arXiv subjects

Tikhon Pshenitsyn

Publications and source records attributed to Tikhon Pshenitsyn.

17 recordsLinked to original sources

arXiv2026

Wider systems for linear logic with fixed points: proof theory and complexity

We investigate infinitary wellfounded systems for linear logic with fixed points, with transfinite branching rules indexed by some closure ordinal $\alpha$ for fixed points. Our main result is that provability in the system for some computable ordinal $\alpha$ is complete for the $\omega^{\alpha^\omega}$ level of the hyperarithmetical hierarchy. To this end we first develop proof theoretic foundations, namely cut elimination and focussing results, to control both the upper and lower bound analysis. Our arguments employ a carefully calibrated notion of formula rank, calculating a tight bound on the height of the (cut-free) proof search space.

math.LO↗

arXiv2025

Extending Action Logic with Omega Iteration

We present a proof system that extends action logic by omega iteration, which is viewed as infinitary multiplicative conjunction. We prove cut admissibility and establish complexity bounds for the provability predicate.

cs.LO↗

arXiv2025

First-Order Intuitionistic Linear Logic and Hypergraph Languages

The Lambek calculus is a substructural logic known to be closely related to the formal language theory: on the one hand, it is used for generating formal languages by means of categorial grammars and, on the other hand, it has formal language semantics, with respect to which it is sound and complete. This paper studies a similar relation between first-order intuitionistic linear logic ILL1 along with its multiplicative fragment MILL1 on the one hand and the hypergraph grammar theory on the other. In the first part, we introduce a novel concept of hypergraph first-order logic categorial grammar, which is a generalisation of string MILL1 grammars studied e.g. in Richard Moot's 2014 works. We prove that hypergraph ILL1 grammars generate all recursively enumerable hypergraph languages and that hypergraph MILL1 grammars are as powerful as linear-time hypergraph transformation systems. In addition, we show that the class of languages generated by string MILL1 grammars is closed under intersection and that it includes a non-semilinear language as well as an NP-complete one. This shows how much more powerful string MILL1 grammars are as compared to Lambek categorial grammars. In the second part, we develop hypergraph language models for MILL1. In such models, formulae of the logic are interpreted as hypergraph languages and multiplicative conjunction is interpreted using parallel composition, which is one of the operations of HR-algebras introduced by Courcelle. We prove completeness of the universal-implicative fragment of MILL1 with respect to these models and thus present a new kind of semantics for a fragment of first-order linear logic.

math.LO↗

arXiv2025

On Decidability and Expressive Power of Fusion Grammars

We study algorithmic complexity and expressive power of fusion grammars, a novel formalism introduced in [Kreowski, Kuske, and Lye 2017], which extends hyperedge replacement grammars. In the first part of the work, we prove that the non-emptiness problem for fusion grammars and the membership problem for fusion grammars without markers and connectors are decidable and are in NEXPTIME. We introduce fusion grammars with bounded usage of markers and connectors and prove decidability of the membership problem for them as well. In the proofs, we develop the technique of hypergraph vertex colourings encoded in hyperedge labels and also the technique of evidence paths and their encodings. In the second part of the work, we study the class of languages generated by connection-preserving fusion grammars. Namely, we prove Parikh's theorem for them, i.e. we show that these languages are semilinear.

cs.FL↗

arXiv2025

Reasoning from hypotheses in *-continuous action lattices

The class of all $\ast$-continuous Kleene algebras, whose description includes an infinitary condition on the iteration operator, plays an important role in computer science. The complexity of reasoning in such algebras - ranging from the equational theory to the Horn one, with restricted fragments of the latter in between - was analyzed by Kozen (2002). This paper deals with similar problems for $\ast$-continuous residuated Kleene lattices, also called $\ast$-continuous action lattices, where the product operation is augmented by residuals. We prove that, in the presence of residuals, the fragment of the corresponding Horn theory with $\ast$-free hypotheses has the same complexity as the $ω^ω$ iteration of the halting problem, and hence is properly hyperarithmetical. We also prove that if only commutativity conditions are allowed as hypotheses, then the complexity drops down to $Π^0_1$ (i.e. the complement of the halting problem), which is the same as that for $\ast$-continuous Kleene algebras. In fact, we get stronger upper bound results: the fragments under consideration are translated into suitable fragments of infinitary action logic with exponentiation, and our upper bounds are obtained for the latter ones.

