Michael Moortgat

Chapter 2 – Categorial Type Logics

Publisher Summary This chapter describes the framework of categorial type logic—that is, grammar architecture that can be seen as the logical development of the categorial approach to natural language analysis initiated in the 1930s. For the reader with a background in linguistics, it tries to provide a useful compendium of the logical tools and results one needs to appreciate the categorial research. The reader with a logic background is justified in classifying the grammar formalism discussed in the chapter under the rubric applied logic. In organizing the material, one can opt for a “historical” mode of development or for a state-of-the-art presentation of the internal dynamics of the field. The chapter presents an interpretation of the categorial formalism in terms of structural composition of grammatical resources. The central objective of the type-logical approach is to develop a uniform deductive account of the composition of form and meaning in natural language: formal grammar is presented as logic—a system for reasoning about structured linguistic resources.

Symmetric Categorial Grammar

The Lambek-Grishin calculus is a symmetric version of categorial grammar obtained by augmenting the standard inventory of type-forming operations (product and residual left and right division) with a dual family: coproduct, left and right difference. Interaction between these two families is provided by distributivity laws. These distributivity laws have pleasant invariance properties: stability of interpretations for the Curry-Howard derivational semantics, and structure-preservation at the syntactic end. The move to symmetry thus offers novel ways of reconciling the demands of natural language form and meaning.

Syntactic Annotation for the Spoken Dutch Corpus Project (CGN)

Of the ten million words of contemporary standard Dutch in the Spoken Dutch Corpus (Corpus Gesproken Nederlands, CGN), a selection of one million words of natural spoken language will be annotated syntactically. In the present paper we discuss the tag sets and the annotation procedures that are currently being developed and tested. The annotation tags provide information about syntactic constituents and about the semantic relations (dependencies) between these constituents. The annotation graphs allow crossing branches, which makes it possible to represent dependencies independently of surface word order. Moreover, constituents can carry multiple dependency roles, a feature that is exploited in the annotation of non-local dependencies and ellipsis. The annotation process is carried out semi-automatically, using an interactive annotation environment developed within the NEGRA project, a syntactically annotated corpus of German newspaper texts. We illustrate the approach with some real life examples from the CGN corpus, focusing on how some typical spoken language phenomena are dealt with.

Symmetries in natural language syntax and semantics: the Lambek-Grishin calculus

In this paper, we explore the Lambek-Grishin calculus LG: a symmetric version of categorial grammar based on the generalizations of Lambek calculus studied in Grishin [1]. The vocabulary of LG complements the Lambek product and its left and right residuals with a dual family of type-forming operations: coproduct, left and right difference. The two families interact by means of structure-preserving distributivity principles. We present an axiomatization of LG in the style of Currys combinatory logic and establish its decidability. We discuss Kripke models and Curry-Howard interpretation for LG and characterize its notion of type similarity in comparison with the other categorial systems. From the linguistic point of view, we show that LG naturally accommodates non-local semantic construal and displacement -- phenomena that are problematic for the original Lambek calculi.

Multimodal Linguistic Inference

In this paper we compare grammatical inference in the context of simple and of mixed Lambek systems. Simple Lambek systems are obtained by taking the logic of residuation for a family of multiplicative connectives /,•,\, together with a package of structural postulates characterizing the resource management properties of the • connective.Different choices for Associativity and Commutativity yield the familiar logics NL, L, NLP, LP. Semantically, a simple Lambek system is a unimodal logic: the connectives get a Kripke style interpretation in terms of a single ternary accessibility relation modeling the notion of linguistic composition for each individual system.

