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<Paper uid="C90-2041">
  <Title>THE COMPLEXITY OF PARSING WITH EXTENDED CATEGOP IAL GP AMMARS</Title>
  <Section position="1" start_page="0" end_page="0" type="metho">
    <SectionTitle>
THE COMPLEXITY OF PARSING WITH
EXTENDED CATEGOP IAL GP AMMARS
</SectionTitle>
    <Paragraph position="0"/>
  </Section>
  <Section position="2" start_page="0" end_page="0" type="metho">
    <SectionTitle>
Abstract
</SectionTitle>
    <Paragraph position="0"> Instead of incorporating a gap-percolation mechanism for handling certain &amp;quot;movement&amp;quot; phenomena, the extended categorial grammars contain special inference rules for treating these problems. The Lambek categorial grammar is one representative of the grammar family under consideration. It allows for a restricted use of hypothetical reasoning. We define a modification of the Cocke-Younger-Kasami (CKY) parsing algorithm which covers this additional deductive power and analyze its time complexity.</Paragraph>
  </Section>
  <Section position="3" start_page="0" end_page="0" type="metho">
    <SectionTitle>
1 Introduction
</SectionTitle>
    <Paragraph position="0"> Categorial grammars have become attractive in the last decade. One reason might be the rediscovery of their conceptual elegance: the combinetory potential of a lexeme is stated in direct association with the lexeme itself- in the lexicon. A basic (bidirectional) categorial grammar B is defined by a set of categories C := C0U{zl ~ = x/Y or z =x\y; x, yC C} (C0 a finite set of basic categories, the category y is referred to as the argument category, the category x is called value category, complex categories are named functor categories), a goal category g (the start symbol) which is a basic category, a lexicon L which is a function from a finite set of lexemes onto a set of finite sets of categories, and the two combination rules &amp;quot;leftward application&amp;quot; (app\) and &amp;quot;rightward application&amp;quot; (app/) which state how argument positions are filled: (app\) y, xky ~ x An object U --* x where U is a sequence of categories is called a sequent.</Paragraph>
    <Paragraph position="1"> This basic concept of categorial grammar may be extended by adding more combinatory rules. E.g.</Paragraph>
    <Paragraph position="2"> the rule of functional composition is the incarnation of the idea how to handle certain phenomena of unbounded dependencies: if a funetor category x/y finds only an incomplete argument category y/z, i.e. which, itself, still lacks an argument z, then these two partial categories can be, nevertheless, combined into the category x/z, which expresses the fact that z is still missing.</Paragraph>
    <Paragraph position="3"> In general, functional composition cannot do its job by itself. One needs rules which allow for changing the order in which a functor category takes its left arguments with regard to its right arguments.</Paragraph>
    <Paragraph position="4"> The use of the type raising rules is one way to obtain this goal: (tr\) y-~ x\(x/y) (tr/) y xl( \y) The first extensive use of the concept of functional composition in syntax appears in the paper lAdes, Steedman 1982\]. The rules of functional coinposition and type raising have already been mentioned in \[Lambek 1958\] as being theorems of the Lambek calculus.</Paragraph>
    <Paragraph position="5"> To obtain a Lambek categorial grammar L from a basic categorial grammar, one has to add the two rules of functional abstraction (T is a non-empty sequence of categories): :/Y' T --+ x (ab tr\) --~ x\y (abstr/) T, y --~ x T--~ x/y The rule (abstr\) reads: If x can be derived from the category sequence 7' under the assumption that y has been put in front of T then it holds that x\y is derivable from T alone. An example of an L-derivation in a somewhat abbreviated notation 1 is given in figure 22. This analysis is an adaptation of ideas due to \[Hepple 1990a\].</Paragraph>
    <Paragraph position="6"> In the remainder of the paper, we first give a brief overview on other approaches to parsing with extended categorial grammars. Then the modified CKY-algorithm is presented.</Paragraph>
  </Section>
  <Section position="4" start_page="0" end_page="0" type="metho">
    <SectionTitle>
2 General Processing Issues
</SectionTitle>
    <Paragraph position="0"> in connection with parsers for extended categorial grammars, the problem of &amp;quot;spurious ambiguities&amp;quot; or derivational equivalence has appeared in a massive way: One syntactic (or semantic) reading of a sentence can be derived in many, many ways. There have been several proposals in tile literature how to tackle this problem. The first group of approaches uses repeated equivalence tests such that the spurious ambiguities stay local and do not carry over</Paragraph>
    <Paragraph position="2"> globally to the final parse results (\[Karttunen 1989\], \[Hepple, Morrill 1989\]). But the presence of local spurious ambiguities means that still some superfluous work has to be done by the parser. The second group of :methods uses top-down information to avoid equivalent derivations completely. This has been illustrated for the method of prediclive combination in \[Wall, Wittenburg 1989\] where, instead of functional composition, rules like the following one are used: (PfCl) xll(x~lx3),x~lx4-+ xll(x41x3) This :method has the shortcoming of changing the set of derivable sequents with regard to the original rule set. Check e.g. the sequent xil(x2lx3), x~l(x~lxs), x~lx6, (x61xs)lxa --+ x~.</Paragraph>
