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<Paper uid="C96-2109">
  <Title>An Evaluation Semantics for DATR Theories</Title>
  <Section position="4" start_page="0" end_page="646" type="metho">
    <SectionTitle>
2 Syntax of DATR Theories
</SectionTitle>
    <Paragraph position="0"> Let NODE and ATOM be finite sets of symbols. E1eInents of NODE are called nodes and denoted by N. Elements of ATOM are called atoms and denoted by a. Elements of ATOM* are called values and denoted by a, /3, 7- The set DESC of DATR value descriptors (or simply descriptors) is built up from the nodes and atoms as shown below. \[n the following, sequences of descriptors in DESC* are denoted C/, ~/J.</Paragraph>
    <Paragraph position="1">  1. a C DESC for any a E ATOM 2. For any N C NODE and 66 C DESC*:</Paragraph>
    <Paragraph position="3"> (1) or else inheritance descriptors (2). Inheritance descriptors are fllrther distinguished as being local (unquoted) or global (quoted). For C/ C PESO* a sequence of descriptors, an expression (C/) is called a path descriptor. When each element of C/ is atomic, then (C/) is called a path, and denoted P.</Paragraph>
    <Paragraph position="4"> For N a node, P a path and C/ a (possibly empty) sequence of value descriptors, an equation of the form N : P == 66 is called a (definitional)  sentence. InforInally, N : P .... C/ specifies a prolmrty of the node N, nalnely that the value of the path P is given by the sequence of value descriptors C/. A DATR theory &amp;quot;Y is a finite se, t of definitional sentences subject to the fbllowing re(luirelnent of functionality: if N : 1' == C/ 6 T &amp; N : P =-- '(; 6 T th(!n ~/) -~ ~t/J t)Smctionality thus correst)onds to a semantic requirement that node/path pairs are associated with (at most) one value.</Paragraph>
  </Section>
  <Section position="5" start_page="646" end_page="646" type="metho">
    <SectionTitle>
3 Inference in DATR
</SectionTitle>
    <Paragraph position="0"> The probh'm of constructing an explicit theory of infhrence for DATR was originally addressed in (Evans and Gazdar, 1989a). In this work, an attempt is made to set out a logic of DATR statemerits. Consider for eXaml)le the following rule of in%rence, adapted from (Evans and Gazdar,</Paragraph>
    <Paragraph position="2"> The prelnises are detinitional sentences which can be read: &amp;quot;the value of path 1'~ at node Nj ix (inherited fl'om) the value of path P'2 at; N2&amp;quot; and &amp;quot;the vahle of path 1~,2 at, node N2 is el&amp;quot;, respectively. Given the premise, s, the rule lieenees the conclusion &amp;quot;the value of path \['l at node Nj is (t&amp;quot;. Thus, the rule captures a logical relationship between DATR sentences. For a given DATR theory T, rules of this kind lllay /)e used to deduce additional sentences as theorems of '\]-.</Paragraph>
    <Paragraph position="3"> In contrast, the system of inR;renee described ill this pal)er characterizes a relationship between DATR expressions (i.e. sequences of descriptors) and the vahles they may be used to cOlnlmte.</Paragraph>
    <Paragraph position="4"> As an example, consider the following (simpliiied) rule of the operational semantics: if N1 : l'~ == 4) G T the, n</Paragraph>
    <Paragraph position="6"> The rule is applieatfle just in case the theory T contains a detinitional sentence N, : t~l - - eft. It states that if the sequence of value descril)tors (/) on the right of the sentence evaluates to (--&gt;) the sequence of atoms tt, then it may be concluded that the node/1)ath pair NI : I~ also evaluates to a. Rules of this kind may be used to provide, an inductive detinition of an evaluation relation between DATR expresskms and their values.</Paragraph>
