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 languages. The basic ingredient are tensor terms, which can be seen as encoding and generalizing proof-nets. Using tensor terms makes the syntax extremely simple and a direct geometric meaning becomes transparent. Then we address the problem of encoding noncommutative operations in our setting. This turns out possible after enriching the system with new unary operators. The resulting system allows representing both ACG and Lambek grammars as conservative fragments, while the formalism remains, as it seems to us, rather simple and intuitive.
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 both systems, we resolve issues connected with the cut rule and provide necessary modifications, after which we prove admissibility of cut (cut elimination theorem). We also prove algorithmic undecidabi…
Nancy Université（南锡大学）；Université Paris 8（巴黎第八大学）；Université de Bordeaux 1（波尔多第一大学）
We first recall some basic notions on minimalist grammars and on categorial grammars. Next we shortly introduce partially commutative linear logic, and our representation of minimalist grammars within this categorial system, the so-called categorial minimalist grammars. Thereafter we briefly present λμ-DRT (Discourse Representation Theory) an extension of λ-DRT (compositional DRT) in the framework of λμcalculus: it avoids type raising and derives different readings from a single semantic representation, in a setting which follows discourse structure. We run a complete example which illustrates…
In addition to their limpid interface with semantics, categorial grammars enjoy another important property: learnability. This was first noticed by Buskowsky and Penn and further studied by Kanazawa, for Bar-Hillel categorial grammars. What about Lambek categorial grammars? In a previous paper we showed that product free Lambek grammars where learnable from structured sentences, the structures being incomplete natural deductions. These grammars were shown to be unlearnable from strings by Foret and Le Nir. In the present paper we show that Lambek grammars, possibly with product, are learnable …
National Research University Higher School of Economics（国立研究大学高等经济学院）
We propose a categorial grammar based on classical multiplicative linear logic. This can be seen as an extension of abstract categorial grammars (ACG) and is at least as expressive. However, constituents of {\it linear logic grammars (LLG)} are not abstract $λ$-terms, but simply tuples of words with labeled endpoints and supplied with specific {\it plugging instructions}: the sets of endpoints are subdivided into the {\it incoming} and the {\it outgoing} parts. We call such objects {\it word cobordisms}. A key observation is that word cobordisms can be organized in a category, very similar to …
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 $\mathbf{Lb}^{\boldsymbol{*}}$ to $\mathbf{Lb}^{\boldsymbol{*}}_{\mathbf{1}}$: (1) languages generated by categorial grammars based on the Lambek calculus with brackets are context-free (Kanazawa 2017); (2) the polynomial-time algorithm for deciding derivability of bounded dep…
National Research University Higher School of Economics（俄罗斯国立高等经济大学）
We propose a categorial grammar based on classical multiplicative linear logic. This can be seen as an extension of abstract categorial grammars (ACG) and is at least as expressive. However, constituents of {\it linear logic grammars (LLG)} are not abstract $λ$-terms, but simply tuples of words with labeled endpoints, we call them {\it multiwords}. At least, this gives a concrete and intuitive representation of ACG. A key observation is that the class of multiwords has a fundamental algebraic structure. Namely, multiwords can be organized in a category, very similar to the category of topologi…
A well-known approach to treating syntactic island constraints in the setting of Lambek grammars consists in adding specific bracket modalities to the logic. We adapt this approach to abstract categorial grammars (ACG). Thus we define bracketed (implicational) linear logic, bracketed lambda-calculus, and, eventually, bracketed ACG based on bracketed $λ$-calculus. This allows us modeling at least simplest island constraints, typically, in the context of relativization. Next we identify specific safely bracketed ACG which, just like ordinary (bracket-free) second order ACG generate effectively d…
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…
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 calcul…
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 ($\ma…
LIRMM（蒙彼利埃大学与法国国家科学研究中心联合计算机科学、机器人与微电子实验室）；Université de Montpellier（蒙彼利埃大学）；CNRS（法国国家科学研究中心）；The Ohio State University（俄亥俄州立大学）
This paper explores proof-theoretic aspects of hybrid type-logical grammars , a logic combining Lambek grammars with lambda grammars. We prove some basic properties of the calculus, such as normalisation and the subformula property and also present both a sequent and a proof net calculus for hybrid type-logical grammars. In addition to clarifying the logical foundations of hybrid type-logical grammars, the current study opens the way to variants and extensions of the original system, including but not limited to a non-associative version and a multimodal version incorporating structural rules …
The Distributional Compositional Categorical (DisCoCat) model is a mathematical framework that provides compositional semantics for meanings of natural language sentences. It consists of a computational procedure for constructing meanings of sentences, given their grammatical structure in terms of compositional type-logic, and given the empirically derived meanings of their words. For the particular case that the meaning of words is modelled within a distributional vector space model, its experimental predictions, derived from real large scale data, have outperformed other empirically validate…
We develop a vector space semantics for verb phrase ellipsis with anaphora using type-driven compositional distributional semantics based on the Lambek calculus with limited contraction (LCC) of Jäger (2006). Distributional semantics has a lot to say about the statistical collocation-based meanings of content words, but provides little guidance on how to treat function words. Formal semantics on the other hand, has powerful mechanisms for dealing with relative pronouns, coordinators, and the like. Type-driven compositional distributional semantics brings these two models together. We review pr…
