On embedding Lambek calculus into commutative categorial grammars
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Abstract categorial grammars (ACG), as well as some other, closely related systems, are based on the ordinary, commutative implicational linear logic and linear λ-calculus in contrast to the better known "noncommutative" Lambek grammars and their variations. ACG seem attractive in many ways, not the least of which is the simplicity of the underlying logic. Yet it is known that ACG and their relatives behave poorly in modeling many natural language phenomena (such as, for example, coordination) compared to "noncommutative" formalisms. Therefore different solutions have been proposed in order to enrich ACG with noncommutative constructions. Tensor grammars of this work are another example of "commutative" grammars, based on the classical, rather than intuitionistic linear logic. They can be seen as a surface representation of ACG in the sense that derivations of ACG translate to derivations of tensor grammars and this translation is isomorphic on the level of string languages. An advantage of this representation, as it seems to us, is that the syntax becomes extremely simple and a direct geometric meaning is transparent. We address the problem of encoding noncommutative operations in our setting. This turns out possible after enriching the system with new unary operators. The resulting system allows representing both ACG and Lambek grammars as conservative fragments, while the formalism remains, as it seems to us, rather simple and intuitive.
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