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<Paper uid="P93-1045">
  <Title>Backward Application (&lt;): Y X\Y --~ X - Generalized Forward Composition</Title>
  <Section position="1" start_page="0" end_page="0" type="metho">
    <SectionTitle>
THE FORMAL CONSEQUENCES OF USING VARIABLES
IN CCG CATEGORIES
</SectionTitle>
    <Paragraph position="0"/>
  </Section>
  <Section position="2" start_page="0" end_page="299" type="metho">
    <SectionTitle>
Abstract
</SectionTitle>
    <Paragraph position="0"> Combinatory Categorial Grammars, CCGs, (Steedman 1985) have been shown by Weir and loshi (1988) to generate the same class of languages as Tree-Adjoining Grammars (TAG), Head Grammars (HG), and Linear Indexed Grammars (LIG). In this paper, I will discuss the effect of using variables in lexical category assignments in CCGs. It will be shown that using variables in lexical categories can increase the weak generative capacity of CCGs beyond the class of grammars listed above.</Paragraph>
    <Paragraph position="1"> A Formal Definition for CCGs In categorial grammars, grammatical entities are of two types: basic categories and functions. A basic category such as NP serves as a shorthand for a set of syntactic and semantic features. A category such as S\NP is a function representing an intransitive verb; the function looks for an argument of type NP on its left and results in the category S. A small set of combinatory rules serve to combine these categories while preserving a transparent relation between syntax and semantics. Application rules allow functions to combine with their arguments, while composition rules allow two functions to combine together.</Paragraph>
    <Paragraph position="2"> Based on the formal definition of CCGs in (Weir-Joshi 1988), a CCG, G, is denoted by (VT , VN , S, f, R), where  - VN C C(VN) and - if el and c2 * C(VN), then (el\c2) and (cl/c2) * *I would like to thank Mark Steedman, Libby Levison, Owen Rambow, and the anonymous referees for their valuable advice. This work was partially supported by DARPA N00014-90-J- 1863, ARO DAAL03-89-C-0031, NSF IR19016592, Ben Franklin 91S.3078C-1.</Paragraph>
    <Paragraph position="3"> * R is a finite set of combinatory rules where X, Y, Z1, * *., Zn are variables over the set of categories C(VN), and the slash variable \]i can bind to \ or/. Certain restrictions may be placed on the possible instantiations of the variables in the rules. - Forward Application (&gt;):  The derives relation in a CCG is defined as act =~ cecl c2fl if R contains the rule Cl c2 ---+ c. The language generated by this grammar is defined as</Paragraph>
    <Paragraph position="5"> Under these assumptions, Weir and Joshi (1988) prove that CCGs are weakly equivalent to TAGs, HGs, and LIGs. Their conversion of a CCG to a LIG l relies on the fact that the combinatory rules in the CCG are linear. To preserve linearity in CCGs, only the category X in the combinatory rules can be unbounded in size; the variables Y and Z must be bounded in their possible instantiations. In other words, only a finite number of categories can fill the secondary constituent of each combinatory rule. The secondary constituent is the second of the pair of categories being combined in the forward rules and the first of the pair in the backward rules (e.g. Y\]Z1...IZn).</Paragraph>
    <Paragraph position="6"> Weir and Joshi do not restrict the size of the secondary constituents in the formal definition of the CCG rules, but they prove that the following lemma holds of the grammar.</Paragraph>
    <Paragraph position="7"> ILinear Indexed Grammars are a restricted version of Indexed Grammars in which no rule can copy a stack of unbounded size to more than one daughter (Gazdar 1985).  Lemma: There is a bound (determined by the grammar G) on the number of useful categories that can match the secondary constituent of a rule.</Paragraph>
    <Paragraph position="8"> There are an infinite number of derivable categories in CCGs, however Weir and Joshi show that the number of components that derivable categories have is bounded. The components of a category c = (col,cll2...I, c,d are its immediate components co, ..., c,~ and the components of these immediate components. A finite set Dc(G) can be defined that contains all derivable components of every useful category where a category c is useful if c ~ w for some w in VT*: c 6 De(G) ifc is a component ofd where c' 6 f(a) for some a 6 VT U {E}.</Paragraph>
    <Paragraph position="9"> Given that every useful category matching the secondary constituents Y and Y IZ1...IZ,~ in the combinatory rules has components which are in Dc(G), the lemma given above holds.</Paragraph>
    <Paragraph position="10"> However, this lemma does not hold if there are variables in the lexical categories in VT. Variables can bind a category of any size, and thus useful categories containing variables do not necessarily have all of their derivable components in the finite set Dc (G).</Paragraph>
    <Section position="1" start_page="298" end_page="299" type="sub_section">
      <SectionTitle>
The Use of Variables
Linguistic Use
</SectionTitle>
      <Paragraph position="0"> In CCGs, a type-raising rule can be used in the lexicon to convert basic elements into functions; for example, an NP category can be type-raised to the category S/(S\NP) representing a function looking for an intransitive verb on its right. Steedman uses type-raising of NPs to capture syntactic coordination and extraction facts. In Steedman's Dutch grammar (1985), variables are used in the lexical category for type-raised NPs, i.e. the variable v in the category v/(v\NP) generalizes across all possible verbal categories. The use of variables allows the type-raised NPs in the following coordinated sentence to easily combine together, using the forward composition rule, even though they are arguments of different verbs.</Paragraph>
      <Paragraph position="1">  I will show that the use of variables in assigned lexical categories increases the weak generative capacity of CCGs. VAR-CCGs, CCGs using variables, can generate languages that are known not to be Tree-Adjoining Languages; therefore VAR-CCGs are more powerful than the weakly equivalent TAG and CCG formalisms.</Paragraph>
