Lexica as theories of minimal language objects

It is rather common to describe a grammar as a theory of the structure of a language, and the linguistic theory on which it is based as a theory of grammar. Many contemporary linguistic theories are lexicalist, in that they do not regard the smallest units of grammar as atoms, but as complex objects, usually represented by feature structures (attribute-value structures), whose internal representation determines the structures into which they can enter. For instance, in such theories, the grammatical category PASSIVE is not regarded as naming a grammatical rule or structure, but as a combinatorial property of a class of transitive verbs. Typical lexicalist theories are Lexical Functional Grammar, Head-Driven Phrase Structure Grammar, the lexicalist variety of Tree Adjunction Grammar, the many flavours of Categorial Grammar and, to a lesser extent, Government and Binding Theory.

We go a small step further with the following claims in the context of the ILEX model:

Theory: A lexicon is a theory, though perhaps in many cases an informal textual one. A computational lexicon, however, has to be formally well-defined.

Model: A lexicon is based on specific modelling conventions which define the ontology of language in terms of finite sets of lexical units -- in effect, the macrostructure and microstructure of the lexicon. The lexicon as a theory is interpreted in terms of the semantic and surface modelling conventions, and these, if formulated rigourously, constitute a formal model.

Formalism: A lexicon theory is formulated within a formal language, a formalism, or `metasyntax', and can be characterised as a set of sentences in the formalism which are interpretable by a given lexicon model. A formalism has three essential components:

1. Syntax: a definition of the categories of the formalism in terms of the symbols which represent them, and their combinatorial properties.
2. Procedural semantics: a definition of how to derive further theorems from axioms, the basic theorems, of theories defined within the formalism (often seen as an extension of syntax, rather than as semantics).
3. Denotational semantics: a definition of objects in some universe in terms of which categories of the syntax are interpreted (sometimes callled declarative semantics).

Notation: A notation is a vocabulary of characters and symbols used to encode a formalism; for example, dots, crosses or asterisks are different notations for multiplication, and SAMPA characters, IPA symbols and numerical codes are different notations for phonetic transcription categories (cf. [Gibbon, Moore & Winski 1997], appendices).

Implementation: The computer program is a model for the theory, and the runtime environment or virtual machine is an operational model (also known as operational semantics) corresponding to the procedural semantics of the theory. The programming language thus corresponds to the formalism in which the theory is defined. Sometimes a programming language with a particularly close relation to a logical or algebraic formalism is also called a formalism (e.g. DATR, ALEP, ALE, CUF, STUF, ...); in cases like these, the theory is the `programme' and the software is the `procedural semantics' for the theory. This software functions as a theorem prover, in that the input consists of axioms of the theory and specific statements about specific objects, and the output consists of derived theorems which are `predicted' by the theory.

 
Figure 3: Lexical formalisms and theories.