Continuation semantics for symmetric categorial grammar

Categorial grammars in the tradition of Lambek [1,2] are asymmetric: sequent statements are of the form Γ ⇒ A, where the succedent is a single formula A, the antecedent a structured configuration of formulas A1, ...,An. The absence of structural context in the succedent makes the analysis of a number of phenomena in natural language semantics problematic. A case in point is scope construal: the different possibilities to build an interpretation for sentences containing generalized quantifiers and related expressions. In this paper, we explore a symmetric version of categorial grammar based on work by Grishin [3]. In addition to the Lambek product, left and right division, we consider a dual family of type-forming operations: coproduct, left and right difference. Communication between the two families is established by means of structure-preserving distributivity principles. We call the resulting system LG.We present a Curry-Howard interpretation for LG(/, \,???,???) derivations. Our starting point is Curien and Herbelins sequent system for λµ calculus [4] which capitalizes on the duality between logical implication (i.e. the Lambek divisions under the formulas-as-types perspective) and the difference operation. Importing this system into categorial grammar requires two adaptations: we restrict to the subsystemwhere linearity conditions are in effect, and we refine the interpretation to take the left-right symmetry and absence of associativity/commutativity into account. We discuss the continuation-passing-style (CPS) translation, comparing the call-by-value and call-by-name evaluation regimes. We show that in the latter (but not in the former) the types ofLGare associated with appropriate denotational domains to enable a proper treatment of scope construal.

Mixed Composition and Discontinuous Dependencies

Consider a directional categorial grammar with the rule of functional application A and, in addition, the two type-shifting rules below, lifting L and division D. For obvious reasons I will call this system L after Lambek’s (1958) syntactic calculus, where these reduction rules have the status of theorems. I single out lifting and division among the theorems of Lambek’s calculus because of their prominent role in linguistic discussion. As will appear from the format, fractional categories are projected from the vertical into the horizontal mode by giving them a quarter turn clockwise. This gives the fractional categories a domain-range structure, which makes it easy to think of them either as syntactic or semantic types: X | Y is a functor which combines syntactically with an expression of type X to give an expression of type Y; semantically it is interpreted as a function f: type (X) → type(Y). (In what follows, I shall refer to the type of a syntactic or semantic object X as ‘t(X)’.) Directionality is encoded in the form of the fraction sign, back-slash standing for left-concatenation, slash for right-concatenation. The vertical slant is used to conflate the modes of concatenation, so that for example one interpretation schema can serve for a pair of directional reduction rules.

Relational semantics for the Lambek-Grishin calculus

We study ternary relational semantics for LG: a symmetric version of the Lambek calculus with interaction principles due to Grishin [10]. We obtain completeness on the basis of a Henkin-style weak filter construction.

Structural Equations in Language Learning

In categorial systems with a fixed structural component, the learning problem comes down to finding the solution for a set of typeassignment equations. A hard-wired structural component is problematic if one want to address issues of structural variation. Our starting point is a type-logical architecture with separate modules for the logical and the structural components of the computational system. The logical component expresses invariants of grammatical composition; the structural component captures variation in the realization of the correspondence between form and meaning. Learning in this setting involves finding the solution to both the type-assignment equations and the structural equations of the language at hand. We develop a view on these two subtasks which pictures learning as a process moving through a two-stage cycle. In the first phase of the cycle, type assignments are computed statically from structures. In the second phase, the lexicon is enhanced with facilities for structural reasoning. These make it possible to dynamically relate structures during on-line computation, or to establish off-line lexical generalizations. We report on the initial experiments in [15] to apply this method in the context of the Spoken Dutch Corpus. For the general type-logical background, we refer to [12]; §1 has a brief recap of some key features.

Assessment of Odour Nuisance Problems in Flanders: a Practical Approach

Abstract The strategy of the Laboratory for Organic Chemistry, University of Gent, for the assessment of odour pollution problems consists of three main parts; 1. Sensory evaluation of the immission using the technique of plotting out odour perception areas and calculation of the total odour emission rate (OER) based on the maximum odour perception distance. 2. Calculation of the impact of the emission on a yearly basis using a frequency-distribution model based on small sampling times. 3. Chemical analysis of important emission points. A practical example shows the application of the strategy in the food industry.