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<Paper uid="J88-4001">
  <Title>LFP: A LOGIC FOR LINGUISTIC DESCRIPTIONS AND AN ANALYSIS OF ITS COMPLEXITY</Title>
  <Section position="12" start_page="0" end_page="0" type="concl">
    <SectionTitle>
5 APPLICATIONS TO HEAD GRAMMARS
</SectionTitle>
    <Paragraph position="0"> In this section we express head grammars (Pollard 1984) in ILFP, and thus show that head languages can be recognized in polynomial time. Since the class of head languages is the same as the class of tree adjunct languages (Vijayashankar, Joshi 1985), we get the same result for this class. We will actually give only a simplified version of head grammars to make our ILFP formulas easy to write. This version corresponds exactly to the Modified Head Grammars of Vijayashankar and Joshi (1985), and differs only from the original version in that it does not treat the empty string. (Roach (1988) has an extended discussion of head languages.) We define a head grammar as a tuple G = (N,~,,P,S), where N and E are finite nonterminal and terminal alphabets, P is a finite set of productions, and S is the start nonterminal. The productions are of the form C Op(A,B), where A,B, and C are nonterminals and Op is chosen from a fixed set of head-wrapping operations.</Paragraph>
    <Paragraph position="1"> Productions can also have the form C ~ (x,y), where x and y are terminal strings.</Paragraph>
    <Paragraph position="2"> We view nonterminals in N as deriving pairs of strings (u, v). In the original formulation, this meant that the head of the string uv occurred at the end of u. The wrapping operations come from the set {LLI,LL2,LC ~, LC2}. We consider LL 2 and LC ! as examples. We define LL2((w,x),(u,v)) = (wu, vx). Thus if A derives (w,x) and B derives (u, v), and C ~ LL2(A,B) is a production, then C derives (wu,vx). Similarly, LCl((W,X),(u,v)) = (w,xuv), so in the corresponding case, we would have C derives (w,xuv) if C ~ LCI(A,B) were a production. A string t is in L(G) iff for some u and v, t = uv and S derives (u,v).</Paragraph>
    <Paragraph position="3"> Given a head grammar, we write an ILFP recursion scheme as follows. For each nonterminal C, we introduce a predicate C(ij, k,l). We think of these four integers as indexing the positions of symbols in a string, starting at the left with 0. Then C(ij, k,l) means that the nonterminal symbol C can derive the pair of substrings of the input string between i and j, and between k and l inclusive. Thus, if C ~ LL2(A,B) is a production, our scheme would include a clause C(ij, k,l) C/:~ (3pq)(A(i,p,q + 1,l) A B(p + ld',k,q)) Similarly, if C --~ LCI(A,B) were a production, we would have C(ij, k,l) C/~ (3pq)(A(ij, k,p) /~ B(p + 1,q,q + 1,/)) Finally, if C--* (a, bb)were a terminating production, we would have C(ij, k,l) C/:~ a(i) /~ i = j/~ k = i + 1/~ b(k)</Paragraph>
    <Paragraph position="5"> The grammar would be defined by the recursion scheme and the assertion 3jS(0jj + 1 ,last), where S is the start symbol of G.</Paragraph>
    <Paragraph position="6"> It can be seen from this formulation that every head grammar can be written as an ILFP scheme with at most six total variables. Section 4 thus gives us an O(n 18) algorithm. However, the algorithm of Vijayashanker and Joshi (1985) is at most n 6. It would seem that a rule of thumb for the order of the polynomial algorithm is to use the number y(th) for the ILFP scheme th, but we have no proof for this conjecture.</Paragraph>
  </Section>
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