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This paper describes SPINDLE - an open source Python module implementing an efficient and accurate parser for written Dutch that transforms raw text input to programs for meaning composition, expressed as {\lambda} terms. The parser integrates a number of breakthrough advances made in recent years. Its output consists of hi-res derivations of a multimodal type-logical grammar, capturing two orthogonal axes of syntax, namely deep function-argument structures and dependency relations. These are produced by three interdependent systems: a static type-checker asserting the well-formedness of grammatical analyses, a state-of-the-art, structurally-aware supertagger based on heterogeneous graph convolutions, and a massively parallel proof search component based on Sinkhorn iterations. Packed in the software are also handy utilities and extras for proof visualization and inference, intended to facilitate end-user utilization., Comment: EACL23 System Demonstrations

In this paper I will present a novel way of combining proof net proof search with neural networks. It contrasts with the 'standard' approach which has been applied to proof search in type-logical grammars in various different forms. In the standard approach, we first transform words to formulas (supertagging) then match atomic formulas to obtain a proof. I will introduce an alternative way to split the task into two: first, we generate the graph structure in a way which guarantees it corresponds to a lambda-term, then we obtain the detailed structure using vertex labelling. Vertex labelling is a well-studied task in graph neural networks, and different ways of implementing graph generation using neural networks will be explored., Comment: This is an extended version of an invited talk for the workshop End-to-End Compositional Models of Vector-Based Semantics

We present a proof-theoretic analysis of the logic NL$\lambda$ (Barker \& Shan 2014, Barker 2019). We notably introduce a novel calculus of proof nets and prove it is sound and complete with respect to the sequent calculus for the logic. We study decidability and complexity of the logic using this new calculus, proving a new upper bound for complexity of the logic (showing it is in NP) and a new lower bound for the class of formal language generated by the formalism (mildly context-sensitive languages extended with a permutation closure operation). Finally, thanks to this new calculus, we present a novel comparison between NL$\lambda$ and the hybrid type-logical grammars of Kubota \& Levine (2020). We show there is an unexpected convergence of the natural language analyses proposed in the two formalism. In addition to studying the proof-theoretic properties of NL$\lambda$, we greatly extends its linguistic coverage.

Linear logic and the linear {\lambda}-calculus have a long standing tradition in the study of natural language form and meaning. Among the proof calculi of linear logic, proof nets are of particular interest, offering an attractive geometric representation of derivations that is unburdened by the bureaucratic complications of conventional prooftheoretic formats. Building on recent advances in set-theoretic learning, we propose a neural variant of proof nets based on Sinkhorn networks, which allows us to translate parsing as the problem of extracting syntactic primitives and permuting them into alignment. Our methodology induces a batch-efficient, end-to-end differentiable architecture that actualizes a formally grounded yet highly efficient neuro-symbolic parser. We test our approach on {\AE}Thel, a dataset of type-logical derivations for written Dutch, where it manages to correctly transcribe raw text sentences into proofs and terms of the linear {\lambda}-calculus with an accuracy of as high as 70%., Comment: 14 pages, CoNLL2020

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 and unary modes.

In this chapter, we introduce a new dialogical system for first order classical logic which is close to natural language argumentation, and we prove its completeness with respect to usual classical validity. We combine our dialogical system with the Grail syntactic and semantic parser developed by the second author in order to address automated textual entailment, that is, we use it for deciding whether or not a sentence is a consequence of a short text. This work-which connects natural language semantics and argumentation with dialogical logic-can be viewed as a step towards an inferentialist view of natural language semantics., Comment: Natural Language Processing in Artificial Intelligence, In press

We will investigate proof-theoretic and linguistic aspects of first-order linear logic. We will show that adding partial order constraints in such a way that each sequent defines a unique linear order on the antecedent formulas of a sequent allows us to define many useful logical operators. In addition, the partial order constraints improve the efficiency of proof search., Comment: 33 pages

We present {\AE}THEL, a semantic compositionality dataset for written Dutch. {\AE}THEL consists of two parts. First, it contains a lexicon of supertags for about 900 000 words in context. The supertags correspond to types of the simply typed linear lambda-calculus, enhanced with dependency decorations that capture grammatical roles supplementary to function-argument structures. On the basis of these types, {\AE}THEL further provides 72 192 validated derivations, presented in four formats: natural-deduction and sequent-style proofs, linear logic proofnets and the associated programs (lambda terms) for meaning composition. {\AE}THEL's types and derivations are obtained by means of an extraction algorithm applied to the syntactic analyses of LASSY Small, the gold standard corpus of written Dutch. We discuss the extraction algorithm and show how `virtual elements' in the original LASSY annotation of unbounded dependencies and coordination phenomena give rise to higher-order types. We suggest some example usecases highlighting the benefits of a type-driven approach at the syntax semantics interface. The following resources are open-sourced with {\AE}THEL: the lexical mappings between words and types, a subset of the dataset consisting of 7 924 semantic parses, and the Python code that implements the extraction algorithm., Comment: 8 pages plus abstract, LREC 2020

The short note describes the chart parser for multimodal type-logical grammars which has been developed in conjunction with the type-logical treebank for French. The chart parser presents an incomplete but fast implementation of proof search for multimodal type-logical grammars using the "deductive parsing" framework. Proofs found can be transformed to natural deduction proofs.

