Glyn Morrill

Categorial Grammar: Logical Syntax, Semantics, and Processing

The title of this book, Categorial Grammar: Logical Syntax, Semantics, and Processing, indicates that this will be a book about categorial grammar. In the preface, however, several strains of categorial grammar are discussed, including combinatory categorial grammar (Steedman 2000), pregroup grammar (Lambek 1999), and abstract categorial grammar (de Groote 2001), and it is clarified that the book investigates only the tradition of type-logical grammar, or rather, those grammars based on Lambek categorial grammar (Lambek 1958). That being said, this book contains a good introduction to type-logical grammar and its first part would make a good textbook in an advanced course on the theory of type-logical grammar. In particular, exercises are sprinkled throughout the book that will be illuminating to the uninitiated reader. The book is neatly divided into three parts that are likely to be of varying levels of interest depending on the specific audience. Part I, titled “Lambek Categorial Grammar,” gives a concise introduction to a number of aspects of Lambek categorial grammar, which is suitable for an audience interested in the basic intuitions and mechanics of that grammar. Part II, titled “Logical Categorial Grammar,” introduces a number of extensions of Lambek’s grammar, each of which are motivated by linguistic considerations. This part is likely to be of relevance to linguists who are interested in the descriptive capabilities of type-logical grammar, but little attention is paid to computational aspects in this part. Part III is a collection of remaining topics that are loosely connected by their attention to processing. This part is directed towards those interested in psycho-linguistics and its connections to type-logical grammars. Section 1 of Part I introduces the origin of type-logical grammar, which is found in the grammar of Ajdukiewicz (1935) and connected to Montague’s semantics (Montague and Thomason 1974). Section 2 introduces the syntax of Lambek categorial grammar, including the introduction of both a proof theory and a model theory. This section then discusses the cut elimination proof for the Lambek calculus, the logical system behind Lambek categorial grammar. Section 3 introduces the semantics for Lambek categorial grammar, including a discussion of the Curry–Howard isomorphism and its relevance to Lambek categorial grammar. Both Sections 2 and 3 include example sentences from English and analyses for those sentences that give the reader an insight into how typelogical grammar is applied to linguistic data. Section 4, titled “Processing,” introduces proof nets for Lambek categorial grammar as a more natural representation of typelogical syntax and semantics. Human performance on garden-path sentences is used to motivate the use of both proof nets, in particular, and type-logical grammar, more generally.

The Displacement Calculus

If all dependent expressions were adjacent some variety of immediate constituent analysis would suffice for grammar, but syntactic and semantic mismatches are characteristic of natural language; indeed this is a, or the, central problem in grammar. Logical categorial grammar reduces grammar to logic: an expression is well-formed if and only if an associated sequent is a theorem of a categorial logic. The paradigmatic categorial logic is the Lambek calculus, but being a logic of concatenation the Lambek calculus can only capture discontinuous dependencies when they are peripheral. In this paper we present the displacement calculus, which is a logic of intercalation as well as concatenation and which subsumes the Lambek calculus. On the empirical side, we apply the new calculus to discontinuous idioms, quantification, VP ellipsis, medial extraction, pied-piping, appositive relativisation, parentheticals, gapping, comparative subdeletion, cross-serial dependencies, reflexivization, anaphora, dative alternation, and particle shift. On the technical side, we prove that the calculus enjoys Cut-elimination.

Incremental processing and acceptability

We describe a left-to-right incremental procedure for the processing of Lambek categorial grammar by proof net construction. A simple metric of complexity, the profile in time of the number of unresolved valencies, correctly predicts a wide variety of performance phenomena including garden pathing, the unacceptability of center embedding, preference for lower attachment, left-to-right quantifier scope preference, and heavy noun phrase shift.

Parsing and derivational equivalence

It is a tacit assumption of much linguistic inquiry that all distinct derivations of a string should assign distinct meanings. But despite the tidiness of such derivational uniqueness, there seems to be no a priori reason to assume that a grammar must have this property. If a grammar exhibits derivational equivalence, whereby distinct derivations of a string assign the same meanings, naive exhaustive search for all derivations will be redundant, and quite possibly intractable. In this paper we show how notions of derivation-reduction and normal form can be used to avoid unnecessary work while parsing with grammars exhibiting derivational equivalence. With grammar regarded as analogous to logic, derivations are proofs; what we are advocating is proof-reduction, and normal form proof; the invocation of these logical techniques adds a further paragraph to the story of parsing-as-deduction.

