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.Comment: In Proceedings TERMGRAPH 2022, arXiv:2303.1421

Making first order linear logic a generating grammar

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

Advances in Abstract Categorial Grammars: Language Theory and Linguistic Modeling. ESSLLI 2009 Lecture Notes, Part II

These are the lecture notes of the ESSLLI 2009 second week course on Abstract Categorial Grammar. This was an advanced course, while an introductory course was given the first week: Introduction to Abstract Categorial Grammars: Foundations and main properties, delivered by Philippe de Groote and Sylvain Salvati. An up-to-date version of these notes, possible errata, and generally ACG papers, can be found at the ACG homepage: http://calligramme.loria.fr/acg. At this URL, the ACG Development Toolkit is also available and downloadable as free software. It might be useful to run some of the examples given in these notes. In particular, some of the latter are gathered in a special file. The Abstract Categorial Grammar (ACG) framework, a grammar formalism based on the typed lambda calculus, elegantly generalizes and unifies a variety of grammar formalisms that have been proposed for the description of formal and natural languages. The first part of the course investigated formal-language-theoretic properties of "second-order" ACGs, a subclass of ACGs that have "context-free" derivations. Their generative capacity was precisely characterized. Lecture notes for this part are available at http://research.nii.ac.jp/~kanazawa/publications/esslli2009_lectures.pdf. An efficient Earley-style algorithm, suitable both for parsing and generation, was then presented. The second part of the course (these present notes) presents linguistic applications of ACGs and gives various illustrations of how ACGs provide flexible and explicit ways to model the syntax-semantics interface of natural language