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<Paper uid="P96-1011">
  <Title>Efficient Normal-Form Parsing for Combinatory Categorial Grammar*</Title>
  <Section position="3" start_page="0" end_page="80" type="intro">
    <SectionTitle>
2 Definitions and Related Work
</SectionTitle>
    <Paragraph position="0"> CCG may be regarded as a generalization of context-free grammar (CFG)--one where a grammar has infinitely many nonterminals and phrase-structure rules. In addition to the familiar atomic nonterminal categories (typically S for sentences, N for 1Namely, Mary pretends to like the galoot in 168 parses and the corner in 84. One might try a statistical approach to ambiguity resolution, discarding the low-probability parses, but it is unclear how to model and train any probabilities when no single parse can be taken as the standard of correctness.</Paragraph>
    <Paragraph position="1">  nouns, NP for noun phrases, etc.), CCG allows infinitely many slashed categories. If z and y are categories, then x/y (respectively z\y) is the category of an incomplete x that is missing a y at its right (respectively left). Thus verb phrases are analyzed as subjectless sentences S\NP, while &amp;quot;John likes&amp;quot; is an objectless sentence or S/NP. A complex category like ((S\NP) \ (S\NP))/N may be written as S\NP\(S\NP)/N, under a convention that slashes are left-associative.</Paragraph>
    <Paragraph position="2"> The results herein apply to the TAG-equivalent CCG formalization given in (Joshi et M., 1991). 2 In this variety of CCG, every (non-lexical) phrase-structure rule is an instance of one of the following binary-rule templates (where n &gt; 0): (4) Forward generalized composition &gt;Bn: ;~/y y\[nzn''&amp;quot; \[2Z211Zl -'+ ;~\[nZn''&amp;quot; \]2Z211Zl Backward generalized composition &lt;Bn: yl.z...- 12z2 Ilzl x\y x I.z.... 12z llzl Instances with n -- 0 are called application rules, and instances with n &gt; 1 are called composition rules. In a given rule, x, y, zl... z~ would be instantiated as categories like NP, S/I~P, or S\NP\(S\NP)/N. Each of \]1 through In would be instantiated as either / or \. A fixed CCG grammar need not include every phrase-structure rule matching these templates. Indeed, (Joshi et al., 1991) place certain restrictions on the rule set of a CCG grammar, including a requirement that the rule degree n is bounded over the set. The results of the present paper apply to such restricted grammars and also more generally, to any CCG-style grammar with a decidable rule set.</Paragraph>
    <Paragraph position="3"> Even as restricted by (Joshi et al., 1991), CCGs have the &amp;quot;mildly context-sensitive&amp;quot; expressive power of Tree Adjoining Grammars (TAGs). Most work on spurious ambiguity has focused on categorial formalisms with substantially less power. (Hepple, 1990) and (Hendriks, 1993), the most rigorous pieces of work, each establish a normal form for the syntactic calculus of (Lambek, 1958), which is weakly context-free. (Kbnig, 1989; Moortgat, 1990) have also studied the Lambek calculus case. (Hepple &amp; Morrill, 1989), who introduced the idea of normal-form parsing, consider only a small CCG fragment that lacks backward or order-changing composition; (Niv, 1994) extends this result but does not show completeness. (Wittenburg, 1987) assumes a CCG fragment lacking order-changing or higher-order composition; furthermore, his revision of the combinators creates new, conjoinable constituents that conventional CCG rejects. (Bouma, 1989) proposes to replace composition with a new combinator, but the resulting product-grammar scheme as2This formalization sweeps any type-raising into the lexicon, as has been proposed on linguistic grounds (Dowty, 1988; Steedman, 1991, and others). It also treats conjunction lexically, by giving &amp;quot;and&amp;quot; the generalized category x\x/x and barring it from composition.  signs different types to &amp;quot;John likes&amp;quot; and &amp;quot;Mary pretends to like,&amp;quot; thus losing the ability to conjoin such constituents or subcategorize for them as a class.</Paragraph>
    <Paragraph position="4"> (Pareschi &amp; Steedman, 1987) do tackle the CCG case, but (Hepple, 1987) shows their algorithm to be incomplete.</Paragraph>
    <Paragraph position="5"> 3 Overview of the Parsing Strategy As is well known, general CFG parsing methods can be applied directly to CCG. Any sort of chart parser or non-deterministic shift-reduce parser will do. Such a parser repeatedly decides whether two adjacent constituents, such as S/NP and I~P/N, should be combined into a larger constituent such as S/N.</Paragraph>
    <Paragraph position="6"> The role of the grammar is to state which combinations are allowed. The key to efficiency, we will see, is for the parser to be less permissive than the grammar--for it to say &amp;quot;no, redundant&amp;quot; in some cases where the grammar says &amp;quot;yes, grammatical.&amp;quot; (5) shows the constituents that untrammeled CCG will find in the course of parsing &amp;quot;John likes Mary.&amp;quot; The spurious ambiguity problem is not that the grammar allows (5c), but that the grammar allows both (5f) and (5g)--distinct parses of the same string, with the same meaning.</Paragraph>
    <Paragraph position="7">  (5) a. \[John\]s/(s\sp) b. \[likes\](S\NP)/Np C. \[John likes\]s/N P d. \[Mary\]N P e. \[likes Mary\]s\N P f. \[\[John likes\] Mary\]s ~ to be disallowed  g, \[John \[likes Mary\]Is The proposal is to construct all constituents shown in (5) except for (5f). If we slightly constrain the use of the grammar rules, the parser will still produce (5c) and (5d)--constituents that are indispensable in contexts like (1)--while refusing to combine those constituents into (5f). The relevant rule S/I~P NP --* S will actually be blocked when it attempts to construct (5f). Although rule-blocking may eliminate an analysis of the sentence, as it does here, a semantically equivalent analysis such as (5g) will always be derivable along some other route.</Paragraph>
    <Paragraph position="8"> In general, our goal is to discover exactly one analysis for each &lt;substring, meaning&gt; pair. By practicing &amp;quot;birth control&amp;quot; for each bottom-up generation of constituents in this way, we avoid a population explosion of parsing options. &amp;quot;John likes Mary&amp;quot; has only one reading semantically, so just one of its analyses (5f)-(5g) is discovered while parsing (6). Only that analysis, and not the other, is allowed to continue on and be built into the final parse of (6). (6) that galoot in the corner that thinks \[John likes Mary\]s For a chart parser, where each chart cell stores the analyses of some substring, this strategy says that all analyses in a cell are to be semantically distinct. (Karttunen, 1986) suggests enforcing that property directly--by comparing each new analysis semantically with existing analyses in the cell, and refusing to add it if redundant--but (Hepple &amp; Morrill, 1989) observe briefly that this is inefficient for large charts. 3 The following sections show how to obtain effectively the same result without doing any semantic interpretation or comparison at all.</Paragraph>
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