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<Paper uid="C96-2111">
  <Title>Top-Down Predictive Linking and Complex-Feature-Based Formalisms</Title>
  <Section position="3" start_page="658" end_page="661" type="metho">
    <SectionTitle>
2 Left-Corner Parsing and Linking
</SectionTitle>
    <Paragraph position="0"/>
    <Section position="1" start_page="658" end_page="658" type="sub_section">
      <SectionTitle>
2.1 The Left-Corner Parsing Algorithm
</SectionTitle>
      <Paragraph position="0"> The so-called left-corner (LC) parsing algorithm is generally credited to Rosenkrantz/ Lewis (1970). It has been presented so often since and is now so welbknown that a brief informal statement of the algorithm should sufrice here: The algorithm applies to CF grammars in general; it is both correct and, with the exception of derivations of the form A --+* A, where A is a nonterminal, is complete. It can be used either to compile a given CF grammar into a parser or to interpret it.</Paragraph>
      <Paragraph position="1"> The principle is simple. To parse a string, the current word form is first parsed, i.e. looked up in the lexicon. Whenever a constituent, be it a word form or a phrase, is successfully parsed, the syntax rules are chosen which have the category of the identified constituent as their left corner, i.e. the left-most category in the right-hand side of the rule. If the remaining sister categories of the left corner can be parsed, then the mother category of the rule is the result tbr the corresponding substring, and the algorithm continues recursively until the entire string is covered by a category; if the category of an expectation was specified, it must match the category found.</Paragraph>
    </Section>
    <Section position="2" start_page="658" end_page="658" type="sub_section">
      <SectionTitle>
2.2 The Linking Relation
</SectionTitle>
      <Paragraph position="0"> It was soon noted that the efficiency of the algorithm could be improved significantly through the use of a reachability or linking relation compiled out of the grammar before parsing; this consists of the reflexive and transitive closure of the relation defined by the pairs of mother and LC categories specified in the set of syntax rules of the grammar. Whenever a lexical entry is found that assigns a category C to a given word form, the linking relation is used to determine whether C can be useful in reaching the goal category C', i.e. whether C is a (transitive) left corner of C'; this test is carried out before the parser looks for rules having C as the left corner. Of course, C may itself be the category C' sought, hence the reflexive closure. Likewise, when a rule is found in which C is the left corner, the relation tests whether the mother Co of C can be used to reach the goal C' before an attempt is made to parse the sisters of C.</Paragraph>
      <Paragraph position="1"> Computation of the linking relation from a set of CF syntax rules is straightforward. Since the nonterminal symbols are atomic, one merely needs to check for left recursive symbols so that the computation terminates.</Paragraph>
    </Section>
    <Section position="3" start_page="658" end_page="659" type="sub_section">
      <SectionTitle>
2.3 Complex-Feature-Based Formalisms
</SectionTitle>
      <Paragraph position="0"> As noted above, the extension of the LC algorithm to a potentially infinite nonterminal domain, i.e. complex feature structures, is nontrivial. An example of the pitfalls awaiting naive attempts at such an extension is provided by the grammars illustrating the list technique for sub-categorization introduced by Shieber (1986: 32, 77-78); also see the similar example of Haas (1989: 227). We quote Shieber's syntax rules for his second analysis of subcategorization (p.84), in which the subject of a verb appears as the first element of the subcategorization list:</Paragraph>
      <Paragraph position="2"> &lt;VP 1 subcat rest&gt;.</Paragraph>
      <Paragraph position="3"> The difficulties clearly lie in the last syntax rule: VP (or VP_I and VP2) seems to be left recursive--whatever we may mean by that at this point--so perhaps no link arises at all from this rule. On the other hand, the two feature structures are unifiable, but their unification produces a cyclic feature structure, which raises additional problems for the definition of linking and possibly for implementation. The difficulty, of course, is that VP 1 and VP 2 are schematic and that these rules recursively generate a denumerably infinite set of VP-type categories, all of which may give rise to distinct elements in the linking relation. Whether this is linguistically important, which is improbable, or merely a mathematical game is beside the point: the forreal problem is there, and we cannot individually specify infinitely many links.</Paragraph>
