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<?xml version="1.0" standalone="yes"?> <Paper uid="P97-1069"> <Title>Generative Power of CCGs with Generalized Type-Raised Categories</Title> <Section position="4" start_page="0" end_page="514" type="metho"> <SectionTitle> 2 CCGs with Generalized Type-Raised Categories </SectionTitle> <Paragraph position="0"> In languages like Japanese, multiple NPs can easily form a non-traditional constituent as in &quot;\[(Subj I Objl) & (Subj2 Obj2)\] Verb&quot;. The proposed ~ammars (CCG-GTRC) admit lexical type-raised categories (LTRC) of the form &quot;1&quot;/(T\a) or'l'\ (T/a) where T is a variable over categories and a is a constant category (Const). 2 Then, composition of LTRCs can give rise to a class of categories having the formT/(T\a .... \at) or T\ (T/a .... /at), representing a multiple-NP constituent exemplified by &quot;Subjl Objt&quot;.</Paragraph> <Paragraph position="1"> We call these categories generalized type-raised categories (GTRC) and each ai of a GTRC an argument (of the GTRC).</Paragraph> <Paragraph position="2"> The introduction of GTRCs affects the use of combinatory rules: functional application &quot;>: z/y + y ---, z&quot; and generalized functional composition &quot;>B ~ (x) : z/y + ylzt ...\[zk --- zlzl ...\[z~&quot; where k is bounded by a grammar-dependent kma~ as in CCG-Std. 3 This paper assumes two constraints defined for the grammars and one condition stipulated to control the formal properties.</Paragraph> <Paragraph position="3"> The following order-preserving constraint, which follows more primitive directionality features (Steedman, 1991), limits the directions of the slashes in GTRCs.</Paragraph> <Paragraph position="4"> (1) In a GTRC &quot;1&quot;\[o (T\[,a .... Ira,), the direction of \[0 must be the opposite to any of In, ..., \]b This prohibits functional composition '>Bx' on 'GTRC+GTRC' pairs so that &quot;T/(T\A\B) + U\(U/C/D)&quot; does not result in T\ (T\A\B/C/D) or U/(UIC/D\A\B). That is, no movement of arguments across the functor is allowed.</Paragraph> <Paragraph position="5"> The variable constraint states that: (2) Variables are limited to the defined positions in GTRCs.</Paragraph> <Paragraph position="6"> This prohibits '>B k (x)' with k > I on the pair 2Categories are in the &quot;result-leftmost&quot; representation and associate left. Thus a/b/c should be read as (a/b)/c and returns a/b when an argument c is applied to its right. A ..... Z stand for nonterminals and a,...,z for complex, constant categories.</Paragraph> <Paragraph position="7"> 3There are also backward rules (<) that are analogous to forward rules (>). Crossing rules where zt is found in the direction opposite of that of y are labelled with 'x'. 'k' represents the number of arguments being passed. '\[' stands for a directional meta-variable for {/, \}.</Paragraph> <Paragraph position="8"> 'Const+GTRC'. For example, '>B 2' on &quot;A/B + T/(TkC)&quot; cannot realize the unification of the form &quot;A/B + TrITe./(TtITz\C)&quot; (with T = TilT,_) resulting in &quot;AIT,./(BITz\C)&quot;.</Paragraph> <Paragraph position="9"> In order to assure the expected generative capacity, we place a condition on the use of rules. The condition can be viewed in a way comparable to those on rewriting rules to define, say, context-free grammars. The bounded argument condition ensures that every argument category is bounded as follows: (3) '>B (x)' should not apply to the pair 'Const+GTRC'.</Paragraph> <Paragraph position="10"> For example, this prohibits &quot;A/ B + T~ (TkC....\Ct) --A/(B\C,...\Cl)&quot;, where the underlined argument can be unboundedly large. These constraints and condition also tell us how we can implement a CCG-GTRC system without overgeneration.</Paragraph> <Paragraph position="11"> The possible cases of combinatory rule application are summarized as follows: (4) a. For 'Const+Const', the same rules as in CCG-Std are applicable.</Paragraph> <Paragraph position="12"> b. For 'GTRC+Const', the applicable rules are: (i) >: e.g., &quot;T/(TkAkB) + SkAkB -- S&quot; (ii) >B k (x): e.g., &quot;T/(TkA\B) + SkA\BkC/D -. S\C/D'&quot; c. For 'Const+GTRC', only '>' is possible: e.g., &quot;S/ (S/ (S\B)) +r/(T\B) --, S&quot; d. For 'GTRC+GTRC', the possibilities are: (i) >: e.g., &quot;T/(mx (S/A/B)) + Tk (T/A/B) (ii) >B: e.g., &quot;T/(T\A\B) + T/(T\C\D) -.</Paragraph> <Paragraph position="13"> T/(TkAkB\C\D)&quot; CCG-GTRC is defined below where g, ta and ~a,rc represent the classes of the instances of CCG-Std and CCG-GTRC, respectively: Definition 1 Gatrc is the collection of G's (extension of a G E G, ta) such that: l. For the lexical function f of G (from terminals to sets of categories), if a E f (a), f' may additionally include { (a, T/(T\a)), (a, T\ (T/a)) }.</Paragraph> <Paragraph position="14"> 2. G' may include the rule schemata in (4).</Paragraph> <Paragraph position="15"> The main claim of the paper is the following: Proposition 1 ~9*~e is weakly equivalent with ~,ta.</Paragraph> <Paragraph position="16"> We show the non-trivial direction: for any G' E Ggt~c, there is a G&quot; 6 ~,,a such that L (G') = L (G&quot;). As G' corresponds to a unique G E ~,ta, we extend G&quot; from G to simulate G', then show that the languages are exactly the same.