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Making first order linear logic a generating grammar
Logical Methods in Computer Science. 2023. Vol. 19. No. 4.
It is known that different categorial grammars have surface representation in a fragment of first order multiplicative linear logic (MLL1). We show that the fragment of interest is equivalent to the recently introduced extended tensor type calculus (ETTC). ETTC is a calculus of specific typed terms, which represent tuples of strings, more precisely bipartite graphs decorated with strings. Types are derived from linear logic formulas, and rules correspond to concrete operations on these string-labeled graphs, so that they can be conveniently visualized. This provides the above mentioned fragment of MLL1 that is relevant for language modeling not only with some alternative syntax and intuitive geometric representation, but also with an intrinsic deductive system, which has been absent. In this work we consider a non-trivial notationally enriched variation of the previously introduced ETTC, which allows more concise and transparent computations. We present both a cut-free sequent calculus and a natural deduction formalism.
Stepan L. Kuznetsov, Journal of Logic and Computation 2026 Vol. 36 No. 1 Article exaf078
The linguistic applications of the Lambek calculus suggest its semantics over algebras of formal languages. A straightforward approach to construct such semantics indeed yields a brilliant completeness theorem (Pentus 1995, Ann. Pure Appl. Logic, 75, 179–213). However, extending the calculus with extra operations ruins completeness. In order to mitigate this issue, Wurm (2017, J. Logic Lang. Inf., ...
Added: January 14, 2026
Kuznetsov S., Speranski S. O., Annals of Pure and Applied Logic 2022 Vol. 173 No. 2 Article 103057
We introduce infinitary action logic with exponentiation — that is, the multiplicative-additive Lambek calculus extended with Kleene star and with a family of subexponential modalities, which allow some of the structural rules (contraction, weakening, permutation). The logic is presented in the form of an infinitary sequent calculus. We prove cut elimination and, in the case ...
Added: December 26, 2025
Kuznetsov S., Speranski S. O., Studia Logica 2023 Vol. 111 No. 2 P. 251–280
Infinitary action logic can be naturally expanded by adding exponential and subexponential modalities from linear logic. In this article we shall develop infinitary action logic with a subexponential that allows multiplexing (instead of contraction). Both non-commutative and commutative versions of this logic will be considered, presented as infinitary sequent calculi. We shall prove cut admissibility ...
Added: December 26, 2025
Kuznetsov S., Lugovaya V., Ryzhova A., Logic Journal of the IGPL 2019 Vol. 27 No. 3 P. 252–266
Added: May 1, 2025
Stepan L. Kuznetsov, , in: Logic, Language, Information, and Computation: 30th International Workshop, WoLLIC 2024, Bern, Switzerland, June 10–13, 2024, ProceedingsVol. 14672: Lecture Notes in Computer Science.: Cham: Springer, 2024. P. 93–107.
Added: June 12, 2024
S. M. Dudakov, Karlov B. N., S. L. Kuznetsov et al., Algebra and Logic 2021 Vol. 60 No. 5 P. 308–326
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 ...
Added: November 12, 2023
Blaisdell E., Kanovich M., Stepan L. Kuznetsov et al., , in: Automated Reasoning: 11th International Joint Conference, IJCAR 2022, Haifa, Israel, August 8–10, 2022, ProceedingsVol. 13385.: Cham: Springer, 2022. P. 449–467.
Added: August 7, 2022
Kanovich M., Kuznetsov Stepan G., Kuznetsov S. et al., Studies in Computational Intelligence 2021 Vol. 999 P. 1–24
CatLog is a categorial grammar parser/theorem-prover
developed by Glyn Morrill and his co-authors. CatLog is based on an
extension of Lambek calculus. A distinctive feature of this extension is
the usage of brackets for controlled non-associativity and a subexponential
modality whose contraction rule interacts with bracketing in a
sophisticated way. We consider two variants of the calculus, appearing in
different versions ...
Added: December 14, 2021
Kanovich M., Kuznetsov S., Scedrov A., Information and Computation 2022 Vol. 287 Article 104760
We investigate language interpretations of two extensions of the Lambek calculus: with additive conjunction and disjunction and with additive conjunction and the unit constant. For extensions with additive connectives, we show that conjunction and disjunction behave differently. Adding both of them leads to incompleteness due to the distributivity law. We show that with conjunction only ...
Added: December 4, 2021
Sergey Slavnov, Mathematical Structures in Computer Science 2021 Vol. 31 No. 5 P. 495–534
Ehrhard et al. (2018. Proceedings of the ACM on Programming Languages, POPL 2, Article 59.) proposed a model of probabilistic functional programming in a category of normed positive cones and stable measurable cone maps, which can be seen as a coordinate-free generalization of probabilistic coherence spaces (PCSs). However, unlike the case of PCSs, it remained unclear ...
Added: November 16, 2021
Slavnov S. A., Journal of Logic and Computation 2022 Vol. 32 No. 3 P. 479–517
We consider tensor grammars, which are an example of ‘commutative’ grammars, based on the classical (rather than intuitionistic) linear logic. They can be seen as a surface representation of abstract categorial grammars (ACG) in the sense that derivations of ACG translate to derivations of tensor grammars and this translation is isomorphic on the level of string ...