      <Paragraph position="4">  Consider the LC analysis of the sentence John loves Mary. After analyzing \[John\]Np, the parser expects a VP\[1\], where VP\[n\] is used as an informal alias for a VP that subcategorizes for n complements. Now, there is an entry for \[loves\]v, so a link &lt;VP\[1\], V\[2\]&gt; is needed since loves subcategorizes for a subject and an object. Indeed, since the grammar allows verbs to subcategorize for any finite number of complements, we need an infinite number of links between VP\[1\] and V\[n\] categories. Moreover, once we have these links we need the same number between VP\[1\] and VP\[n\] categories since the first VP expansion simply unifies the subcategorization of the Vdaughter with that of the VP mother.</Paragraph>
      <Paragraph position="5"> Other problems arise in grammars with indirect left recursion. This is linguistically plausible in the following example, where both NP and Det are indirectly left recursive:</Paragraph>
      <Paragraph position="7"> The rules account for sentences like The child's father sleeps. We must take care not to exclude A mother's children sleep, which will happen if the linking relation is defined so that any determiner--also in a possessive construction-must have the same agreement features as the sentence subject of which it is a left corner.</Paragraph>
    </Section>
    <Section position="4" start_page="659" end_page="660" type="sub_section">
      <SectionTitle>
2.4 Top-Down Predictive Linking
</SectionTitle>
      <Paragraph position="0"> The aim of our proposal is to define equivalence relations that keep the linking relation finite while also preventing it from being too restrictive; this turns the linking relation into a weakpredietion table in the sense of Haas (1989: 227ff). Like Shieber (1985, 1992) with the notion of restriction, we confine our attention to a subset of specifications; in particular, we can define a feature structure that subsumes all VP-type feature structures of Shieber's recursive subcategorization rules. But unlike Shieber, our restrictors are computed automatically by building the generalization of the occurrences ofleftrecursive categories in a grammar.</Paragraph>
      <Paragraph position="1"> The intuitive idea is that we consider categories to be left recursive if their tokens can be unified (rather than being identical, as in the case of atoms); we then use their generalization, or greatest lower bound, as a common denominator defining an equivalence relation.</Paragraph>
      <Paragraph position="2"> We shall say that two categories build a left-recursive link, i.e. &lt;X, X'&gt; e L 1 iffon the basis of the given grammar there is a derivation A --&gt;* A't~' (where o~'is a string of categories and terminal symbols)such that the unification A u A' exists, whether or not it is cyclic, and there is no A&amp;quot; such that A ---&gt;' A&amp;quot; t~&amp;quot; --&gt;' A' t~', where A u A&amp;quot; and A&amp;quot; u A' exist. Let Age, be the generalization A rq A' of A and A'; then we define X and X' as distinct copies of Ag~, such that for every path n where A@n = A '@~, it also holds that X@n = X'@n, where F@n is &amp;quot;the value of the feature structure F at some path n at which it is defined&amp;quot; (Carpenter 1992:38) and '=' denotes token identity. We thus expressly allow reentrancies between the distinct feature structures X and X'; as we shall see below, this is essential in order for us to use the linking relation to instantiate information during parsing.</Paragraph>
      <Paragraph position="3"> A second relation, corresponding directly to the conventional notion of links as compiled directly from rules, can now be defined: two categories build a rule link, i.e. &lt;Ao', A,~ ~ L 2 iff on the basis of the given grammar there is a finite derivation A o --&gt; AIo~ ~ ... A,,vo~,v ---&gt; A,t~, with 1 &lt; n such that for all i with 1 _&lt; i &lt; n it is the case that A o u Aj is undefined (i.e. the derivation is nonrecursive), and for all i with 0 &lt; i &lt; n it is the case that if &lt;X, X~&gt; ~ L 1 and A i u X exists, then A/also exists and is the extension X'E A/that arises when A~ is unified with X. Intuitively, L 2 is given by the set of nonrecursive derivations licensed by the grammar, where each left-recursive left-most category in the derivation is replaced by (i.e. restricted to) the generalization of the instances of that left-recursive category.</Paragraph>
      <Paragraph position="4"> The overall linking relation L is then defined so that &lt;B, B~&gt; E L iff(1) there is a &lt;X, X'&gt; (L~ u L2) such that B u X and B' m X' exist, or else (2) B u B' exists (the reflexive case where B' satisfies the parse goal B).</Paragraph>
      <Paragraph position="5"> Moore and Alshawi (1992: 134ff) describe a similar algorithm that compiles the linking relation for complex-feature-based formalisms.</Paragraph>