math.LO↗

arXiv2025

Uniform Membership for Hyperedge Replacement Grammars and Related Decision Problems

This paper investigates complexity of the uniform membership problem for hyperedge replacement grammars in comparison with other mildly context-sensitive grammar formalisms. It turns out that the complexity of this problem depends on how one defines a hypergraph. There are two commonly used definitions in the field, which differ in whether repetitions of attachment nodes of a hyperedge are allowed in a hypergraph or not. We show that, in general, the problem under consideration is EXPTIME-complete, even for string-generating hyperedge replacement grammars, but it is NP-complete if repetitions are not allowed. We extend the developed proof techniques in order to prove a general meta-theorem: checking whether a given hyperedge replacement grammar generates a hypergraph satisfying a non-Parikh property is EXPTIME-hard. Non-Parikh properties are those that are not preimages of properties on Parikh vectors of hypergraphs. This includes any graph property relying significantly on structure of graphs, e.g. connectivity, Eulerianity, Hamiltonianity, acyclicity. A tight upper bound is established for EXPTIME-compatible properties via Filter Theorem.

cs.FL↗

arXiv2024

Bonding Grammars

We introduce bonding grammars, a graph grammar formalism developed to model DNA computation by means of graph transformations. It is a modification of fusion grammars introduced by Kreowski, Kuske and Lye in 2017. Bonding is a graph transformation that consists of merging two hyperedges into a single larger one. We show why bonding models interaction between DNA molecules better than fusion. Then, we investigate formal properties of this formalism. Firstly, we study the relation between bonding grammars and hyperedge replacement grammars proving that each of these kinds of grammars generates a language the other one cannot generate. Secondly, we prove that bonding grammars naturally generalise regular sticker systems. Finally, we prove that the membership problem for bonding grammars is NP-complete and, moreover, that some bonding grammar generates an NP-complete set.

cs.FL↗

arXiv2023

Hyperarithmetical Complexity of Infinitary Action Logic with Multiplexing

In 2023, Kuznetsov and Speranski introduced infinitary action logic with multiplexing $!^m\nabla \mathrm{ACT}_ω$ and proved that the derivability problem for it lies between the $ω$ and $ω^ω$ levels of the hyperarithmetical hierarchy. We prove that this problem is $Δ^0_{ω^ω}$-complete under Turing reductions. Namely, we show that it is recursively isomorphic to the satisfaction predicate for computable infinitary formulas of rank less than $ω^ω$ in the language of arithmetic. As a consequence we prove that the closure ordinal for $!^m\nabla \mathrm{ACT}_ω$ equals $ω^ω$. We also prove that the fragment of $!^m\nabla \mathrm{ACT}_ω$ where Kleene star is not allowed to be in the scope of the subexponential is $Δ^0_{ω^ω}$-complete. Finally, we present a family of logics, which are fragments of $!^m\nabla \mathrm{ACT}_ω$, such that the complexity of the $k$-th logic lies between $Δ^0_{ω^k}$ and $Δ^0_{ω^{k+1}}$.

math.LO↗

arXiv2023

Expressive Power of Hypergraph Lambek Grammars

Hypergraph Lambek grammars (HL-grammars) is a novel logical approach to generating graph languages based on the hypergraph Lambek calculus. In this paper, we establish a precise relation between HL-grammars and hypergraph grammars based on the double pushout (DPO) approach: we prove that HL-grammars generate the same class of languages as DPO grammars with the linear restriction on lengths of derivations. This can be viewed as a complete description of the expressive power of HL-grammars and also as an analogue of the Pentus theorem, which states that Lambek grammars generate the same class of languages as context-free grammars. As a corollary, we prove that HL-grammars subsume contextual hyperedge replacement grammars.

