A theory such as the AVM-based account of English compounds sketched above, may simultaneously describe any number of models. A model may be formal, such as a set-theoretic representation of an empirical domain, or more informal, as is generally the case in descriptive linguistics, formulated in plain text enriched with symbols and line drawings. A theory is simply a subset of sentences in a formalism for which a model exists in terms of which the sentences can be interpreted.
One kind of formal model for a theory is an `implementation', i.e. an interpretation of the theory in terms of an operational knowledge representation language or programming language. This is actually a special case of a more general kind of formal interpretation; interpretations for AVMs have been given, for example, in terms of finite state automata (see [Kasper & Rounds 1986]). An interpretation of a theory in terms of a different but perhaps more well-known formalism permits conclusions to be drawn about whether the theory is complete (describes all it is supposed to describe) and sound (does not describe anything it is not supposed to describe). If the interpretation function is bijective, then in principle the model could be regarded as the theory and the theory as the model; this is then just a question of perspective.
In this sense, the lexical representation formalism DATR will be used to provide an operational model for the theory which permits quick consistency checking of complex theories by the automatic deduction of hypotheses. Descriptions in DATR are, however, generally referred to as `theories'.
DATR `theories', used here as `operational models' for AVM theories, are sets of DATR sentences. DATR sentences are pairs of a node and a set of equations, each of which is a pair of an attribute path and a value.
In the ILEX approach, therefore, a lexicon is an AVM theory which describes an operational model formulated in DATR; this model can itself be seen as an empirical theory which is interpreted (like the AVM theory) in terms of an empirical model with observationally identifiable categories.