This section provides the notational conventions and syntactic framework for the subsequent formal discussion of DATR's semantics (Section 4) and theory of inference (Section 5).
Let NODE and ATOM be finite sets of symbols. Elements of NODE
are called nodes and denoted by N. Elements of ATOM are
called atoms and denoted by a. Elements of 
are
called values and denoted by 
, 
, 
.
The set DESC of DATR value descriptors (or simply
descriptors), and denoted by d, is built up from
the nodes and atoms as shown below. In the following, sequences of
descriptors ( 
),in 
are denoted 
, 
.
for any 
and 
:
Value descriptors are either atoms or inheritance descriptors, where an inheritance descriptor is further distinguished as either local (unquoted) or global (quoted). There is just one kind of local descriptor (node/path), but three kinds of global descriptor (node/path, path and node). The syntax presented informally in Section 3.1.2, above, and in E&G (1989a, 1989b) permits nodes and paths to stand as local descriptors. However, these additional forms can be viewed as conventional abbreviations, in the appropriate syntactic context, for node/path pairs.
A path 
is a (possibly empty)
sequence of atoms enclosed in angle brackets. Paths are denoted by P.
For N a node, P a path and 
a (possibly empty)
sequence of atoms, an equation of the form 
is
called an extensional sentence. Intuitively, an extensional
sentence states that the value associated with the
path P at node N is . For a (possibly empty) sequence
of value descriptors, an equation of the form 
is
called a definitional sentence. A definitional sentence
specifies a property of the node N, namely that
the path P is associated with the value defined by the sequence of
value descriptors .
A collection of equations can be used to specify the properties of
different nodes in terms of one another, and a finite set of DATR\
sentences 
is called a DATR theory. In principle, a
DATR theory may consist of any combination of DATR\
sentences, either definitional or extensional, but in practice, DATR\
theories are more restricted than this. The theory is said to
be definitional if it consists solely of definitional sentences
and it is said to be functional if it meets the following
condition:
Functionality for DATR theories, as defined above, is
really a syntactic notion. However, it approximates a deeper, semantic
requirement that the nodes should correspond to (partial) functions from
paths to values
. Functionality is discussed in greater detail
in Section 3.1.5, below.
In the formal semantics (4) and inference
(5) sections of this document,
we will use the term (DATR)
theory always in the sense functional, definitional (DATR)
theory. For a given DATR theory and node N of
, we write 
to denote that subset of the sentences in
that relate to the node N. That is:
The set is referred to as the definition of N
(in ).