      <Paragraph position="6"> But rather than computing the generalization of left-recursive categories to avoid the possibility of generating an infinite relation, they instead &amp;quot;impose a cutoff after two nested occurrences of the same functor in a feature specification,  substituting new unique variables for the arguments of the inner occurrence of the functor, so that any constituent with a more complex feature description will be accepted.&amp;quot; Moreover, they discuss linking only with regard to filtering. null Our use of the linking relation with destructive unification in parsing requires special comment. IfB is a goal and B' is a parsed left corner such that &lt;X, X'&gt; e L and B u X and B' u X' exist, then there is a link between B and B'; we can stop here with a mere test of unification if we only want to use linking as a filter to reduce the search space. But if B and B' are actually unified with X and X', respectively, information may become shared between B and B'. In the reflexive case of linking for completion, B and B' are unified with each other.</Paragraph>
      <Paragraph position="7"> Since the information that becomes shared via L 1 is subsumed by that of the reflexive case, completion works correctly for left-recursive categories, but B and B' must still be unified in the actual completion step.</Paragraph>
      <Paragraph position="8"> We have employed generalization in the definition of linking to make a kind of mask allowing just the appropriate information to become shared between B and B'. Thus, linking ceases to be a mere test or filter and can instead function as an independent device for the transmission of information in parsing.</Paragraph>
      <Paragraph position="9"> Returning to Shieber's rules for subcategorization, the definition of L given here allows instantiated head features of a VP goal to be passed top-down to a verb. Moreover, since the treatment of subcategorization of Shieber (1986: 84) is adopted in which the subject NP appears consistently as thefirst element of the subcategorization list in all projections of V, then instantiated agreement features and other information about the subject can be passed top-down as well. The linking relation compiled from the grammar in (2.3) above is listed here: ll(\[cat:vp,head:A, subcat:\[first:B, rest:C\]\], \[cat:vp,head:A, subcat:\[first:B, rest:D\]\]).</Paragraph>
      <Paragraph position="10"> 12(\[cat:s\], \[cat:np\]).</Paragraph>
      <Paragraph position="11"> 12(\[cat:vp,head:A, subcat:\[first:B, rest:C\]\], \[cat:v, head:A, subcat:\[first:B, rest:D\]\]).</Paragraph>
      <Paragraph position="12"> Finally, we list the linking relation compiled from the example with left-recursive categories  \[cat:np\], \[cat:np\]).</Paragraph>
      <Paragraph position="13"> \[cat:det\], \[cat:det\]).</Paragraph>
      <Paragraph position="14"> \[cat:s\], \[cat:np\]).</Paragraph>
      <Paragraph position="15"> \[cat:np\], \[cat:det\]).</Paragraph>
      <Paragraph position="16"> \[cat:det\], \[cat:np\]).</Paragraph>
      <Paragraph position="17"> \[cat:s\], \[cat:det\]).</Paragraph>
      <Paragraph position="18"> In contrast to the previous example, agreement specifications have been compiled out of the relation, but no additional convention whereby eat specifications define a context-free skeleton is involved here.</Paragraph>
    </Section>
    <Section position="5" start_page="660" end_page="661" type="sub_section">
      <SectionTitle>
2.5 Notes on Implementation in Prolog
</SectionTitle>
      <Paragraph position="0"> Implementation of the LC algorithm in Prolog has been discussed by Matsumoto et al. (1982) for the BUP system, by Pereira/Shieber (1987), Kilbury (1990), and Covington (1994). Here we present, with minor changes, the LC-based interpreter with linking for a modified DCG formalism as formulated by Pereira/Shieber (1987: 180ff); note that the interpreter itself is encoded as a DCG:  right(\[Cat\[Cs\]) --&gt; parse(Cat), right(Cs). Our version stated here tranfers the call to link from the definition of parse to that of leaf; the motivation for this change steins from our use of top-down information in the morphological analysis and in the treatment of missing lexical entries.</Paragraph>
      <Paragraph position="1"> Moreover, our implementation uses the open Prolog lists of Eisele/D6rre (1986) to encode the feature structures of PATR-II (also see Gazdar/Mellish 1989: 228ff). Since simple variable sharing does not capture the unification of these objects, we instead employ unify/2.</Paragraph>
      <Paragraph position="2"> The transitive subset of the linking relation can be implemented as follows:</Paragraph>
      <Paragraph position="4"> Of course, such clauses lead to a highly inefficient search for Prolog. A better solution, which we have adopted from Kilbury (1990), is to introduce rule numbers, which are then used to define a purely filtering linking relation. This amounts to the simplest case of the restriction technique of Shieber (1985). Only when a link between numerical pointers is first found is the linking relation between feature structures used to instantiate information.</Paragraph>
