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The length of joins in Lambek calculus. (English. Russian original) Zbl 1304.03056

Mosc. Univ. Math. Bull. 66, No. 3, 101-104 (2011); translation from Vest. Mosk. Univ. Mat. Mekh. 66, No. 3, 10-14 (2011).
Summary: M. Pentus [“Equivalent types in Lambek calculus and linear logic”, Preprint No. 2, Dept. Math. Logic, Steklov Math. Inst., Ser. Logic and Comput. Sci., Moscow (1992)] established a criterion for the existence of a type \(C\) such that for given types \(A\) and \(B\) the sequences \(A \to C\) and \(B \to C\) are derivable in the Lambek calculus. In this paper we give an algorithm for construction of such a type \(C\) (provided it exists) and prove a quadratic upper bound for its length.

MSC:

03B47 Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics)
Full Text: DOI

References:

[1] J. Lambek, ”The Mathematics of Sentence Structure,” Amer. Math. Monthly, 65, 154 (1958). · Zbl 0080.00702 · doi:10.2307/2310058
[2] A. Foret, ”Conjoinability and Unification in Lambek Categorial Grammars,” in New Perspectives in Logic and Formal Linguistics: Proc. V Roma Workshop (Bulzoni Editore, Roma, 2001). · Zbl 0990.68151
[3] M. Pentus, Equivalent Types in Lambek Calculus and Linear Logic Preprint N 2 (Dept. Math. Logic, Steklov Math. Inst., Ser. Logic and Comput. Sci., Moscow, 1992).
[4] A. Foret, ”On the Computation of Joins for Non-Associative Lambek Categorial Grammars,” in Proc. 17th Int. Workshop on Unification, Valencia, Spain, June 8–9, 2003 (Universidad Politécnica de Valencia, Valencia, 2003), pp. 25–37.
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