From Proof Nets to the Free *-Autonomous Category

Abstract

Abstract. In the first part of this paper we present a theory of proof nets for full multiplicative linear logic, including the two units. It naturally extends the well-known theory of unit-free multiplicative proof nets. A linking is no longer a set of axiom links but a tree in which the axiom links are subtrees. These trees will be identified according to an equivalence relation based on a simple form of graph rewriting. We show the standard results of sequentialization and strong normalization of cut elimination. In the second part of the paper we show that the identifications enforced on proofs are such that the class of two-conclusion proof nets defines the free * -autonomous category.

Canonical Sequent Proofs via Multi-Focusing

“…MLL − proof nets are certainly the most concise canonical structures for this fragment. There are candidates to extend MLL − proof nets to broader fragments (MLL with units [14], MALL [12] or MELL) but they are not as satisfactory as for MLL − . The analysis we just made could be carried to MALL proof nets as introduced by Hughes and van Glabbeek [12] for the appropriate extension of definition 17 (in particular to take into account the fact that with MALL proof nets there is not only one linking but a set of linkings corresponding to the additive slices of the proof net).…”

Some Observations on the Proof Theory of Second Order Propositional Multiplicative Linear Logic

“…For defining proof nets for MLL2, we follow the ideas presented in [21,17] where the axiom linking of multiplicative proof nets has been replaced by a linking formula to accommodate the units 1 and ⊥. In such a linking formula, the ordinary axiom links are replaced by -nodes, which are then connected by s. A unit can then be attached to a sublinking by another , and so on.…”

“…In [21,17] a proof net consists of the sequent forest and the linking formula. The presence of the quantifiers, in particular, the presence of instantiation and substitution, makes it necessary to expand the structure of the sequent in the proof net.…”

“…It is a combination of the sequentialization proofs in [17] and [10], and it makes crucial use of the "splitting tensor lemma" which in our case also needs the well-nestedness condition.…”

“…In [21,17], it was shown how boxes for the multiplicative units can be avoided by entirely dropping the connection between nodes in the proof net and rules in the sequent calculus. Here, these ideas are pushed further to accommodate the quantifiers.…”