    </Section>
  </Section>
  <Section position="4" start_page="661" end_page="662" type="metho">
    <SectionTitle>
3 Consequences of Predictive Linking
</SectionTitle>
    <Paragraph position="0"> What is the advantage of predictive linking as discussed above in (2)? In the previous parsing literature attention has been drawn mainly to linking as a filter employed to reduce the search space as early as possible in a syntactic analysis. But we have seen that linking as defined above in terms of feature structures and used in parsing with destructive unification leads to the top-down instantiation of information. This has far-reaching consequences for the analysis of inflectional morphology and lexical items for which no entry at all or no adequate entry is found in the parser's lexicon. We briefly address these areas separately.</Paragraph>
    <Section position="1" start_page="661" end_page="661" type="sub_section">
      <SectionTitle>
3.1 Inflectional Morphology
</SectionTitle>
      <Paragraph position="0"> If we ignore capitalization, the German word form runden 'round' can be assigned the category N, A, or V; the corresponding inflected forms are too numerous to list here but include e.g. accusative plural for N, genitive singular weak inflection for A, and first person plural present indicative for V. The facts of German inflection lead to massively disjunctive analyses in conventional systems of morphological analysis that simply take an isolated inflected word form and consider what it might be. But if we are given a top-down prediction or expectation of the lexical category from the linking relation, then we can first find a lexical entry for the stem, use linking to confirm the appropriateness of the entry (in our example, the appropriateness of the category N, A, or V) for the context, simultaneously use linking to further instantiate features (in particular for agreement) on the category, and then check the given inflection within a highly restricted search space. This captures our intuition that an inflection like -n in German in itself bears practically no information and is functional only because we normally have expectations and can greatly reduce the range of inflections possible in a given context. null</Paragraph>
    </Section>
    <Section position="2" start_page="661" end_page="662" type="sub_section">
      <SectionTitle>
3.2 Analysis of Unknown Words
</SectionTitle>
      <Paragraph position="0"> A fundamental issue for parsing lies in the area of &amp;quot;unknown&amp;quot; or &amp;quot;new&amp;quot; words. This involves cases where no entry at all is found for a given word form, but also cases where an entry for the form is found, which, however, does not fit the given context; the missing lexical entries may simply have been omitted from the lexicon of a system, or may reflect novel lexical creations. The theoretical and practical significance of such unknown forms is great; see the discussion in Kilbury/Barg/Renz (1994).</Paragraph>
      <Paragraph position="1"> Even without the linking relation, unification, together with backtracking through a space of possible analyses (or corresponding use of a chart) gives us information for the missing entry; this in itself is not novel. But simple extensions of well-known parsing algorithms presuppose a random search through the space of open lexical classes to get the candidate category. The procedure is roughly as follows: (1) once it is established that no appropriate entry is in the lexicon, (2) arbitrarily select a category for an open lexical class, (3) check whether it fits the given context, and (4) if it fits, take the final feature structure for the form which is instantiated in the course of a successfully completed parse.</Paragraph>
      <Paragraph position="2"> in contrast, top-down predictive linking provides for a directed search since top-down instantiation can propose a category. The efficiency can be further increased by partitioning the linking relation according to lexical and phrasal categories for the left corner and, among the lexical left corners, open and closed lexical classes.</Paragraph>
      <Paragraph position="3"> In implementation, missing lexical entries can be dealt with in a first approximation by extending the interpreter with a second clause in the definition of leaf:</Paragraph>
      <Paragraph position="5"> open lexical link(NT,Cat), new word(Wor~,Cat\]}.</Paragraph>
      <Paragraph position="6">  Obviously, more needs to be said about the control strategy of the modified interpreter since garden paths and structural ambiguity must be dealt with before new entries are postulated, but that goes beyond the scope of this paper.</Paragraph>
    </Section>
  </Section>
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