Section: Scientific Foundations

Proof Nets and Combinatorial Characterization of Proofs

Proof nets are abstract (graph-like) presentations of proofs such that all "trivial rule permutations" are quotiented away. More generally, we investigate combinatoric objects and correctness criteria for studying proofs independently from syntax. Ideally the notion of proof net should be independent from any syntactic formalism. But due to the almost absolute monopoly of the sequent calculus, most notions of proof nets proposed in the past were formulated in terms of their relation to the sequent calculus. Consequently we could observe features like “boxes” and explicit “contraction links”. The latter appeared not only in Girard's proof nets [42] for linear logic but also in Robinson's proof nets [60] for classical logic. In this kind of proof nets every link in the net corresponds to a rule application in the sequent calculus.

The concept of deep inference allows to design entirely new kinds of proof nets. Recent work by Lamarche and Straßburger [62] and [5] have extended the theory of proof nets for multiplicative linear logic to multiplicative linear logic with units. This seemingly small step—just adding the units—had for long been an open problem, and the solution was found only by consequently exploiting the new insights coming from deep inference. A proof net no longer just mimics the sequent calculus proof tree, but rather an additional graph structure that is put on top of the formula tree (or sequent forest) of the conclusion. The work on proof nets within the team is focused on the following two directions

• Extend the work of Lamarche and Straßburger to larger fragments of linear logic, containing the additives, the exponentials, and the quantifiers.

• Finding (for classical logic) a notion of proof nets that is deductive, i.e., can effectively be used for doing proof search. An important property of deductive proof nets must be that the correctness can be checked in linear time. For the classical logic proof nets by Lamarche and Straßburger [4] this takes exponential time (in the size of the net). We hope that eventually deductive proof nets will provide a “bureaucracy-free” formalism for proof search.

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