Discontinuity in categorial grammar

Discontinuity refers to the character of many natural language constructions wherein signs differ markedly in their prosodic and semantic forms. As such it presents interesting demands on monostratal computational formalisms which aspire to descriptive adequacy. Pied piping, in particular, is argued by Pollard (1988) to motivate phrase structure-style feature percolation. In the context of categorial grammar, Bach (1981, 1984), Moortgat (1988, 1990, 1991) and others have sought to provide categorial operators suited to discontinuity. These attempts encounter certain difficulties with respect to model theory and/or proof theory, difficulties which the current proposals are intended to resolve.Lambek calculus is complete for interpretation byresiduation with respect to the adjunction operation of groupoid algebras (Buszkowski 1986). In Moortgat and Morrill (1991) it is shown how to give calculi for families of categorial operators, each defined by residuation with respect to an operation of prosodic adjunction (associative, non-associative, or with interactive axioms). The present paper treats discontinuity in this way, by residuation with respect to three adjunctions: + (associative), (.,.) (split-point marking), andW (wrapping) related by the equations1+s2+s3=(s1,s3)Ws2. We show how the resulting methods apply to discontinuous functors, quantifier scope and quantifier scope ambiguity, pied piping, and subject and object antecedent reflexivisation.

Higher-order linear logic programming of categorial deduction

We show how categorial deduction can be implemented in higher-order (linear) logic programming, thereby realising parsing as deduction for the associative and non-associative Lambek calculi. This provides a method of solution to the parsing problem of Lambek categorial grammar applicable to a variety of its extensions.

Dutch Grammar and Processing: A Case Study in TLG

The aim of this paper is to see type logical grammar (TLG) at work on an interesting linguistic case: the incremental processing of Dutch subordinate clause word order, namely the so-called cross-serial dependencies. With the help of proof net machinery adapted for the continuous and discontinuous Lambek calculus we are able to account for the increasing unacceptability of cross-serial dependencies with increasingly multiple embeddings.

Logic programming of the displacement calculus

The displacement calculus of Morrill, Valentin and Fadda (2011)[12] forms a foundation for type logical categorial grammar in which discontinuity is accommodated alongside continuity in a logic which is free of structural rules and which enjoys Cut-elimination, the subformula property, decidability, and the finite reading property. The calculus deploys a new kind of sequent calculus which we call hypersequent calculus in which types and configurations have not only external context but also internal context, in the case that they are discontinuous. In this paper we consider the logic programming of backward chaining hypersequent proof search for the displacement calculus. We show how focusing eliminates all spurious ambiguity in the fragment without antecedent tensors and we illustrate coding of the essential features of displacement. In this way we lay a basis for parsing/theorem proving for this calculus, which is being used and extended in a system CatLog currently under development.

Proof Nets for Basic Discontinuous Lambek Calculus

The theory of proof nets for continuity based on the Lambek calculus is well-developed, but we need a compatible extension to include discontinuity. Earlier work set out ingredients: hypersequent calculus and proof nets expanded with parameter edges. This article completes a preliminary line by finalizing a version of proof nets for the basic discontinuous Lambek calculus BDLC (the minimal system with one point of discontinuity) and proving correctness with respect to the hypersequent calculus.

A Count Invariant for Lambek Calculus with Additives and Bracket Modalities

The count invariance of van Benthem (1991[16]) is that for a sequent to be a theorem of the Lambek calculus, for each atom, the number of positive occurrences equals the number of negative occurrences. (The same is true for multiplicative linear logic.) The count invariance provides for extensive pruning of the sequent proof search space. In this paper we generalize count invariance to categorial grammar (or linear logic) with additives and bracket modalities. We define by mutual recursion two counts, minimum count and maximum count, and we prove that if a multiplicative-additive sequent is a theorem, then for every atom, the minimum count is less than or equal to zero and the maximum count is greater than or equal to zero; in the case of a purely multiplicative sequent, minimum count and maximum count coincide in such a way as to together reconstitute the van Benthem count criterion. We then define in the same way a bracket count providing a count check for bracket modalities. This allows for efficient pruning of the sequent proof search space in parsing categorial grammar with additives and bracket modalities.