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COMPLEXITY OF THE INFINITARY LAMBEK CALCULUS WITH KLEENE STAR

Published online by Cambridge University Press:  22 July 2020

STEPAN KUZNETSOV*
Affiliation:
STEKLOV MATHEMATICAL INSTITUTE OF RUSSIAN ACADEMY OF SCIENCES 8 GUBKINA STREET, 119991MOSCOW, RUSSIA E-mail: [email protected]

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).

Type
Research Article
Copyright
© Association for Symbolic Logic, 2020

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References

BIBLIOGRAPHY

Abramsky, S., & Tzevelekos, N. (2010). Introduction to categories and categorical logic. In Coecke, B., editor. New Structures for Physics. Lecture Notes in Physics, Vol. 813. New York: Springer, pp. 394. CrossRefGoogle Scholar
Ajdukiewicz, K. (1935). Die syntaktische Konnexität. Studia Philosophica1, 127. Google Scholar
Andréka, H., & Mikulás, S. (1994). Lambek calculus and its relational semantics: Completeness and incompleteness. Journal of Logic, Language, and Information3(1), 137.CrossRefGoogle Scholar
Bar-Hillel, Y. (1953). A quasi-arithmetical notation for syntactic description. Language29, 4758.CrossRefGoogle Scholar
Bar-Hillel, Y., Gaifman, C., & Shamir, E. (1960). On the categorial and phrase-structure grammars. Bulletin of the Research Council of Israel9F, 116.Google Scholar
Buszkowski, W. (1982a). Compatibility of a categorial grammar with an associated category system. Zeitschrift für mathematische Logik und Grundlagen der Mathematik28, 229237.CrossRefGoogle Scholar
Buszkowski, W. (1982b). Some decision problems in the theory of syntactic categories. Zeitschrift für mathematische Logik und Grundlagen der Mathematik28, 539548.CrossRefGoogle Scholar
Buszkowski, W. (1985). The equivalence of unidirectional lambek categorial grammars and context-free grammars. Zeitschrift für mathematische Logik und Grundlagen der Mathematik31, 369384.CrossRefGoogle Scholar
Buszkowski, W. (2006). On the complexity of the equational theory of relational action algebras. In RelMiCS 2006: Relations and Kleene Algebra in Computer Science. Lecture Notes in Computer Science, Vol. 4136. New York: Springer, pp. 106119.Google Scholar
Buszkowski, W. (2007). On action logic: Equational theories of action algebras. Journal of Logic and Computation17(1), 199217.CrossRefGoogle Scholar
Buszkowski, W., & Palka, E. (2008). Infinitary action logic: Complexity, models and grammars. Studia Logica89(1), 118.CrossRefGoogle Scholar
Carpenter, B. (1998). Type-logical Semantics. Cambridge, MA: MIT Press.CrossRefGoogle Scholar
Du, D.-Z., & Ko, K.-I. (2001). Problem Solving in Automata, Languages, and Complexity. New York: John Wiley & Sons.CrossRefGoogle Scholar
Galatos, N., Jipsen, P., Kowalski, T., & Ono, H. (2007). Residuated Lattices: An Algebraic Glimpse at Substructural Logics . Amsterdam: Elsevier.Google Scholar
Girard, J.-Y. (1987). Linear logic. Theoretical Computer Science50(1), 1102.CrossRefGoogle Scholar
Greibach, S. A. (1965). A new normal-form theorem for context-free phrase structure grammars. Journal of the ACM12(1), 4252.CrossRefGoogle Scholar
Jipsen, P., & Tsinakis, C. (2002). A survey of residuated lattices. In Martinez, J., editor. Ordered Algebraic Structures. New York: Springer, pp. 1956.CrossRefGoogle Scholar
Kanovich, M., Kuznetsov, S., & Scedrov, A. (2019). The complexity of multiplicative-additive Lambek calculus: 25 years later. In Iemhoff, R., Moortgat, M., & de Queiroz, R., editors. WoLLIC 2019: Logic, Language, Information, and Computation. Lecture Notes in Computer Science, Vol. 11541. New York: Springer, pp. 356372.Google Scholar
Kleene, S. C. (1956). Representation of events in nerve nets and finite automata. In Automata Studies. Princeton, NJ: Princeton University Press, pp. 341.Google Scholar