    <Paragraph position="7"> Both approaches to inference in DATR aim to provide a system of deductioi~ that makes it possible to (teterlnine formally, for a given DATR theory 7~, what; follows fl'om the stateulellts in 7. The primary interest lies in deducing statements about the vahles associated with particular node/path pairs defined within the theory. UnRn'tunately, the proof rules described in (Ewms and Gazdar, 1989a) are not su\[\[iciently general to support all of the required inferenees, and it is not obvious that the approach can be extended appropriately to (:over all of the available DATR constructs. A partieuiar t)rot)hnn (:on(:erns DATR's notion of non-local or global inheritance. The value (~xi)resse(l t)y a global inheritan(:e descriptor (lep(,nds on more than just the proi)ertie.s of the nodes sl)eeified by {;he definitional sentences of a theory. In fact, it only makes sense to talk about the value of a global descriptor relative to a given context of evaluation, or 91obal context. Because the proof rules of (Evans and C, azdar, 1989a) just talk about DATR sentences, which do not make explicit reii'~rence to contexl., it is not possible to give. a satisfactory a(:comlt of the global inheritance mechanism. The evaluation semantics described ill the following sections provides a perspicuous treatment of both local and global inheritance in DATR. The, rules eaptme the essential details of the t)roeess of evaluating DATR expressions, and for this reason silouhl prove, of use to the language imt)lementer.</Paragraph>
  </Section>
  <Section position="6" start_page="646" end_page="647" type="metho">
    <SectionTitle>
4 Local Inheritance
</SectionTitle>
    <Paragraph position="0"> As a point of departure, this section provides rules of inference for a restricted variant of DATR which lacks both global inheritance and tilt default me(:hmfism. This varianl; will be refl'ared to as DATRL. The syntax of DATRI, is as given in see|ion 2, except of course that the three forms of global inheritance descriptor are omitted. An exami&gt;le of a simph; DATR~, theory is shown next.</Paragraph>
    <Paragraph position="1"> Noun: (cat) .... nouu</Paragraph>
    <Paragraph position="3"> In this and all subsequent examples, a nun&gt; her of standard abbreviatory devices are adopted.</Paragraph>
    <Paragraph position="4"> Sets of definitional sentences with the same node on the left-hand side are groupe.d together and the node h;ft implicit in all but the tirst given sentence. Als% a definitional sent;en(:e such its Dog: (c.at) --=== Noun: (,'at), where the path on the right is identical to that on the left, is written more succinctly as Dog : (cat) --= Noun. Similarly, nodes oil the right of a sentence a.re suppressed whe.n identical to the node on the left.</Paragraph>
    <Paragraph position="5"> The DATRL theory defines the propertie~s of two nodes, Noun and Dog. The detinitional sente, iices specify values for node/path l)airs, where the st)eeitication is either direct (a particular value is exhitfited), or indirect (the wflue is obtained by local inheritance), l%r exalnpte, the value of the node/path pair Noun : {eat} is specitied directly as noun. Ill contrast, the node/path pair Dog : (cat} obtains its value indirectly, by local  inheritance from the value of Noun : (cat). Thus Dog : &lt;cat) also has the value noun. The value of Dog : (plur) is specified indirectly by a sequence of descriptors Dog: (root) Noun: (suiT). Intuitively, the required value is obtained by concatenating the values of the descriptors Dog : (root) and Noun : (surf}, yielding dog s.</Paragraph>
    <Paragraph position="6"> We wish to provide an inductive definition of an evaluation relation (denoted ~) between sequences of DATR descriptors in DESC* and sequences of atoms (i.e. values) in ATOM*. We write to mean that the sequences of descriptors C/ evaluates to the sequence of atoms a. With respect to the DATR/ theory above we should expect that Dog : (cat) ~ noun and that Dog : (root) Noun: (surf) ~ dog s, amongst other things.</Paragraph>
    <Paragraph position="7"> The formal definition of ==v for DATRL is provided by just four rules of inference, as shown in figure 1. The rule for Values states simply that a sequence of atoms evaluates to itself. Another way of thinking about this is that atom sequences are basic, and thus cannot be evaluated further.</Paragraph>
    <Paragraph position="8"> The rule for Definitions was briefly discussed in the previous section. It permits inferences to be made about the values associated with node/path pairs, provided that the theory T contains the appropriate definitional sentences. The third rule deals with the evaluation of sequences of descriptors, by breaking them up into shorter sequences. Given that the values of the sequences C/ and C/ are known, then the value of C/C/ can be obtained simply by concatenation. Note that this rule introduces some non-determinism, since in general there is more than one way to break up a sequence of value descriptors. However, whichever way the sequence is broken up, the result (i.e.</Paragraph>
