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COMPLEXITY OF THE INFINITARY LAMBEK CALCULUS WITH KLEENE STAR

Part of: Proof theory and constructive mathematics

Published online by Cambridge University Press:  22 July 2020

STEPAN KUZNETSOV
STEPAN KUZNETSOV*
Affiliation:
STEKLOV MATHEMATICAL INSTITUTE OF RUSSIAN ACADEMY OF SCIENCES 8 GUBKINA STREET, 119991MOSCOW, RUSSIA E-mail: sk@mi-ras.ru
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Abstract

We consider the Lambek calculus, or noncommutative multiplicative intuitionistic linear logic, extended with iteration, or Kleene star, axiomatised by means of an $\omega $ -rule, and prove that the derivability problem in this calculus is $\Pi _1^0$ -hard. This solves a problem left open by Buszkowski (2007), who obtained the same complexity bound for infinitary action logic, which additionally includes additive conjunction and disjunction. As a by-product, we prove that any context-free language without the empty word can be generated by a Lambek grammar with unique type assignment, without Lambek’s nonemptiness restriction imposed (cf. Safiullin, 2007).

Keywords

Kleene starLambek calculusnoncommutative linear logicinfinitary action logiccomplexitycategorial grammars

MSC classification

Primary: 03F52: Linear logic and other substructural logics
Type
Research Article
Information
The Review of Symbolic Logic , Volume 14 , Issue 4 , December 2021 , pp. 946 - 972
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Copyright
© Association for Symbolic Logic, 2020

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