# Stepan KuznetsovRussian Academy of Sciences | RAS · Steklov Mathematical Institute

Stepan Kuznetsov

C. Sc. (equivalent to Ph. D.)

## About

62

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Introduction

Substructural logics with structural (subexponential) and fixpoint modalities; their applications to linguistics and computer science

## Publications

Publications (62)

Adding multi-modalities (called subexponentials) to linear logic enhances its power as a logical framework, which has been extensively used in the specification of e.g. proof systems, programming languages and bigraphs. Initially, subexponentials allowed for classical, linear, affine or relevant behaviors. Recently, this framework was enhanced so t...

Adding multi-modalities (called subexponentials) to linear logic enhances its power as a logical framework, which has been extensively used in the specification of e.g. proof systems, programming languages and bigraphs. Initially, subexponentials allowed for classical, linear, affine or relevant behaviors. Recently, this framework was enhanced so t...

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 the Lambek calculus extended with intersection (meet) operation. For its variant which does not allow empty antecedents, Andréka and Mikulás (1994) prove strong completeness w.r.t. relational models (R-models). Without the antecedent non-emptiness restriction, however, only weak completeness w.r.t. R-models (so-called square ones) holds...

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 allows some of the structural rules (contraction, weakening, permutation). The logic is presented in the form of an infinitary sequent calculus. We prove cut el...

Infinitary action logic (\(\mathbf {ACT}_\omega \)) can be viewed as an extension of the multiplicative-additive Lambek calculus (\(\mathbf {MALC}\)) with iteration (Kleene star) governed by an omega-rule (Buszkowski, Palka 2007). An alternative formulation utilizes non-well-founded proofs instead of the omega-rule (Das, Pous 2017). Another unary o...

Action logic is the algebraic logic (inequational theory) of residuated Kleene lattices. One of the operations of this logic is the Kleene star, which is axiomatized by an induction scheme. For a stronger system that uses an -rule instead (infinitary action logic), Buszkowski and Palka (2007) proved -completeness (thus, undecidability). Decidabilit...

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...

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 mod...

In the original publication, the affiliation of the author Max Kanovich was processed incorrectly. It has been updated in this correction.

We prove undecidability and pinpoint the place in the arithmetical hierarchy for commutative action logic, that is, the equational theory of commutative residuated Kleene lattices (action lattices), and infinitary commutative action logic, the equational theory of *-continuous action lattices. Namely, we prove that the former is $\Sigma_1^0$-comple...

We consider commutative infinitary action logic, that is, the equational theory of commutative *-continuous action lattices, and show that its derivability problem is \(\varPi _1^0\)-complete. Thus, we obtain a commutative version of \(\varPi _1^0\)-completeness for non-commutative infinitary action logic by Buszkowski and Palka (2007). The proof o...

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 Π10-complete. As shown recently by Kuznetsov and Speranski, the extension of infinitary action logic with the exponential modality is much harder: Π11-complete. T...

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...

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 mod...

The Lambek calculus (a variant of intuitionistic linear logic initially introduced for mathematical linguistics) enjoys natural interpretations over the algebra of formal languages (L-models) and over the algebra of binary relations which are subsets of a given transitive relation (R-models). For both classes of models there are completeness theore...

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...

Linear logic and its refinements have been used as a specification language for a number of deductive systems. This has been accomplished by carefully studying the structural restrictions of linear logic modalities. Examples of such refinements are subexponentials, light linear logic, and soft linear logic. We bring together these refinements of li...

We consider the Lambek calculus, or non-commutative 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 fo...

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 allows some of the structural rules (contraction, weakening, permutation). The logic is presented in the form of an infinitary sequent calculus. We prove cut...

This Festschrift is in honor of Prof. Andre Scedrov at the University of Pennsylvania. Scedrov has laid the foundations for a number of now well-established domains in mathematics and computer science including Proof Theory, Logic in Computer Science, Foundations in Computer Security, and Linguistics.
This combination of breadth and penetrating ori...

Action logic is the algebraic logic (inequational theory) of residuated Kleene lattices. This logic involves Kleene star, axiomatized by an induction scheme. For a stronger system which uses an $\omega$-rule instead (infinitary action logic) Buszkowski and Palka (2007) have proved $\Pi_1^0$-completeness (thus, undecidability). Decidability of actio...

In his recent papers “Parsing/theorem-proving for logical grammar CatLog3” and “A note on movement in logical grammar”, Glyn Morrill proposes a new substructural calculus to be used as the basis for the categorial grammar parser CatLog3. In this paper we prove that the derivability problem for a fragment of this calculus is algorithmically undecida...

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 s...

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 distribu...

We show that Craig’s trick is not valid for the Lambek calculus, i.e. there exists such a recursively enumerable theory (set of sequents) over the Lambek calculus, which does not have a decidable axiomatization. We show that Lambek’s non-emptiness restriction (the constraint that left-hand sides of all sequents should be non-empty) and an infinite...

Relativisation involves dependencies which, although unbounded, are constrained with respect to certain island domains. The Lambek calculus L can provide a very rudimentary account of relativisation limited to unbounded peripheral extraction; the Lambek calculus with bracket modalities Lb can further condition this account according to island domai...

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...

