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<Paper uid="E89-1009">
  <Title>Inference in DATR</Title>
  <Section position="4" start_page="0" end_page="0" type="metho">
    <SectionTitle>
EN MOR:
</SectionTitle>
    <Paragraph position="0"> &lt; &gt; == VERB &lt;past participle&gt; == (coneat &amp;quot;&lt;root&gt;&amp;quot; en). Take: &lt; &gt; == EN MOR.</Paragraph>
    <Paragraph position="1"> &lt;root&gt; ~ ~ake.</Paragraph>
    <Paragraph position="2"> 3 Rule-based inference DATR has seven syntactic rules of inference falling into three groups. The first rule just provides us with a trivial route from definitional to extensional sentences: (I) N:P ~ V.</Paragraph>
    <Paragraph position="3"> N:P = V.</Paragraph>
    <Paragraph position="4"> For example, from: VERB: &lt;past&gt; ~--- ed.</Paragraph>
    <Paragraph position="5"> one can infer: VERB: &lt;past&gt; = ed.</Paragraph>
    <Paragraph position="6">  Note that V must be a value (not an lvalue) here, otherwise the consequent would not be wellformed. null The next three rules implement local inheritance of values, and use the following additional meta-notational device: the expression EO{E2/E1} is well-formed iff EO, E1 and E2 are Ivalues and E1 occurs as a subexpression of EO. In that case, the expression denotes the result of substituting E2 for all occurrences of E1 in EO.</Paragraph>
    <Paragraph position="8"> Rule II says that if we have a theorem NI:P1 == L. where L contains N2:P2 as a subexpression, and we also have a theorem N2:P2 == G., then we can derive a theorem in which all occurrences of N2:P2 in L are replaced by G. In the simplest case, this means that we can interpret a sentence of the form NI:P1 ~ N2:P2.</Paragraph>
    <Paragraph position="9"> as an inheritance specification meaning &amp;quot;the value of P1 at N1 is inherited from P2 at N2&amp;quot;. So for example, from: NOUN: &lt;sing gen&gt; ~--- s.</Paragraph>
    <Paragraph position="10"> PRON: &lt;sing gen&gt; == NOUN:&lt;sing gen&gt;.</Paragraph>
    <Paragraph position="11"> one can infer: PRON: &lt;sing gen&gt; ~ s.</Paragraph>
    <Paragraph position="12"> Rules III and IV are similar, but specify only a new node or path (not both) to inherit from. The other component (path or node) is unchanged, that is, it is the same as the corresponding component on the left-hand-side of the rule specifying the inheritance. In fact, the following two sentence schemas are entirely equivalent:</Paragraph>
    <Paragraph position="14"> Rules II, III, and IV implement a local notion of inheritance in the sense that the new node or path specifications are interpreted in the current local context. The three remaining inference rules implement a non-local notion of inheritance: quoted descriptors specify values to be - 68 interpreted in the context in which the original query was made (the global context), rather than the current context.</Paragraph>
    <Paragraph position="16"> To see how the operation of these rules differs from the earlier unquoted cases, consider the following theory:  CAT: &lt;sing&gt; == &lt;plur&gt;.</Paragraph>
    <Paragraph position="17"> V: &lt;sing&gt; == CAT &lt;plur&gt; ~ er.</Paragraph>
    <Paragraph position="18"> AI: &lt;sing&gt; == CAT &lt;plur&gt; ~ ern.</Paragraph>
    <Paragraph position="19"> A2: &lt;sing&gt; == en</Paragraph>
    <Paragraph position="21"> The intention here is that the CAT node expresses the generalisation that by default plural is the same as singular, v and A1 inherit this, but A2, while inheriting its plural form from A1, has an exceptional singular form, overriding inheritance from CAT (via A1). Now from this theory we can derive all the following theorems concerning plural:  V: &lt;plur&gt; = er.</Paragraph>
    <Paragraph position="22"> AI: &lt;plur&gt; = ern.</Paragraph>
    <Paragraph position="23"> A2: &lt;plur&gt; = ern.</Paragraph>
    <Paragraph position="24">  and the following theorem concerning singular: A2: &lt;sing&gt; = en.</Paragraph>
    <Paragraph position="25"> But we cannot derive a theorem for V:&lt;sing&gt;, for example. This is because v:&lt;sing&gt; inherits from CAT:&lt;sing&gt;, which inherits (locally) from CAT:&lt;plur&gt;, which is not defined. What we wanted was for CAT:&lt;sing&gt; to inherit from v:&lt;plur&gt;, that is, from the global initial context. To achieve this we change the CAT definition to be: CAT: &lt;sing&gt; == &amp;quot;&lt;plur&gt;&amp;quot;.</Paragraph>
    <Paragraph position="26"> Now we find that we can still derive the same plural theorems, but now in addition we get all these theorems conceming singular:  V: &lt;sing&gt; = er.</Paragraph>
    <Paragraph position="27"> AI: &lt;sing&gt; = ern.</Paragraph>
    <Paragraph position="28"> A2: &lt;sing&gt; = en.</Paragraph>