Added: October 21, 2021
Sergey Slavnov, Journal of Cognitive Science 2021 Vol. 22 No. 2 P. 68–91
We propose a concrete surface representation of abstract categorial grammars in the category of word cobordisms or cowordisms
for short, which are certain bipartite graphs decorated withwords in a given alphabet, generalizing linear logic proof-nets.
We also introduce and study linear logic grammars, directly based on cobordisms and using classical multiplicative linear logic
as a typing system. ...
Added: October 2, 2021
Kanovich M., Kuznetsov S., Scedrov A., Journal of Logic, Language and Information 2021 Vol. 30 No. 1 P. 31–88
We give a proof-theoretic and algorithmic complexity analysis for systems introduced by Morrill to serve as the core of the CatLog categorial grammar parser. We consider two recent versions of Morrill’s calculi, and focus on their fragments including multiplicative (Lambek) connectives, additive conjunction and disjunction, brackets and bracket modalities, and the ! subexponential modality. For ...
Added: November 25, 2020
Kuznetsov S., , in: Logic, Language, and Security. Essays Dedicated to Andre Scedrov on the Occasion of His 65th BirthdayIssue 12300.: Cham: Springer, 2020. P. 3–16.
Infinitary action logic is an extension of the multiplicative-additive Lambek calculus with Kleene iteration, axiomatized by an 𝜔-rule. Buszkowski and Palka (2007) show that this logic is \(\Pi^0_1\)-complete. As shown recently by Kuznetsov and Speranski, the extension of infinitary action logic with the exponential modality is much harder: \(\Pi^1_1\)-complete. The raise of complexity is of ...
Added: November 25, 2020
Kanovich M., Kuznetsov S., Nigam V. et al., , in: Automated Reasoning, 10th International Joint Conference, IJCAR 2020, Part IVol. 12166.: Cham: Springer, 2020. P. 500–517.
Added: July 1, 2020
Kanovich M., Kuznetsov S., Scedrov A., Journal of Logic and Computation 2020 Vol. 30 No. 1 P. 239–256
The Lambek calculus can be considered as a version of non-commutative intuitionistic linear logic. One of the interesting features of the Lambek calculus is the so-called ‘Lambek’s restriction’, i.e. the antecedent of any provable sequent should be non-empty. In this paper, we discuss ways of extending the Lambek calculus with the linear logic exponential modality ...
Added: July 1, 2020
Sergey Slavnov, Logical Methods in Computer Science 2019 Vol. 15 No. 3 P. 1–25
Pomset logic introduced by Retoré is an extension of linear logic with a self-dual noncommutative connective. The logic is defined by means of proof-nets, rather than a sequent calculus. Later a deep inference system BV was developed with an eye to capturing Pomset logic, but equivalence of system has not been proven up to now. ...
Added: October 23, 2019
Kanovich M., Kuznetsov S., Scedrov A., , in: Logic, Language, Information, and Computation: 26th International Workshop, WoLLIC 2019, Utrecht, The Netherlands, July 2-5, 2019, ProceedingsVol. 11541: Lecture Notes in Computer Science.: Berlin, Heidelberg: Springer, 2019. P. 356–372.
The Lambek calculus was introduced as a mathematical description of natural languages. The original Lambek calculus is NP-complete (Pentus), while its product-free fragment with only one implication is polynomially decidable (Savateev). We consider Lambek calculus with the additional connectives: conjunction and disjunction. It is known that this system is PSPACE-complete (Kanovich, Kanazawa). We prove, in ...
Added: September 4, 2019
Kanovich M., Kuznetsov S., Scedrov A., , in: Logic, Language, Information, and Computation: 26th International Workshop, WoLLIC 2019, Utrecht, The Netherlands, July 2-5, 2019, ProceedingsVol. 11541: Lecture Notes in Computer Science.: Berlin, Heidelberg: Springer, 2019. P. 373–391.
Language and relational models, or L-models and R-models, are two natural classes of models for the Lambek calculus. Completeness w.r.t. L-models was proved by Pentus and completeness w.r.t. R-models by Andréka and Mikulás. It is well known that adding both additive conjunction and disjunction together yields incompleteness, because of the distributive law. The product-free Lambek ...
Added: September 4, 2019
Kanovich M., Kuznetsov S., Nigam V. et al., , in: 9TH INTERNATIONAL JOINT CONFERENCE ON AUTOMATED REASONINGIssue 10900.: Springer International Publishing AG, part of Springer Nature 2018, 2018. P. 228–245.
Logical frameworks allow the specification of deductive systems using the same logical machinery. Linear logical frameworks have been successfully used for the specification of a number of computational, logics and proof systems. Its success relies on the fact that formulas can be distinguished as linear, which behave intuitively as resources, and unbounded, which behave intuitionistically. ...
Added: July 11, 2018