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Path extensions and defaults

  In DATR, values may be associated with particular node/path pairs either explicitly, in terms of local or global inheritance, or implicitly (by default). The basic idea underlying DATR's default mechanism is as follows: any definitional sentence is applicable not only to the path specified on its left-hand side, but also for any rightward extension of that path for which no more specific definitional sentence exists. Making use of defaults, the ${\tt DATR}_G$ theory given above can be expressed more succinctly as shown below:

    NOUN:
        <cat> == noun
        <suff> == s.
        <sing> == "<root>"
        <plur> == "<root>" <suff>.
    Llama:
        <> == NOUN
        <root> == llama.
Here, the relationship between the nodes Llama and NOUN has effectively been collapsed into just a single statement Llama:<> == NOUN. This is possible because the sentence now corresponds to a whole class of implicit definitional sentences, each of which is obtained by extending the paths found on the left- and right-hand sides in the same way. Accordingly, the value of Llama:<cat> is specified implicitly as the value of NOUN:<cat>, and similarly for Llama:<sing> and Llama:<suff>. In contrast, the specification Llama:<root> == NOUN:<root> does not follow by default from the definition of Llama, even though it can be obtained by extending left and right paths in the required manner. The reason is that the theory already contains an explicit statement about the value of Llama:<root>.

The evaluation relation is now defined as a mapping from elements of $\mbox{\sc cont}\times \mbox{\sc desc}^* \times \mbox{\sc atom}^*$(i.e., context/descriptor sequence/path extension triples) to $\mbox{\sc atom}^*$. We write:

\begin{displaymath}
C \vdash\phi \Longrightarrow_\gamma \alpha\end{displaymath}

to mean that $\phi$ evaluates to $\alpha$ in context C given path extension $\gamma$. When $\gamma = \epsilon$ is the empty path extension, we will continue to write $C\vdash\phi\Longrightarrow\alpha$.


  
Figure 5: The evaluation semantics for DATR
\begin{figure}
{\small
\begin{displaymath}
\begin{array}
{lc}
\multicolumn{2}{l}...
 ... <\alpha\gamma\gt}\Longrightarrow\beta}\end{array}\end{displaymath}}\end{figure}

A complete set of inference rules for DATR is shown in Figure 5. The rules for Values, Sequences and Evaluable Paths require only slight modification as the path extension is simply passed through from premises to consequent. The rules for Quoted Descriptors are also much as before. Here, however, the path extension $\gamma$ appears as part of the global context in the premise of each rule. This means that when a global descriptor is encountered, any path extension present is treated globally rather than locally. The main change in the Definitions rule lies in the conditions under which it is applicable. The amended rule just captures the ``most specific sentence wins'' default mechanism. Finally, the new rule for Path Extensions serves as a way of making any path extension explicit. For example, if Llama:<cat> evaluates to noun, then Llama:<> also evaluates to noun given the (explicit) path extension cat.

An example proof showing that Llama:<plur> evaluates to llama s given the DATR theory presented above is shown in Figure 6.

  
Figure 6: Proof utilising defaults
\begin{figure*}
{\small
\begin{displaymath}
 \infer[\mbox{\em Def\/}]{(\mbox {\t...
 ...ghtarrow_{\mbox{\tt plur}} \mbox{\tt llama s}}{}}\end{displaymath}}\end{figure*}

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Left: Global inheritance Up: Formal theory of inference Right: Comments on the theory
Copyright © Roger Evans, Gerald Gazdar & Bill Keller, Tuesday 10 November 1998