    <Paragraph position="9"> value obtained) should be the same. The following proof serves to illustrate the use of the rules Val, Def and Seq. It establishes formally that the node/path pair Dog : (plur) does indeed evaluate to dog s given the DATRL theory above.</Paragraph>
    <Paragraph position="10"> dog==~dog Val s ~ s Val Dcf Dog: &lt;root) :=~ dog Noun: &lt;suiT) ~ s Def Seq Dog: &lt;root) Noun: (surf) ==~ dog s Dcf Dog: (plur) ~ dog s The final rule of figure 1 deals with DATR's evaluable path construct. Consider a value descriptor of the form A: (B: 0). To determine the value of the descriptor it is first necessary to establish what path is specified by the path descriptor (B : 0)-This involves evaluating the descriptor B : 0 and then 'plugging in' the resultant value a to obtain the path (a). The required value is then obtained by evaluating A : (a). The rule for Evaluable Paths provides a general statement of this process: if a sequence of value descriptors C/ evaluates to a and N: (a) evaluates to/3, then N: (C/) also evaluates to ~.</Paragraph>
  </Section>
  <Section position="7" start_page="647" end_page="648" type="metho">
    <SectionTitle>
5 Global Inheritance
</SectionTitle>
    <Paragraph position="0"> DATR's local inheritance mechanism provides for a simple kind of data abstraction. Thus, in the DATRL theory above, information about the plural suffix is stated once and for all at the abstract Noun node. It is then available to any instance of Noun such as Dog via local inheritance. On the other hand, information about the formation of singular and plural forms of dog must still be located at the Dog node, even though the processes involved are entirely regular. To overcome this problem, DATR provides a second form of inheritance: global inheritance. This section provides an evaluation semantics for a default-free variant of DATR with both local and global inheritance (DATRG). A simple DATRG theory is shown below.</Paragraph>
    <Paragraph position="2"> The new theory is equivalent to that given previously in the sense that it associates exactly the same values with node/path pairs. However, in the DATRa theory global inheritance is used to capture the relevant generalizations about the singular and plural forms of nouns in English. Thus, the sent~ence Noun : &lt;sing) == &amp;quot;&lt;root)&amp;quot; states that the singular form of any noun is identical to its root (whatever that may be). The sentence Noun: (plur) == &amp;quot;(root)&amp;quot; (surf) states that the plural is obtained by attaching the (plural) suffix to the root.</Paragraph>
    <Paragraph position="3"> To understand the way in which global inheritance works, it is necessary to introduce DATR's  notion of global contea't. Suppose that we wish to determine the value of Dog : (sing) in the exalnt)le DATRc; theory. Initially, the global context will be the pair (Dog, sing), bS&amp;quot;om tile theory, the value of Dog : (sing} is to be inherited (locally) fl'om Noun : (sing), which ill turn inherits its value (globally) from the quoted path &amp;quot;(root)&amp;quot;. rio evaluate the quoted path, the global context is examined to find the current global node (this is Dog) and the vahle of &amp;quot;(root)&amp;quot; is then obtained by evaluating Dog : (root), which yields dog as required.</Paragraph>
    <Paragraph position="4"> More generally, the global context is used to fill in the missing node (t/ath) when a quoted path (node) is encountered. In addition, as a side effect of evahlating a global inheritance descriptor the global context is updated. Thus, after encountering the quoted path &amp;quot;(root}&amp;quot; in the preceding example, tile global context is changed from (Dog, sing) to (Dog, root). That is, the path component of the context is set to tile new global path root.</Paragraph>
    <Paragraph position="5"> Let T be a DATRa theory defined with respect to the set of nodes NODE and the set of atoms ATOM. The set (:ON'X' of (.qlobal) contexts of 7- is defined as the set of all pairs of the form (N, (t), for N G NODE and (.~ G ATOM*. Contexts are denoted t)y C. The evaluation relation ~ is now taken to be a mapping from elements of CeNT X \])ESC* to ATOM*. We write cF4)~ to mean that C/ evaluates to fl in the global context C.</Paragraph>
