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. Most versions of categorial grammar analyze sentence structure in terms of constituencies (as opposed to dependencies) and are therefore phrase structure grammars (as opposed to dependency grammars).
Contents
Basics
A categorial grammar consists of two parts: a lexicon, which assigns a set of types (also called categories) to each basic symbol, and some type inference rules, which determine how the type of a string of symbols follows from the types of the constituent symbols. It has the advantage that the type inference rules can be fixed once and for all, so that the specification of a particular language grammar is entirely determined by the lexicon.
A categorial grammar shares some features with the simply typed lambda calculus. Whereas the lambda calculus has only one function type
The notation is based upon algebra. A fraction when multiplied by (i.e. concatenated with) its denominator yields its numerator. As concatenation is not commutative, it makes a difference whether the denominator occurs to the left or right. The concatenation must be on the same side as the denominator for it to cancel out.
The first and simplest kind of categorial grammar is called a basic categorial grammar, or sometimes an AB-grammar (after Ajdukiewicz and Bar-Hillel). Given a set of primitive types
A basic categorial grammar is a tuple
The relation
Such a grammar for English might have three basic types
For example, take the string "the bad boy made that mess". Now "the" and "that" are determiners, "boy" and "mess" are nouns, "bad" is an adjective, and "made" is a transitive verb, so the lexicon is {
and the sequence of types in the string is
now find functions and appropriate arguments and reduce them according to the two inference rules
The fact that the result is
Categorial grammars of this form (having only function application rules) are equivalent in generative capacity to context-free grammars 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.
Lambek calculus
A Lambek grammar is an elaboration of this idea that has a concatenation operator for types, and several other inference rules. Pentus has shown that these still have the generative capacity of context-free grammars.
For the Lambek calculus, there is a type concatenation operator
The Lambek calculus consists of several deduction rules, which specify how type inclusion assertions can be derived. In the following rules, upper case roman letters stand for types, upper case Greek letters stand for sequences of types. A sequent of the form
The process is begun by the Axiom rule, which has no antecedents and just says that any type includes itself.
The Cut rule says that inclusions can be composed.
The other rules come in pairs, one pair for each type construction operator, each pair consisting of one rule for the operator in the target, one in the source, of the arrow. The name of a rule consists of the operator and an arrow, with the operator on the side of the arrow on which it occurs in the conclusion.
For an example, here is a derivation of "type raising", which says that
Relation to context-free grammars
Recall that a context-free grammar is a 4-tuple:
1.
2.
3.
4.
From the point of view of categorial grammars, a context-free grammar can be seen as a calculus with a set of special purpose axioms for each language, but with no type construction operators and no inference rules except Cut.
Specifically, given a context-free grammar as above, define a categorial grammar
Of course, this is not a basic categorial grammar, since it has special axioms that depend upon the language; i.e. it is not lexicalized. Also, it makes no use at all of non-primitive types.
To show that any context-free language can be generated by a basic categorial grammar, recall that any context-free language can be generated by a context-free grammar in Greibach normal form.
The grammar is in Greibach normal form if every production rule is of the form
Now given a CFG in Greibach normal form, define a basic categorial grammar with a primitive type for each non-terminal variable
The same construction works for Lambek grammars, since they are an extension of basic categorial grammars. It is necessary to verify that the extra inference rules do not change the generated language. This can be done and shows that every context-free language is generated by some Lambek grammar.
To show the converse, that every language generated by a Lambek grammar is context-free, is much more difficult. It was an open problem for nearly thirty years, from the early 1960s until about 1991 when it was proven by Pentus.
The basic idea is, given a Lambek grammar,
Of course, there are infinitely many types and infinitely many derivable sequents, so in order to make a finite grammar it is necessary put a bound on the size of the types and sequents that are needed. The heart of Pentus's proof is to show that there is such a finite bound.
Notation
The notation in this field is not standardized. The notations used in formal language theory, logic, category theory, and linguistics, conflict with each other. In logic, arrows point to the more general from the more particular, that is, to the conclusion from the hypotheses. In this article, this convention is followed, i.e. the target of the arrow is the more general (inclusive) type.
In logic, arrows usually point left to right. In this article this convention is reversed for consistency with the notation of context-free grammars, where the single non-terminal symbol is always on the left. We use the symbol
Some authors on categorial grammars write
Some definitions
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..
Discontinuity
The grammar is extended to handle linguistic phenomena such as discontinuous idioms, gapping and extraction.