Categorial grammar

Categorial grammar is a term used for a family of formalisms in natural language syntax motivated by the principle of compositionality and organized according to the view that syntactic constituents should generally combine as functions or according to a function-argument relationship.

Basics of categorial grammar

A categorial grammar shares some features with the simply-typed lambda calculus.Whereas the lambda calculus has only one function type "A → B", a categorial grammar typically has more. For example, a simple categorial grammar for English might have two function types "A/B" and "AB", depending on whether the function takes its argument from the left or the right. Such a grammar would have only two rules: left and right function application. Such a grammar might have three basic categories ("N","NP", and "S"), putting count nouns in the category "N", adjectives in the category "N/N", determiners in the category "NP/N", names in the category "NP", intransitive verbs in the category "NPS", and transitive verbs in the category "(NPS)/NP". Categorial grammars of this form (having only function application rules) are equivalent in generative capacity to context-free grammar and are thus often considered inadequate for theories of natural language syntax. Unlike CFGs, categorial grammars are lexicalized, meaning that only a small number of (mostly language-independent) rules are employed, and all other syntactic phenomena derive from the lexical entries of specific words.

Another appealing aspect of categorial grammars is that it is often easy to assign them a compositional semantics, by first assigning interpretation types to all the basic categories, and then associating all the derived categories with appropriate function types. The interpretation of any constituent is then simply the value of a function at an argument. With some modifications to handle intensionality and quantification, this approach can be used to cover a wide variety of semantic phenomena.

Historical notes

The basic ideas of categorial grammar date from work by Kazimierz Ajdukiewicz (in 1935) and Yehoshua Bar-Hillel (in 1953). In 1958, Joachim Lambek introduced a syntactic calculus that formalized the function type constructors along with various rules for the combination of functions. This calculus is a forerunner of
linear logic in that it is a substructural logic. Montague grammar uses an ad hoc syntactic system for English that is based on the principles of categorial grammar. Although Montague's work is sometimes regarded as syntactically uninteresting, it helped to bolster interest in categorial grammar by associating it with a highly successful formal treatment of natural language semantics. More recent work in categorial grammar has focused on the improvement of syntactic coverage. One formalism which has received considerable attention in recent years is Steedman and Szabolcsi's combinatory categorial grammar which builds on combinatory logic invented by Moses Schönfinkel and Haskell Curry.

There are a number of related formalisms of this kind in linguistics, such as type logical grammar.

ome definitions

*DerivationA derivation is a binary tree that encodes a proof.
*Parse tree
*Functor and ArgumentIn a left (right) function application, the node of the type AB (A/B) is called the functor, and the node of the type A is called an argument.
*Functor-argument structure

Refinements of categorial grammar

A variety of changes to categorial grammar have been proposed to improve syntactic coverage. Some of the most common ones are listed below.

Features and subcategories

Most systems of categorial grammar subdivide categories. The most common way to do this is by tagging them with features, such as person, gender, number, and tense. Sometimes only atomic categories are tagged in this way. In Montague grammar, it is traditional to subdivide function categories using a multiple slash convention, so "A/B" and "A//B" would be two distinct categories of left-applying functions, that took the same arguments but could be distinguished between by other functions taking them as arguments.

Function composition

Rules of function composition are included in many categorial grammars. An example of such a rule would be one that allowed the concatenation of a constituent of type "A/B" with one of type "B/C" to produce a new constituent of type "A/C". The semantics of such a rule would simply involve the composition of the functions involved. Function composition is important in categorial accounts of conjunction and extraction, especially as they relate to phenomena like right node raising. The introduction of function composition into a categorial grammar leads to many kinds of derivational ambiguity that are vacuous in the sense that they do not correspond to semantic ambiguities.

Conjunction

Many categorial grammars include a typical conjunction rule, of the general form "X CONJ X → X", where "X" is a category. Conjunction can generally be applied to nonstandard constituents resulting from type raising or function composition..

Type raising

Rules of type raising allow one to convert an expression of category "X" into one of category "Y/(YX)" or "Y(Y/X)", for some other category "Y". These rules essentially reverse the function-argument relationship.

References

*Curry, Haskell B. and Richard Feys (1958), Combinatory Logic, Vol. 1. North-Holland.
*Jacobson, Pauline (1999), “Towards a variable-free semantics.” Linguistics and Philosophy 22, 1999. pp. 117-184
*Steedman, Mark (1987),” Combinatory grammars and parasitic gaps”. Natural Language and Linguistic Theory 5, 403-439.
*Steedman, Mark (1996), Surface Structure and Interpretation. The MIT Press.
*Steedman, Mark (2000), The Syntactic Process. The MIT Press.
*Szabolcsi, Anna (1989), "Bound variables in syntax (are there any?)." Semantics and Contextual Expression, ed. by Bartsch, van Benthem, and van Emde Boas. Foris, 294-318.
*Szabolcsi, Anna (1992), "Combinatory grammar and projection from the lexicon." Lexical Matters. CSLI Lecture Notes 24, ed. by Sag and Szabolcsi. Stanford, CSLI Publications. 241-269.
*Szabolcsi, Anna (2003), “Binding on the fly: Cross-sentential anaphora in variable-free semantics”. Resource Sensitivity in Binding and Anaphora, ed. by Kruijff and Oehrle. Kluwer, 215-229.