cs.FL↗

arXiv2023

Multimodality in the Hypergraph Lambek Calculus

The multimodal Lambek calculus is an extension of the Lambek calculus that includes several product operations (some of them being commutative or/and associative), unary modalities, and corresponding residual implications. In this work, we relate this calculus to the hypergraph Lambek calculus HL. The latter is a general pure logic of residuation defined in a sequent form; antecedents of its sequents are hypergraphs, and the rules of HL involve hypergraph transformation. Our main result is the embedding of the multimodal Lambek calculus (with at most one associative product) in HL. It justifies that HL is a very general Lambek-style logic and also provides a novel syntactic interface for the multimodal Lambek calculus: antecedents of sequents of the multimodal Lambek calculus are represented as tree-like hypergraphs in HL, and they are derived from each other by means of hyperedge replacement. The advantage of this embedding is that commutativity and associativity are incorporated in the sequent structure rather than added as separate rules. Besides, modalities of the multimodal Lambek calculus are represented in HL using the product and the division of HL, which explicitizes their residual nature.

cs.LO↗

arXiv2023

From Double Pushout Grammars to Hypergraph Lambek Grammars With and Without Exponential Modality

We study how to relate well-known hypergraph grammars based on the double pushout (DPO) approach and grammars over the hypergraph Lambek calculus HL (called HL-grammars). It turns out that DPO rules can be naturally encoded by types of HL using methods similar to those used by Kanazawa for multiplicative-exponential linear logic. In order to generalize his reasonings we extend the hypergraph Lambek calculus by adding the exponential modality, which results in a new calculus HMEL0; then we prove that any DPO grammar can be converted into an equivalent HMEL0-grammar. We also define the conjunctive Kleene star, which behaves similarly to this exponential modality, and establish a similar result. If we add neither the exponential modality nor the conjunctive Kleene star to HL, then we can still use the same encoding and show that any DPO grammar with a linear restriction on the length of derivations can be converted into an equivalent HL-grammar.

cs.LO↗

arXiv2022

Grammars over the Lambek Calculus with Permutation: Recognizing Power and Connection to Branching Vector Addition Systems with States

In (Van Benthem, 1991) it is proved that all permutation closures of context-free languages can be generated by grammars over the Lambek calculus with the permutation rule (LP-grammars); however, to our best knowledge, it is not established whether the converse holds or not. In this paper, we show that LP-grammars are equivalent to linearly-restricted branching vector addition systems with states and with additional memory (shortly, lBVASSAM), which are modified branching vector addition systems with states. Then an example of such an lBVASSAM is presented, which generates a non-semilinear set of vectors; this yields that LP-grammars generate more than permutation closures of context-free languages. Moreover, equivalence of LP-grammars and lBVASSAM allows us to present a normal form for LP-grammars and, as a consequence, prove that LP-grammars are equivalent to LP-grammars without product. Finally, we prove that the class of languages generated by LP-grammars is closed under intersection.

cs.FL↗

arXiv2021

Grammars Based on a Logic of Hypergraph Languages

The hyperedge replacement grammar (HRG) formalism is a natural and well-known generalization of context-free grammars. HRGs inherit a number of properties of context-free grammars, e.g. the pumping lemma. This lemma turns out to be a strong restriction in the hypergraph case: it implies that languages of unbounded connectivity cannot be generated by HRGs. We introduce a formalism that turns out to be more powerful than HRGs while having the same algorithmic complexity (NP-complete). Namely, we introduce hypergraph Lambek grammars; they are based on the hypergraph Lambek calculus, which may be considered as a logic of hypergraph languages. We explain the underlying principles of hypergraph Lambek grammars, establish their basic properties, and show some languages of unbounded connectivity that can be generated by them (e.g. the language of all graphs, the language of all bipartite graphs, the language of all regular graphs).

