Abstract
We examine some combinatorial properties of parallel cut elimination in multiplicative linear logic (MLL) proof nets. We show that, provided we impose a constraint on some paths, we can bound the size of all the nets satisfying this constraint and reducing to a fixed resultant net. This result gives a sufficient condition for an infinite weighted sum of nets to reduce into another sum of nets, while keeping coefficients finite. We moreover show that our constraints are stable under reduction. Our approach is motivated by the quantitative semantics of linear logic: many models have been proposed, whose structure reflect the Taylor expansion of multiplicative exponential linear logic (MELL) proof nets into infinite sums of differential nets. In order to simulate one cut elimination step in MELL, it is necessary to reduce an arbitrary number of cuts in the differential nets of its Taylor expansion. It turns out our results apply to differential nets, because their cut elimination is essentially multiplicative. We moreover show that the set of differential nets that occur in the Taylor expansion of an MELL net automatically satisfies our constraints. Interestingly, our nets are untyped: we only rely on the sequentiality of linear logic nets and the dynamics of cut elimination. The paths on which we impose bounds are the switching paths involved in the Danos--Regnier criterion for sequentiality. In order to accommodate multiplicative units and weakenings, our nets come equipped with jumps: each weakening node is connected to some other node. Our constraint can then be summed up as a bound on both the length of switching paths, and the number of weakenings that jump to a common node.
Highlights
We examine some combinatorial properties of parallel cut elimination in multiplicative linear logic (MLL) proof nets
Our purpose in the present paper is to develop a similar technique for multiplicative exponential linear logic (MELL) proof nets: we show that one can bound the size of a resource net p by a function of the size of any of its parallel reducts, and of an additional quantity on p, yet to be defined
Observe that we need only to consider the support of Taylor expansion, so we do not formalize the expansion of MELL nets into infinite linear combinations of resource nets: rather, we introduce T (P ) as a set of approximants
Summary
Linear logic takes its roots in the denotational semantics of λ-calculus: it is often presented, by Girard himself [Gir87], as the result of a careful investigation of the model of coherence spaces.
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