Kozen, D. (1994a). A completeness theorem for Kleene algebras and the algebra of regular events. Information and Computation110(2), 366390.CrossRefGoogle Scholar
Kozen, D. (1994b). On action algebras. In van Eijck, J., & Visser, A., editors. Logic and Information Flow. London: MIT Press, pp. 7888.Google Scholar
Krull, W. (1924). Axiomatische Begründung der algemeinen Idealtheorie. Sitzungsberichte der physikalischmedizinischen Societät zu Erlangen56, 4763.Google Scholar
Kuznetsov, S. (2012). Lambek grammars with one division and one primitive type. Logic Journal of the IGPL20(1), 207221.CrossRefGoogle Scholar
Kuznetsov, S. (2017). The Lambek calculus with iteration: Two variants. In Kennedy, J., & de Queiroz, R., editors. WoLLIC 2017: Logic, Language, Information, and Computation. Lecture Notes in Computer Science, Vol. 10388. New York: Springer, pp. 182198.Google Scholar
Kuznetsov, S. (2018). *-continuity vs. induction: Divide and conquer. In Proceedings of AiML '18. Advances in Modal Logic, Vol. 12. London: College Publications, pp. 493510.Google Scholar
Lambek, J. (1958). The mathematics of sentence structure. American Mathematical Monthly65, 154170.CrossRefGoogle Scholar
Lambek, J. (1961). On the calculus of syntactic types. In Jakobson, R., editor. Structure of Language and Its Mathematical Aspects. Providence, RI: AMS, pp. 166178.CrossRefGoogle Scholar
Lambek, J. (1969). Deductive systems and categories II: Standard constructions and closed categories. In Hilton, P., editor. Category Theory, Homology Theory, and Their Applications I. Lecture Notes in Mathematics, Vol. 86. New York: Springer, pp. 76122.CrossRefGoogle Scholar
Moot, R., & Retoré, C. (2012). The Logic of Categorial Grammars: A Deductive Account of Natural Language Syntax and Semantics. New York: Springer.CrossRefGoogle Scholar
Morrill, G. (2011). Categorial Grammar: Logical Syntax, Semantics, and Processing . Oxford: Oxford University Press.Google Scholar
Ono, H. (1993). Semantics for substructural logics. In Schroeder-Heister, P., & Došen, K., editors. Substructural Logics. Oxford: Clarendon Press, pp. 259291.Google Scholar
Ono, H., & Komori, Y. (1985). Logics without contraction rule. Journal of Symbolic Logic50(1), 169201.CrossRefGoogle Scholar
Palka, E. (2007). An infinitary sequent system for the equational theory of *-continuous action lattices. Fundamenta Informaticae78(2), 295309.Google Scholar
Pentus, M. (1993). Lambek grammars are context-free. In Proceedings of LICS '93. IEEE, pp. 429433.Google Scholar
Pentus, M. (1994). The conjoinability relation in Lambek calculus and linear logic. Journal of Logic, Language, and Information3(2), 121140.CrossRefGoogle Scholar
Pentus, M. (1998). Free monoid completeness of the Lambek calculus allowing empty premises. In Proceedings of Logic Colloquium '96. New York: Springer, pp. 171209.CrossRefGoogle Scholar
Pentus, M. (2006). Lambek calculus is NP-complete. Theoretical Computer Science357(1), 186201.CrossRefGoogle Scholar
Pratt, V. (1991). Action logic and pure induction. In JELIA 1990: Logics in AI. Lecture Notes in Artificial Intelligence, Vol. 478. New York: Springer, pp. 97120.Google Scholar
Restall, G. (2000). An Introduction to Substructural Logics. New York: Routledge.CrossRefGoogle Scholar
Safiullin, A. N. (2007). Derivability of admissible rules with simple premises in the Lambek calculus. Moscow University Mathematics Bulletin62(4), 168171.CrossRefGoogle Scholar
Sipser, M. (2012). Introduction to the Theory of Computation (third edition). Boston, MA: Cengage Learning.Google Scholar
Sorokin, A. (2012). On the completeness of the Lambek calculus with respect to cofinite language models. In LACL 2012: Logical Aspects of Computational Linguistics. Lecture Notes in Computer Science, Vol. 7351. New York: Springer, pp. 229233.CrossRefGoogle Scholar
Ward, M., & Dilworth, R. P. (1939). Residuated lattices. Transactions of the AMS45(3), 335354.CrossRefGoogle Scholar