A polynomial algorithm for the membership problem with categorial grammars
Theoretical Computer Science
Basic simple type theory
Descriptive Approach to Language - Theoretic Complexity
Descriptive Approach to Language - Theoretic Complexity
Logic of Information Structures
Logic of Information Structures
Automata and Computability
Deductive systems and grammars: proofs as grammatical structures
Deductive systems and grammars: proofs as grammatical structures
Journal of Computer and System Sciences
Derived Tree Languages of Nonassociative Lambek Categorial Grammars with Product
Fundamenta Informaticae
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We investigate natural deduction proofs of the Lambek calculus from the point of view of tree automata. The main result is that the set of proofs of the Lambek calculus cannot be accepted by a finite tree automaton. The proof is extended to cover the proofs used by grammars based on the Lambek calculus, which typically use only a subset of the set of all proofs. While Lambek grammars can assign regular tree languages as structural descriptions, there exist Lambek grammars that assign non-regular structural descriptions, both when considering normal and non-normal proof trees. Combining the results of Pentus (1993) and Thatcher (1967), we can conclude that Lambek grammars, although generating only context-free languages, can extend the strong generative capacity of context-free grammars. Furthermore, we show that structural descriptions that disregard the use of introduction rules cannot be used for a compositional semantics following the Curry-Howard isomorphism.