</Paragraph> <Paragraph position="17"> Std where the lexicon does not involve GTRCs. In order to (statically) simulate (5b) by a CCG-Std, we add S\BkA to the value of f&quot; (c) in the lexicon of G'. Let us call this type of relation between the original S\A\B and the S\B\]\A\] wrapping, due to its resemblance to the new operation of the same name in (Bach, 1979). There are two potential problems with this simple augmentation. First, wrapping may affect unboundedly long chunks of categories as exemplified in (6). Second, the simulation may overgenerate. We discuss these issues in turn.</Paragraph> <Paragraph position="19"> We need S\~ -- \AXB...kAkB 1 which can be the result of unboundedly-long compositions, to simulate (6) without depending on the GTRCs. Intuitively, this situation is analogous to long-distance movement of C from the position left of SkAkB...kC to the sentence-initial position.</Paragraph> <Paragraph position="20"> In order to deal with the first problem, the following key properties of CCG-GTRC must be observed: (7) a. Any derived category is a combination of lexical categories. For example,</Paragraph> <Paragraph position="22"> b. Wrapping can occur only when GTRCs are involved in the use of'> Bkx ' and can only cross at most km~= arguments. Since there are only finitelymany argument categories, the argument(s) being passed can be encoded in afinite store.</Paragraph> <Paragraph position="23"> For derivable categories bounded by the maximum number of arguments of a lexical category, we add all the instances of wrapping required for simulating the effect of GTRC into the lexicon of G&quot;. For the unbounded case, we extend the lexicon as in the following example: (8) a. For a category S\A\B\C, add S{\c}\AkB to the lexicon.</Paragraph> <Paragraph position="24"> b. For SkA\BkS, add S{\c}\A\BkS{\c}, S\A\B\C\S{\c} ..... S\C~\S{\c}.</Paragraph> <Paragraph position="25"> S{\c} is a new category representing the situation where \C is being passed across categories. Thus \C which originatedin SkAkB\C in (a) may be passed onto another category in (b), after a possibly unbounded number of compositions as follows:</Paragraph> <Paragraph position="27"> Now, both of the permutations in (5) can be derived in this extension of CCG-Std. The finite lexicon with finite extension assures the termination of the process. This covers the case (4bii).</Paragraph> <Paragraph position="28"> Case (4e) can be characterized by a general pattern &quot;cl (hi (b\ak...\a,)) + T/(T\ak...\a,) --* c&quot; where T = b. Since any argument category is bounded, we can add b/(b\ak...\a~) 6 f' (al...a,) in the lexicon as an idiom. The other cases do not require simulation as the same string can be derived in the original grammar.</Paragraph> <Paragraph position="29"> The second problem of overgeneration calls for another step. Suppose that the lexicon includes</Paragraph> <Paragraph position="31"> {E\(S\B\A)} and that S\BF~ is added to f(c) by wrapping. To avoid generating an illegal string &quot;c e&quot; (in addition to the legal &quot;de&quot;), we label the state of wrapping as S\Bt+~o,~pl\[ \A~+,~,.~,p\] t The original entries can be labelled as S\Bt .... p\]\A\[ .... pj and E\ (S\B\[ .... pj\A\[ .... pl). The lexical, argument categories, e.g., A, are underspecified with respect to the feature. Since finite features can be folded into a category, this can be written as a CCG-Std without features.</Paragraph> <Paragraph position="32"> 4 Equivalence of the Two Languages Proposition I can be proved by the following lemma (as a special case where c = S): Lemma 1 For any G' 6 Ggtre (an extension of G), there is a G&quot; 6 ~,td such that a string w is derivable from a constant category c in G' iff (~) w is derivable from c in Gll * The sketch of the proof goes as follows. First, we construct G&quot; from G' as in the previous section. Both directions of the lemma can be proved by induction on the height of derivation. Consider the direction of '---.'. The base (lexical) case holds by definition of the grammars.</Paragraph> <Paragraph position="33"> For the induction step, we consider each case of rule application in (4). Case (4a) allows direct application of the induction hypothesis for the substructure of smaller height starting with a constant category. Other cases involve GTRC and require sublemmas which can be proved by induction on the length of the GTRC. Cases (4hi, di) have a differently-branching derivation in G&quot; but can be derived without simulation. Cases (4bii, c) depend on the simulation of the previous section. Case (4dii) only appears in sublemmas as the result category is GTRC. In each sublemma, the induction hypothesis of Lemma 1 is applied (mutually recursively) to handle the derivations of the smaller substructures from a constant category.</Paragraph> <Paragraph position="34"> A similar proof is applicable to the other direction.</Paragraph> <Paragraph position="35"> The special cases in this direction involves the feature \[+wrap\] and/or the new categories of the form 'z{...}' which record the argument(s) being passed. As before, we need sublemmas to handle each case. The proof of the sublemma involving the 'z{...}' form can be done by induction on the length of the category.</Paragraph> </Section> class="xml-element"></Paper>