Relativisation involves dependencies which, although unbounded, are constrained with respect to certain island domains. The Lambek calculus L can provide a very rudimentary account of relativisation limited to unbounded peripheral extraction; the Lambek calculus with bracket modalities Lb can further condition this account according to island domai...

We present a translation of the Lambek calculus with brackets and the unit constant, $\mathbf{Lb}^{\boldsymbol{*}}_{\mathbf{1}}$, into the Lambek calculus with brackets allowing empty antecedents, but without the unit constant, $\mathbf{Lb}^{\boldsymbol{*}}$. Using this translation, we extend previously known results for $\mathbf{Lb}^{\boldsymbol{*...

Linear logical frameworks with subexponentials have been used for the specification of among other systems, proof systems, concurrent programming languages and linear authorization logics. In these frameworks, subexponentials can be configured to allow or not for the application of the contraction and weakening rules while the exchange rule can alw...

The Lambek calculus is a well-known logical formalism for modelling natural language syntax. The original calculus covered a substantial number of intricate natural language phenomena, but only those restricted to the context-free setting. In order to address more subtle linguistic issues, the Lambek calculus has been extended in various ways. In p...

We define infinitary count-invariance for categorial logic, extending count-invariance for multiplicatives (van Benthem, 1991) and additives and bracket modalities (Valentín et al., 2013) to include exponentials. This provides an e↵ective tool for pruning proof search in categorial parsing/theorem-proving.

We define infinitary count-invariance for categorial logic, extending count-invariance for multiplicatives (van Benthem, 1991) and additives and bracket modalities (Valentín et al., 2013) to include exponentials. This provides an e↵ective tool for pruning proof search in categorial parsing/theorem-proving.

Formulae of the Lambek calculus are constructed using three binary connectives, multiplication and two divisions. We extend it using a unary connective, positive Kleene iteration. For this new operation, following its natural interpretation, we present two lines of calculi. The first one is a fragment of infinitary action logic and includes an omeg...

Lambek calculus is a logical foundation of categorial grammar, a linguistic paradigm of grammar as logic and parsing as deduction. Pentus (2010) gave a polynomial-time algorithm for determ- ining provability of bounded depth formulas in the Lambek calculus with empty antecedents allowed. Pentus' algorithm is based on tabularisation of proof nets. L...

Morrill and Valentín in the paper “Computational coverage of TLG: Nonlinearity” considered an extension of the Lambek calculus enriched by a so-called “exponential” modality. This modality behaves in the “relevant” style, that is, it allows contraction and permutation, but not weakening. Morrill and Valentín stated an open problem whether this syst...

Morrill and Valentin in the paper "Computational coverage of TLG: Nonlinearity" considered two extensions of the Lambek calculus with so-called "exponential" modalities. These calculi serve as a basis for describing fragments of natural language via categorial grammars. In this paper we show undecidability of derivability problems in these two calc...

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," that is, 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 e...

We describe a method of translating a Lambek grammar with one division into an equivalent context-free grammar whose size is bounded by a polynomial in the size of the original grammar. Earlier constructions by Buszkowski and Pentus lead to exponential growth of the grammar size.

Morrill and Valentin in the paper "Computational coverage of TLG: Nonlinearity" considered an extension of the Lambek calculus enriched by a so-called "exponential" modality. This modality behaves in the "relevant" style, that is, it allows contraction and permutation, but not weakening. Morrill and Valentin stated an open problem whether this syst...

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,” that is, 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 e...

We consider context-free grammars and Lambek grammars enriched with semantic labeling. Such grammars do not just answer whether a given word belongs to the language described by the grammar, but, if the answer is positive, also assign the word a λ-term that corresponds to the semantic value (“meaning”) of the word. We present a modification of W. B...

We consider language models for the Lambek calculus that allow empty antecedents and enrich them with constants for the empty language and for the language containing only the empty word. No complete calculi are known with respect to these semantics, and in this paper we consider several trivalent systems that arise as fragments of these models’ lo...

In this paper we prove that the Lambek calculus allowing empty antecedents and enriched with a unary connective corresponding to language reversal is complete with respect to the class of models on subsets of free monoids (L-models).

We prove that any language without the empty word, generated by a conjunctive grammar in Greibach normal form, is generated by a grammar based on the Lambek calculus enriched with additive (“intersection” and “union”) connectives.

We extend the Lambek calculus with rules for a unary operation corresponding to language reversal and prove that this calculus is complete with respect to the class of models on subsets of free semigroups (L-models). We also prove that categorial grammars based on this calculus generate precisely all context-free languages without the empty word.

We prove that a formal language without the empty word is context-free if and only if it is generated by some L(∖; p1)-grammar, where L(∖; p1) is the Lambek calculus with one division and one primitive type. To do that, we use a substitution of types which reduces
derivability in L(∖) to derivability in L(∖; p1). We also prove that a formal languag...

Pentus’ theorem states that any language generated by a Lambek grammar is context-free. We present a substitution that reduces the Lambek calculus enriched with the unit constant to the variant of the Lambek calculus that does not contain the unit (but still allows empty premises), and use this substitution to prove that any language generated by a...

In this paper we present a substitution that reduces the derivability in the Lambek calculus with a unit and one division
to the derivability in the Lambek calculus with one division permitting empty antecedents. Using this substitution, we establish
the existence of an algorithm checking the derivability in the Lambek calculus with a unit and one...