cs.FL↗

arXiv2021

Cyclic Shift in the Lambek Calculus

We enrich the Lambek calculus with the cyclic shift operation, which is expected to model the closure operator of formal languages with respect to cyclic shifts. We introduce a Gentzen-style calculus and prove cut elimination. Secondly, we turn to categorial grammars based on this calculus and show that they can generate non-context-free languages; besides, we consider a related calculus where the cyclic shift is a structural rule, and compare recognizing power of these two calculi. Thirdly, we attempt to embed the Lambek calculus with the cyclic shift operation in the hypergraph Lambek calculus. This results in considering a ``bracelet'' operation, which can be defined through the cyclic shift, union, and the reversal operation.

math.LO↗

arXiv2021

Introduction to a Hypergraph Logic Unifying Different Variants of the Lambek Calculus

In this paper hypergraph Lambek calculus ($\mathrm{HL}$) is presented. This formalism aims to generalize the Lambek calculus ($\mathrm{L}$) to hypergraphs as hyperedge replacement grammars extend context-free grammars. In contrast to the Lambek calculus, $\mathrm{HL}$ deals with hypergraph types and sequents; its axioms and rules naturally generalize those of $\mathrm{L}$. Consequently, certain properties (e.g. the cut elimination) can be lifted from $\mathrm{L}$ to $\mathrm{HL}$. It is shown that $\mathrm{L}$ can be naturally embedded in $\mathrm{HL}$; moreover, a number of its variants ($\mathrm{LP}$, $\mathrm{NL}$, $\mathrm{NLP}$, $\mathrm{L}$ with modalities, $\mathrm{L}^\ast(\mathbf{1})$, $\mathrm{L}^{\mathrm{R}}$) can also be embedded in $\mathrm{HL}$ via different graph constructions. We also establish a connection between $\mathrm{HL}$ and Datalog with embedded implications. It is proved that the parsing problem for $\mathrm{HL}$ is NP-complete.

math.LO↗

arXiv2020

Weak Greibach Normal Form for Hyperedge Replacement Grammars

It is known that hyperedge replacement grammars are similar to string context-free grammars in the sense of definitions and properties. Therefore, we expect that there is a generalization of the well-known Greibach normal form from string grammars to hypergraph grammars. Such generalized normal forms are presented in several papers; however, they do not cover a large class of hypergraph languages (e.g. languages consisting of star graphs). In this paper, we introduce a weak Greibach normal form, whose definition corresponds to the lexicalized normal form for string grammars, and prove that every context-free hypergraph language (with nonsubstantial exceptions) can be generated by a grammar in this normal form. The proof presented in this paper generalizes a corresponding one for string grammars with a few more technicalities.

cs.FL↗

arXiv2020

Hypergraph Lambek Calculus

It is known that context-free grammars can be extended to generating graphs resulting in graph grammars; one of such fundamental approaches is hyperedge replacement grammars. On the other hand there are type-logical grammars which also serve to describe string languages. In this paper, we investigate how to extend the Lambek calculus ($\mathrm{L}$) and grammars based on it to graphs. The resulting approach is called hypergraph Lambek calculus ($\mathrm{HL}$). It is a logical sequential calculus whose sequents are graphs; it naturally extends the Lambek calculus and also allows one to embed its variants (commutative $\mathrm{L}$, $\mathrm{NL\diamondsuit}$, $\mathrm{L}_{\mathbf{1}}^\ast$). Besides, many properties of the Lambek calculus (cut elimination, counters, models) can be lifted to $\mathrm{HL}$. However, while Lambek grammars are equivalent to context-free grammars in the string case, hypergraph Lambek grammars are much more powerful than hyperedge replacement grammars. Particularly, the former can generate the language of all graphs without isolated nodes; the language of all bipartite graphs; finite intersections of languages generated by hyperedge replacement grammars. Nevertheless, the derivability problem in $\mathrm{HL}$ and the membership problem for grammars based on $\mathrm{HL}$ are NP-complete as well as the membership problem for hyperedge replacement grammars.

math.LO↗