    <Paragraph position="29"> For example, the derivation for the first of these is as follows: (1) V: &lt;sing&gt; == CAT. (given) (2) CAT: &lt;sing&gt; == &amp;quot;&lt;phtr&gt;&amp;quot;. (given) (3) V: &lt;sing&gt; == &amp;quot;&lt;plur&gt;&amp;quot;. (III on 1 and 2) (4) V: &lt;plur&gt; ~ er. (given) (5) V: &lt;plur&gt; = er. (I on 4) (6) V: &lt;sing&gt; -- er. (VII on 3 and 5)  Finally, given a set of sentences &amp;quot;/', we define the rule-closure of '/', rcl('/') to be the closure of 'T under finite application of the above inference rules in the conventional fashion.</Paragraph>
  </Section>
  <Section position="5" start_page="0" end_page="0" type="metho">
    <SectionTitle>
4 Default inference
</SectionTitle>
    <Paragraph position="0"> In addition to the conventional inference defined above, I~AI'IC/ has a nonmonotonic notion of inference by default: each definitional sentence about some node/path combination implicitly determines additional sentences about all the extensions to the path at that node for which no more specific definitional sentence exists in the theory.</Paragraph>
    <Paragraph position="1"> Our overall approach follows Moore (1983, 1985), whose treatment of inferences from sets of beliefs can be viewed more generally as a technique for providing a semantics for a declarative notion of inference by default (cf.</Paragraph>
    <Paragraph position="2"> Touretzky 1986, p34; Evans 1987). We begin with some auxiliary definitions.</Paragraph>
    <Paragraph position="3"> The expression P^Q, where P and Q are paths, denotes the path formed by concatenating components of P and Q. A path P2 is an extension of a path P1 iff there is a path Q such that P2 = PI^Q. P2 is a strict extension iff Q is nonempty. We also use the ^ operator to denote extension of all the paths in a DArR sentence, as in the following examples:</Paragraph>
    <Paragraph position="5"> Given a sentence S, we define the root of 5 to be the \[node\]:\[path\] expression appearing to the left of the equality ('==' or '=') in S (for example the root of 'N:P -- V.' is 'N:P)'. The root does not correspond to any syntactic category defined above: it is simply a substring of the sentence.</Paragraph>
    <Paragraph position="6"> Given a set of sentences in DATR, T, a node N and a path P, we say N:P is specified in Tiff T contains a definitional sentence S whose root is N:P.</Paragraph>
    <Paragraph position="7"> Let NI:P1, NI:P2 be such that NI:P1 is specified in T. We say NI:P2 is connected to NI:P1 (relative to T) iff: i) P2 is an extension of P1, and ii) there is no strict extension P3 of P1 of which P2 is an extension such thatNl:P3 is specified in T. So NI:P2 is connected to NI:P1 if P1 is the maximal subpath of P2 that is specified (with N1) in T.</Paragraph>
    <Paragraph position="8"> Now given a set of sentences T, define the path closure pcl(T) of Tto be: pcl(T) = {S:S is an extensional sentence in T } w {S^Q: S is a definitional sentence in T, with root N:P, and N:P^Q is connected to N:P} It is clear from these definitions that any N:P is connected to itself and thus that T is always a subset of pal(T). The path closure contains all those theorems which can be inferred by default from T.</Paragraph>
    <Paragraph position="9"> To illustrate path closure, consider the following example theory:  &lt;past tense singular third&gt; == ed.</Paragraph>
    <Paragraph position="10"> The situation is slightly more complicated with sentences that have paths on their right-hand sides. Such paths are also extended by the sub-path used to extend the left-hand side. So the sentence: A2:&lt;sing&gt; ~ &amp;quot;Al:&lt;phtr&gt;&amp;quot;.</Paragraph>
    <Paragraph position="11"> might give rise Coy default) to sentences such as: A2:&lt;sing fern nom&gt; == &amp;quot;Al:&lt;plur fern nom&gt;&amp;quot;. Using default inference, the example theory we used to illustrate global inference can be phrased more succinctly:  CAT: &lt;sing&gt; == &amp;quot;&lt;plur&gt;&amp;quot;.</Paragraph>
    <Paragraph position="12"> V: &lt; &gt; == CAT &lt;plur&gt; ~ er.</Paragraph>
    <Paragraph position="13"> AI: &lt; &gt; == CAT &lt;plur&gt; ~ ern.</Paragraph>
    <Paragraph position="14"> A2: &lt;sing&gt; == en &lt; &gt; == A1.</Paragraph>
    <Paragraph position="15">  In this version, we state that anything not specifically mentioned for v is inherited Coy default) from CAT, whereas before we had to list cases (only 'sing' in the example) explicitly. Similarly A1 inherits by default from CAT, and A2 from A1. The operation of path closure is non-monotonic: if we add more sentences to our original theory, some of our derived sentences may cease to be true.</Paragraph>
    <Paragraph position="16"> The two forms of inference in DATR are combined by taking the path closure of a theory first, and then applying the inference rules to the result. In other words, given a theory T, and a  sentence S, S is provable from Tiff S rcl(pcl(T)).</Paragraph>
    <Paragraph position="17"> - 70 -</Paragraph>
  </Section>
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