We present a proof net calculus for the Displacement calculus and show its correctness. This is the first proof net calculus which models the Displacement calculus directly and not by some sort of translation into another formalism. The proof net calculus opens up new possibilities for parsing and proof search with the Displacement calculus., Comment: Formal Grammar, Aug 2016, Bolzano, Italy. Springer, Proceedings of Formal Grammar 2016

This paper is a reflexion on the computability of natural language semantics. It does not contain a new model or new results in the formal semantics of natural language: it is rather a computational analysis of the logical models and algorithms currently used in natural language semantics, defined as the mapping of a statement to logical formulas - formulas, because a statement can be ambiguous. We argue that as long as possible world semantics is left out, one can compute the semantic representation(s) of a given statement, including aspects of lexical meaning. We also discuss the algorithmic complexity of this process.

This report first shows the equivalence bewteen several formulations of classical logic in intuitionistic logic (tertium non datur, reductio ad absurdum, Pierce's law). Then it establishes the correctness of the G\"odel-Kolmogorov translation, whose restriction to the propositional case is due to Glivenko. This translation maps a formula $F$ of first order logic to a formula $F^{\lnot\lnot}$ in such a way that $F$ is provable in classical logic if and only if $F^{\lnot\lnot}$ is provable in intuitionistic logic. All formal proofs are presented in natural deduction. These questions are well-known proof theoretical facts, but in textbooks, they are often ignored or left to the reader. Because of the combinatorial difficulty of some of the needed formal proofs, we hope that this report may be useful, in particular to students and colleagues from other areas.

As the name suggests, type-logical grammars are a grammar formalism based on logic and type theory. From the prespective of grammar design, type-logical grammars develop the syntactic and semantic aspects of linguistic phenomena hand-in-hand, letting the desired semantics of an expression inform the syntactic type and vice versa. Prototypical examples of the successful application of type-logical grammars to the syntax-semantics interface include coordination, quantifier scope and extraction.This chapter describes the Grail theorem prover, a series of tools for designing and testing grammars in various modern type-logical grammars which functions as a tool . All tools described in this chapter are freely available.

Lambeks Syntactic Calculus, commonly referred to as the Lambek calculus, was innovative in many ways, notably as a precursor of linear logic. But it also showed that we could treat our grammatical framework as a logic (as opposed to a logical theory). However, though it was successful in giving at least a basic treatment of many linguistic phenomena, it was also clear that a slightly more expressive logical calculus was needed for many other cases. Therefore, many extensions and variants of the Lambek calculus have been proposed, since the eighties and up until the present day. As a result, there is now a large class of calculi, each with its own empirical successes and theoretical results, but also each with its own logical primitives. This raises the question: how do we compare and evaluate these different logical formalisms? To answer this question, I present two unifying frameworks for these extended Lambek calculi. Both are proof net calculi with graph contraction criteria. The first calculus is a very general system: you specify the structure of your sequents and it gives you the connectives and contractions which correspond to it. The calculus can be extended with structural rules, which translate directly into graph rewrite rules. The second calculus is first-order (multiplicative intuitionistic) linear logic, which turns out to have several other, independently proposed extensions of the Lambek calculus as fragments. I will illustrate the use of each calculus in building bridges between analyses proposed in different frameworks, in highlighting differences and in helping to identify problems., Comment: Empirical advances in categorial grammars, Aug 2015, Barcelona, Spain. 2015

In this article we show that hybrid type-logical grammars are a fragment of first-order linear logic. This embedding result has several important consequences: it not only provides a simple new proof theory for the calculus, thereby clarifying the proof-theoretic foundations of hybrid type-logical grammars, but, since the translation is simple and direct, it also provides several new parsing strategies for hybrid type-logical grammars. Second, NP-completeness of hybrid type-logical grammars follows immediately. The main embedding result also sheds new light on problems with lambda grammars/abstract categorial grammars and shows lambda grammars/abstract categorial grammars suffer from problems of over-generation and from problems at the syntax-semantics interface unlike any other categorial grammar.

We developed a type-theoretical framework for natural lan- guage semantics that, in addition to the usual Montagovian treatment of compositional semantics, includes a treatment of some phenomena of lex- ical semantic: coercions, meaning, transfers, (in)felicitous co-predication. In this setting we see how the various readings of plurals (collective, dis- tributive, coverings,...) can be modelled.

First-order multiplicative intuitionistic linear logic (MILL1) can be seen as an extension of the Lambek calculus. In addition to the fragment of MILL1 which corresponds to the Lambek calculus (of Moot & Piazza 2001), I will show fragments of MILL1 which generate the multiple context-free languages and which correspond to the Displacement calculus of Morrilll e.a., Comment: Logic and Language, Allemagne (2013)

Grishin's generalization of Lambek's Syntactic Calculus combines a non-commutative multiplicative conjunction and its residuals (product, left and right division) with a dual family: multiplicative disjunction, right and left difference. Interaction between these two families takes the form of linear distributivity principles. We study proof nets for the Lambek-Grishin calculus and the correspondence between these nets and unfocused and focused versions of its sequent calculus., Comment: Revised version to appear as a chapter in E. Grefenstette, C. Heunen, and M. Sadrzadeh (eds.) 'Compositional Methods in Physics and Linguistics', Oxford University Press

Proof nets are a graph theoretical representation of proofs in various fragments of type-logical grammar. In spite of this basis in graph theory, there has been relatively little attention to the use of graph theoretic algorithms for type-logical proof search. In this paper we will look at several ways in which standard graph theoretic algorithms can be used to restrict the search space. In particular, we will provide an O(n4) algorithm for selecting an optimal axiom link at any stage in the proof search as well as a O(kn3) algorithm for selecting the k best proof candidates.

This paper explores several extensions of proof nets for the Lambek calculus in order to handle the different connectives of display logic in a natural way. The new proof net calculus handles some recent additions to the Lambek vocabulary such as Galois connections and Grishin interactions. It concludes with an exploration of the generative capacity of the Lambek-Grishin calculus, presenting an embedding of lexicalized tree adjoining grammars into the Lambek-Grishin calculus.