Girard (1987) introduced proof nets as a syntax of linear proofs which
eliminates inessential rule ordering manifested by sequent calculus.
Proof nets adapted to the Lambek calculus (Roorda 1991) fulfill a role
in categorial grammar analogous to that of phrase structure trees in
CFG so that categorial proof nets have a central part to play in
computational syntax and semantics; in particular they allow a
reinterpretation of the "problem" of spurious ambiguity as an
opportunity for parallelism. This article aims to make three
contributions: i) provide a tutorial overview of categorial proof
nets, ii) apply and provide motivation for proof nets by showing how
a partial execution eschews the need for semantic evaluation in
language processing, and iii) analyse the intrinsic geometry of
partially commutative proof nets for the kinds of discontinuity
attested in language, offering proof nets for the in situ binder
type-constructor Q(., ., .) of Moortgat (1991/6).
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