      <Paragraph position="2"> The following language is known not to be a TAL:</Paragraph>
      <Paragraph position="4"> The following VAR-CCG, G', generates a language L' which is very similar to L:</Paragraph>
      <Paragraph position="6"> The rules allowed in this grammar are forward and backward application and forward crossing composition with n &lt; 2. The variable v can bind an arbitrarily large category in the infinite set of categories C(Vjv) defined for the grammar.</Paragraph>
      <Paragraph position="7"> In the language generated by this grammar, two characters of the same type can combine together using the forward crossing composition rule &gt;Bx(2). The composition of the types for the character e is shown below. A string of e's can be constructed by allowing the result of this composition to combine with another e category.</Paragraph>
      <Paragraph position="9"> The types for the characters b, c, and d can combine using the same composition rule; these types contain variables (e.g. v and v' below) which can bind to a category of unbounded size.</Paragraph>
      <Paragraph position="11"> By applying the forward crossing composition rule to a string of nb's, we can form the complex category v\A, . . . d,~/(v\Bl . . . Bn) representing this string.</Paragraph>
      <Paragraph position="12"> Thus, during the derivation of anbnc'~d'~e n for n &gt; O, the following complex categories are created: v\A, .. .At/(v\B1. . .B,) ... B /(vkC,</Paragraph>
      <Paragraph position="14"> Once the complex categories for a string of b's, a string of c's, a string of d's, and a string of e's are constructed, we can link one string of a particular character to another using the forward application rule. This rule can only apply to these categories if</Paragraph>
      <Paragraph position="16"> ber of A's generated and i, j, k, l are as in the complex categories listed above. For example,</Paragraph>
      <Paragraph position="18"> With each succesful forward application, we ensure that there are equal numbers of two characters: the E's are linked to the D's, the D's are linked to the C's, etc., so that we have the exact same number of all five characters. In fact, the grammar can be easily extended ,,n .. ~ 0} to generate a language such as {a 1 a 2 . at: In _&gt; for any k.</Paragraph>
      <Paragraph position="19"> The language L' generated by G / intersected with the regular language a*b*c*d*e* gives the language L above. If we assume that L' is a Tree-Adjoining Language (TAL), then L would be a TAL as well since TALs are closed under intersection with Regular languages. However, since we know that L is not a TAL, L' cannot be a TAL either. Thus, G' generates a language that TAGs and CCGs cannot.</Paragraph>
      <Paragraph position="20"> Conclusions We have seen that using variables in the lexical categories of a CCG can increase its weak generative capacity. However, there is some linguistic motivation for looking at the more powerful formalism of VAR-CCGs. As argued by Gazdar (1985), this extra power may be necessary in order to capture coordination in natural languages. We have seen that type-raised categories with variables in CCGs can be used to capture syntactic coordination and extraction facts in Dutch (Steedman 1985). Further research is needed to decide whether this linguistic motivation warrants the move to a more powerful formalism.</Paragraph>
      <Paragraph position="21"> Although VAR-CCGs have a greater weak generative capacity than the class including TAGs, HGs, CCGs, and LIGs, we conjecture that it is still a mildly context-sensitive grammar as defined by Joshi (1985).</Paragraph>
      <Paragraph position="22"> The language discussed above is a mildly context-sensitive language since it observes the constant growth and semilinearity properties. It is an open question whether VAR-CCGs can generate languages which are beyond mildly context-sensitive. Note that MC-TAGs, which are a more powerful extension of TAGs, can also generate languages like L, and they are known to be mildly context-sensitive formalisms (Weir 1988). In future research, we will investigate exactly what the resulting generative capacity of VAR-CCGs is.</Paragraph>
      <Paragraph position="23"> Future Research on Word Order My current research also involves extending the CCG formalism to handle free word order languages. By representing NPs as type-raised categories, we can derive a scrambled sentence in which the NPs do not occur in the order that the verb specifies:  In many free word order languages, an NP can be scrambled an unbounded distance away from its verb, i.e. long distance scrambling. If we allow unrestricted composition rules for any n arguments as well as the use of variables in type-raised categories in a CCG, a string of any number of scrambled NPs followed by a string of verbs can be derived. We first combine any number of verbs together, using backward composition, to get a complex verb category looking for all of the NPs; next, we combine each NPs with this complex verb category. Any type-raised Npi can combine with the complex verb regardless of the order specified by the complex verb. The variable in the type-raised category can bind a verbal category of unbounded size, e.g. (v = S\Up,\...\Upi_l).</Paragraph>
      <Paragraph position="25"> Although we can capture scrambling by using variables in type-raised categories, this analysis is not consistent with incremental processing and cannot account for co-ordination in scrambled sentences; for instance, in the first example given above, NP2 and NP1 cannot combinetogether before combining with the verb. In future research, I will investigate whether VAR-CCGs is an adequate linguistic formalism in capturing all aspects of free word order languages or whether a formalism such as {}-CCGs (Hoffman 1992), which allows sets of arguments in function categories, is better suited.</Paragraph>
    </Section>
  </Section>
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