    <Paragraph position="6"> To axiomatise the IleW evaluation relation, the, DATRc rules m'e modified to incorporate the global context parameter. For example, the rule for Evaluable Paths now becolnes:</Paragraph>
    <Paragraph position="8"> Two sinfilar rules are required for sentences contMning quoted descriptors of the forms &amp;quot;N : {C/)&amp;quot; and &amp;quot;(qS)&amp;quot;. Note that the context (7 plays no sl)ecial role here, but is simply carried unchanged from premises to conclusion. The rules for Values, Definitio'ns and Sequences are modified in an entirely similar inanner. Finally, to capture tile way in which values are derived for quoted descriptors three entirely new rules are required, one for each of the quoted fi/rms. These rules are shown in figure 2.</Paragraph>
    <Paragraph position="9"> Consider for examt)le the Quoted Path rule.</Paragraph>
    <Paragraph position="10"> The premise states that N : (c~) evahmtes to fl in the glohal context (N, a). Given rills, the rule licences the conehlsion that the quoted descriptor' &amp;quot;(oe)&amp;quot; also evaluates to fl in any context with the same node component N. in other words, to ewduate a quoted path &amp;quot;(a)&amp;quot; in a context (N, (f), just evahmte the local descriptor N : (a) in the  updated global context (N,a). The rules (leal~ ing with global node/t)adl pairs, and global nodes work in a similar way.</Paragraph>
    <Paragraph position="11"> The following proof illustrates the use of tt{e Quoted Path rule (Qu%). It demonstrates that Dog : (sing) evaluates to dog, given the DATRo theory, aim when the initial global context is taken to be (Dog, sing).</Paragraph>
    <Paragraph position="12"> Val (Dog, root) t- dog =-=~ dog Def (Dog, root) \[- Dog: (root) ==~ ,log Q'ao2 (Dog, sing) t- &amp;quot;(root)&amp;quot; ==~ dog Def (Dog, sing) F- Not,,,: (sing) :=~ dog Def (Dog, sing) F- Dog: (sing) ~ dog</Paragraph>
  </Section>
  <Section position="8" start_page="648" end_page="649" type="metho">
    <SectionTitle>
6 Path Extensions and Defaults
</SectionTitle>
    <Paragraph position="0"> In DATR, wflues 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 umchanism 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 specitic definitional sentence exists. Making use of defimlts, the DATRc: theory given above can be expressed more succinctly as shown next.</Paragraph>
    <Paragraph position="2"> Ilere, the relationship between the nodes Dog and Noun has effectively been collapsed into just a single statement Dog : 0 == Noun. This is p0ssible 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 ill the same way.</Paragraph>
    <Paragraph position="3"> Accordingly, the value of Dog : {cat) is specified implicitly as the value of Noun : (eat), and similarly for Dog : (sing) and Dog : (surf}. In contrast, the specification Dog : {root} == Noun :  (root} does not follow 'by default' from the definition of Dog, 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 Dog: {root}.</Paragraph>
    <Paragraph position="4"> The evaluation relation is now defined as a mapping from elements of CONT x DESC* X ATOM* (i.e. context/descriptor sequence/path extension triples) to ATOM*. We write: to nman that C/ evaluates to a in context C given path extension 7. When 7 = e is the emi)ty path extension, we will continue to write C ~- C/ ~ c,. A complete set of iifference rules for DATR is shown in figure 3. 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 hefore.</Paragraph>
    <Paragraph position="5"> Here however, the path extension 7 appears as part of the global context in the premise of each rule. This means/;hat 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 Eztensions serves as a way of making any path extension explicit. For example, if Dog : (eat} evaluates to noun, then Dog : (} also evaluates to noun given the (explicit) path extension cat.</Paragraph>
    <Paragraph position="6"> An example proof showing thai; Dog : &lt;plur&gt; evaluates to dog s given the DATR theory presented above is shown in figure 4.</Paragraph>
  </Section>
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