DATR theories can be viewed semantically as collections of definitions of partial functions (nodes in DATR parlance) that map paths onto values. A model of a DATR theory is then an assignment of functions to node symbols that is consistent with the definitions of those nodes within the theory. This picture of DATR\ as a formalism for defining partial functions is complicated by two features of the language, however. First, the meaning of a given node depends, in general, on the global context of interpretation, so that nodes do not correspond directly to mappings from paths to values, but rather to functions from contexts to such mappings. Second, it is necessary to provide an account of DATR's default mechanism. It will be convenient to present our account of the semantics of DATR in two stages. The present section considers a restricted version of DATR without the default mechanism. Section 4.2 then shows how implicit information can be modelled by treating value descriptors as families of values indexed by paths.
Elements of the set U are denoted by u and elements of 
are
denoted by v. Intuitively, is the domain of (semantic)
values/paths. Elements of the set 
are called
contexts and denoted by c. The function 
can be thought of
as mapping global contexts onto (partial) functions from local contexts
to values.
The function F is extended to paths, so that for 
( 
) we write F(P) to denote 
, where 
for each i ( 
).
Figure 1: Denotation function for DATR descriptors
Intuitively, value descriptors denote elements of (as we
shall see, this will need to be revised later in order to account for
DATR's default mechanism). We associate with the interpretation 
a partial denotation function 
and write 
to denote the
meaning (value) of descriptor d in the global context c. The
denotation function is defined as shown in Figure 1.
Note that an atom always denotes the same element of U, regardless of the
context. By contrast, the denotation of an inheritance descriptor is, in
general, sensitive to the global context c in which it appears. Note
also that in the case of a global inheritance descriptor, the global
context is effectively altered to reflect the new local context c'.
The denotation function is extended to sequences of value descriptors in
the obvious way. Thus, for 
( ), we write
to denote 
if 
( ) is defined (and is
undefined otherwise).
Now, let be an
interpretation and 
a theory. We will write
to denote that partial function from to
given by
It is easy to verify that does indeed denote a
partial function (it follows from the functionality of the theory
). Let us also write 
to denote that partial
function from to given by 
, for all 
. Then, I models
just in case the following containment holds for each node N
and context c:
That is, an interpretation is a model of a DATR theory just in case (for each global context) the function it associates with each node respects the definition of that node within the theory.
