The Lambek calculus with the unit can be defined as the atomic theory (algebraic logic) of the class of residuated monoids. This calculus, being a theory of a broader class of algebras than Heyting ones, is weaker than intuitionistic logic. Namely, it lacks structural rules: permutation, contraction, and weakening. We consider two extensions of the Lambek calculus with modalities—the exponential, under which all structural rules are permitted, and the relevant modality, under which only permutation and contraction rules are allowed. The Lambek calculus with a relevant modality is used in mathematical linguistics. Both calculi are algorithmically undecidable. We consider their fragments in which the modality is allowed to be applied to just formulas of Horn depth not greater than 1. We prove that these fragments are decidable and belong to the NP class. To show this, in the case of a relevant modality, we introduce a new notion of ℛ-total derivability in context-free grammars, i.e., existence of a derivation in which each rule is used at least a given number of times. It is stated that the ℛ-totality problem is NP-hard for context-free grammars. Also we pinpoint algorithmic complexity of ℛ-total derivability for more general classes of generative grammars.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
W. Krull, Axiomatische Begründung der allgemeinen Idealtheorie, Sitzber. d. phys.-med. Soc. Erlangen, 56, 47-63 (1924).
J. Lambek, “Deductive systems and categories. II: Standard constructions and closed categories,” Lect. Notes Math., 86, Springer, Berlin (1969), pp. 76-122.
J.-Y. Girard, “Linear logic,” Theor. Comput. Sci., 50, No. 1, 1-102 (1987).
L. L. Maksimova, “On system of axioms of the calculus of rigorous implication,” Algebra Logika, 3, No. 3, 59-68 (1964).
M. Kanovich, S. Kuznetsov, and A. Scedrov, “Undecidability of the Lambek calculus with a relevant modality,” Lect. Notes Comput. Sci., 9804, Springer, Berlin (2016), pp. 240-256.
M. Kanovich, S. Kuznetsov, V. Nigam, and A. Scedrov, “Subexponentials in non-commutative linear logic,” Math. Struct. Comput. Sci., 29, No. 8, 1217-1249 (2019).
P. Lincoln, J. Mitchell, A. Scedrov, and N. Shankar, “Decision problems for propositional linear logic,” Ann. Pure Appl. Log., 56, Nos. 1-3, 239-311 (1992).
M. Pentus, “Lambek calculus is NP-complete,” Theor. Comput. Sci., 357, Nos. 1-3, 186-201 (2006).
Yu. V. Savateev. “Algorithmic complexity of fragments of the Lambek calculus,” Cand. Sci. Dissertation, Moscow State Univ., Moscow (2009).
A. Aho and J. Ullman, The Theory of Parsing, Translation, and Compiling, Vol. 1, Parsing, Prentice-Hall, 1972.
M. R. Garey and D. S. Johnson, “Complexity results for multiprocessor scheduling under resource constraints,” SIAM J. Comput., 4, No. 4, 397-411 (1975).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Algebra i Logika, Vol. 60, No. 5, pp. 471-496, September-October, 2021. Russian DOI: https://doi.org/10.33048/alglog.2021.60.502.
S. M. Dudakov, B. N. Karlov and S. L. Kuznetsov are supported by RFBR, project No. 20-01-00435.
S. L. Kuznetsov is supported by the Council for Grants (under RF President), grant MK-1184.2021.1.1.
Rights and permissions
About this article
Cite this article
Dudakov, S.M., Karlov, B.N., Kuznetsov, S.L. et al. Complexity of Lambek Calculi with Modalities and of Total Derivability in Grammars. Algebra Logic 60, 308–326 (2021). https://doi.org/10.1007/s10469-021-09657-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10469-021-09657-5