Notes in Computer Science, vol 4162. Springer, Berlin, Heidelberg

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 cut-elimination defined on them. We give two translations from deduction graphs for minimal proposition logic to proof nets: a direct one, and a mapping via so-called 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 cut-elimination.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Herman Geuvers
    • 1
  • Iris Loeb
    • 1
  1. 1.Institute for Computing and Information SciencesRadboud University NijmegenNijmegenThe Netherlands

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