Natural deduction via graphs: formal definition and computation rules

We introduce the formalism of deduction graphs as a generalization of both Gentzen-Prawitz style natural deduction and Fitch style flag deduction. The advantage of this formalism is that subproofs can be shared, like in flag deductions (and unlike natural deduction), but also that the linearisation used in flag deductions is avoided. Our deduction graphs have both nodes and boxes, which are collections of nodes that also form a node themselves. This is reminiscent of the bigraphs of Milner, where the link graph describes the nodes and edges and the place graph describes the nesting of nodes. In the paper we give a precise definition of deduction graphs and we give examples to illustrate them. Furthermore we analyse their computational behaviour by studying the process of cut-elimination and by defining translations from deduction graphs to simply typed lambda terms. From a slight variation of this translation we conclude that the process of cut-elimination is strongly normalising. The translation to simple type theory removes quite a lot of structure and we therefore also propose a translation to a context calculus with lets, that faithfully captures the structure of deduction graphs. The proof nets of linear logic also present a graph-like presentation of natural deduction. We point out some similarities of the two formalisms.