Commutative linear logic as a multiple context-free grammar
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The formalism of multiple context-free grammars (MCFG) is a non-trivial generalization of context-free grammars (CFG), where basic constituents on which rules operate are discontinuous tuples of words rather than single words. Just as context-free ones, multiple context-free grammars have polynomial parsing algorithms, but their expressive power is strictly stronger. It is well known that CFG generate the same class of languages as type logical grammars based on Lambek calculus, which is, basically, a variant of noncommutative linear logic. We construct a system of type logical grammars based on ordinary commutative linear logic and show that these grammars are in the same relationship with MCFG as Lambek grammars with CFG. It turns out that tuples of words on which MCFG operate can be organized into a symmetric monoidal category, very similar to the category of topological cobordisms; we call it the category of word cobordisms. In particular, this category is compact closed and, thus, a model of linear logic. Using interpretation of linear logic proofs as word cobordisms allows us to define type logical grammars by adding extra axioms (a lexicon) and interpreting them as cobordisms as well. Such grammars turn out to be equivalent to MCFG.
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