  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.

Logic in Computer Science