Implicit information and default models

  The notion of a model presented in the preceding section is too liberal in that it takes no account of information implicit in a theory. For example, consider again the definition of the node Love from the example theory of Section 2, and repeated below.

    Love:
        <> == VERB
        <mor root> == love.

According to the definition of a model given previously, any model of that theory will associate with the node Love a function from paths to values which respects the above definition. This means that for every global context c, the following containment must hold (in this and subsequent examples in this section, syntactic objects, e.g., love, <mor root>) are used to stand for their semantic counterparts under F (i.e., $F(\mbox{\tt love})$, $F({\tt <mor \: root\gt})$, respectively):

$$\begin{array} {lcl} \left[\!\left[ \mbox {\tt Love} \right]\... ...le {\tt <mor \; root\gt}, \mbox{\tt love} \rangle \}\end{array}$$

On the other hand, there is no guarantee that a given model will also respect the following containment:

$$\begin{array} {lcl} \left[\!\left[ \mbox {\tt Love} \right]\... ...<mor \; root \; root\gt}, \mbox{\tt love} \rangle \}\end{array}$$

In fact, this containment (amongst other things) should hold. It follows by default from the statements made about Love that the path <mor> inherits locally from VERB and that the value associated with any extension of <mor root> is love.

There have been a number of formal treatments of defaults in the setting of attribute-value formalisms. All these approaches formalize a notion of default inheritance by defining appropriate operations (e.g., default unification) for combining strict and default information. Strict information is allowed to over-ride default information where the combination would otherwise lead to inconsistency (i.e., unification failure). In the case of DATR however, the formalism does not draw an explicit distinction between strict and default values for paths. In fact, all of the information given explicitly in a DATR theory is strict. The non-monotonic nature of DATR theories arises from a general, default mechanism which fills in the gaps by supplying values for paths not explicitly specified in a theory. More specifically, DATR's default mechanism ensures that any path that is not explicitly specified for a given node will take its definition from the longest prefix of that path that is specified. Thus, the default mechanism defines a class of implicit, definitional sentences with paths on the left that extend paths found on the left of explicit sentences. Furthermore, this extension of paths is also carried over to paths occurring on the right. In effect, each (explicit) path is associated not just with a single value specification, but with a whole family of specifications indexed by extensions of those paths.

  

Figure 2: Revised denotation for global node/path pairs

This suggests the following approach to the semantics of defaults in DATR. Rather than interpreting node definitions (in a given global context) as partial functions from paths to values (i.e., of type $U^* \rightarrow U^*$) we choose instead to interpret them as partial functions from (explicit) paths, to functions from extensions of those paths to values (i.e., of type $U^* \rightarrow (U^* \rightarrow U^*)$). Now suppose that $f:U^* \rightarrow (U^* \rightarrow U^*)$ is the function associated with the node definition $\mbox{\em T}/N$ in a given DATR interpretation. We can define a partial function $\Delta(f): U^* \rightarrow U^*$ (the default interpretation of $\mbox{\em T}/N$) as follows. For each $v \in U^*$ set

$$\begin{array} {lcl} \Delta(f)(v) & = & f(v_1)(v_2)\end{array}$$

where v = v₁v₂ and v₁ is the longest prefix of v such that f(v₁) is defined. In effect, the function $\Delta(f)$ makes explicit that information about paths and values that is only implicit in f, but just in so far as it does not conflict with explicit information provided by f.

In order to re-interpret node definitions in the manner suggested above, it is necessary to modify the interpretation of value descriptors. In a given global context c, a value descriptor d now corresponds to a total function $\left[\!\left[ d \right]\!\right]_c:U^* \rightarrow U^*$ (intuitively, a function from path extensions to values). For example, atoms now denote constant functions:

$$\begin{array} {lcl} \left[\!\left[ a \right]\!\right]_c(v) & = & F(a) \mbox{ for all } v \in U^*\end{array}$$

More generally, value descriptors will denote different values for different paths. Figure 2 shows the revised clause for global node/path pairs, the other definitions being very similar. Note the way in which the path argument v is used to extend $v_1 \cdots v_n$ in order to define the new local (and in this case also, global) context c'. Similarly, the meaning of each of the d_i is obtained with respect to the path extension v.

As before, the interpretation function is extended to sequences of path descriptors, so that for $\phi = d_1 \cdots d_n$ ($n \ge o$) we have $\left[\!\left[ \phi \right]\!\right]_c(v) = v_1 \cdots v_n \in U^*$, if $v_i = \left[\!\left[ d_i \right]\!\right](v)$ is defined, for each i ($1 \le i \le n$) (and $\left[\!\left[ \phi \right]\!\right]_c(v)$ is undefined otherwise). The definition of the interpretation of node definitions can be taken over unchanged from the previous section. However, for a theory T and node N, the function $\left[\!\left[ \mbox{\em T}/N \right]\!\right]_c$ is now of type $U^* \rightarrow (U^* \rightarrow U^*)$. An interpretation $I = (U,\kappa,F)$ is a default model for theory T just in case for every context c and node N we have:

$$\begin{array} {lcl} \left[\!\left[ N \right]\!\right]_c & \s... ...ta(\left[\!\left[ \mbox{\em T}/N \right]\!\right]_c)\end{array}$$

As an example, consider the default interpretation of the definition of the node Love given above. By definition, any default model of the theory must respect the following containment:

$$\begin{array} {lcl} \left[\!\left[ \mbox {\tt Love} \right]\... ...r \; root\gt}, \lambda v. \mbox{\tt love} \rangle \}\end{array}$$

From the definition of $\Delta$, it follows that for any path v, if v extends <mor root>, then it is mapped onto the value love, and otherwise it is mapped to the value given by $\left[\!\left[ {\tt \mbox {\tt VERB}\!:\!{\tt <\gt}} \right]\!\right]_c(v)$. We have the following picture:

$$\begin{array} {lcl} \left[\!\left[ \mbox {\tt Love} \right]\... ...}, \mbox{\tt love} \rangle, \ & & \:\:\: \ldots \}\end{array}$$

The default models of a theory T constitute a proper subset of the models of T: just those that respect the default interpretations of each of the nodes defined within the theory.