    <Paragraph position="3"> Since those extended categorial grammars which are based on the explicit use of functional composition and type raising seem to be reluctant to leave behind the problem of spurious ambiguity, we have chosen the Lambek calculus as a different setup for an extended categorial grammar where those rules are theorems but not axioms. In the Lambek calculus, the problem of spurious ambiguities disappears since one can define a normal form for derivations which can be carried out on the fly (cf. \[Hepple 1990b\], \[KSnig 1989\]). One essential characteristic of this normal form is the following: If there is a choice between using an application rule and an abstraction rule, the abstraction rule is preferred. Another requirement is that no spontaneous uses of the abstraction rules do occur (cf. Prawitz normal form, \[Prawitz 1965\]). This means that an abstraction rule can only be triggered by lexical material.</Paragraph>
  </Section>
  <Section position="5" start_page="0" end_page="0" type="metho">
    <SectionTitle>
3 A chart parser for extended
</SectionTitle>
    <Paragraph position="0"> categorial grammar The CKY-algorithm (\[Aho, Ullman 1972\]) works bottom.-up. It uses a chart of well-formed substrings in order to avoid redundant work. By defining a parser for the basic categorial grammars, we introduce our particular representation of chart parsing.</Paragraph>
    <Section position="1" start_page="0" end_page="0" type="sub_section">
      <SectionTitle>
3.1 Chart parsing with a basic cate-
</SectionTitle>
      <Paragraph position="0"> gorial grammar A chart is a set of items. An item is a triple \[x, i,j\] which states the fact that the category x has been derived from the continuous part of the input string between the positions i and j. We use a slightly alienated sequent notation where the lefthand side of a sequent is the current chart, and the righthand side is the goal category g of the grammar. The *-sign marks the current scanning position.</Paragraph>
      <Paragraph position="1"> Figure 1 shows the processing steps which can be performed with a chart. The rule (axiom) describes the conditions for successful termination of the parsing process: The chart has been processed until its end and there exists an item which covers the whole input string of length n. The rule (scan) replaces a lexeme in the input string by one of the categories it is assigned to in the lexicon. The rule (complete\) implements leftward functional application: a leftward looking functor which follows immediately its argument causes the addition of a new item with the value category. For rightward functional application, this works symmetrically. The inference rule-style representation of our algorithm abstracts away from details dealing with the appropriate control mechanism which guarantees that e.g. all possible (complete\)-steps are performed with one item.</Paragraph>
      <Paragraph position="2"> The completer step is the part of the CKY-algorithm which determines its cubic time complexity. The applications of the rules (complete\) and (complete/) stay within the same time bound because the search for a pair of reducible items works as in the original algorithm. In particular, the number of items which span the same section of the input string has a constant bound which is the number of subcategories of the categories occuring in the lexicon. This number is proportional to the size of the finite lexicon.</Paragraph>
      <Paragraph position="4"/>
    </Section>
    <Section position="2" start_page="0" end_page="0" type="sub_section">
      <SectionTitle>
3.2 Extending the basic chart parser
</SectionTitle>
      <Paragraph position="0"> In the traditional concept of a chart every item is visible to every other item in the chart. For the Lambek calculus, general accessability is inadequate because of its use of hypothetical reasoning: The ab234 2 t;traction rules assert hypothetical constituents which :must only be used in the subproof for the premise ~equent. Thus, tile interaction of items must be regulated in a more fine-grained way.</Paragraph>
      <Paragraph position="1"> Fig. 3 shows a section of an L-derivation where ~ use of an abstraction rule occurs 3. The complex ~rgurnent of the emitter category */\[...</Paragraph>
      <Paragraph position="2"> ~;riggers the act of a,sserting its own arguments as hypothetical categories (in a fixed order) in addition to a given non-empty category sequence xi...xj. We (:all the sequence of hypothetical categories which are ~tsserted on the lefthand side tile left mini-chart, its symmetric counterpart is a right mini-chart. A subproof is carried out on the basis of these additional premise categories. If this subproof yields a category y, then x spans the whole sequence of premises under consideration.</Paragraph>
      <Paragraph position="3"> To pertbrm all abstraction rule, obviously an intermingling of top-down and bottom-up directed act,ions is needed. The sequence xi...xj, i.e. the po:i;itions where the mini-charts are attached, h~ to be chosen non-determinist\tally. Instead of asserting copies of the mini-charts at all positions in the current chart, it is more convenient to put the mini,:harts apart and to keel) them accessible from all chart positions. An adapted completer rule will built additional kems on the basis of the extended chart.</Paragraph>
      <Paragraph position="4"> Since there might be several emitter categories in lhe original sequent, it is important to provide the a:~serted mini-charts with enough information in of der to control their use. In particular, altachrae'al chai~zs can arise because, mird-charts ca&amp;quot;. be attached I.o other mini-charts.</Paragraph>
      <Paragraph position="5"> We assume that each value category of a cornplex argument of an emitter is marked with a unique number m. Items also have mfique nlHnbers a. A (left) index of an item is now a quadruple &lt;t, m, i, p) where t stands tbr the type of miui-.chart the \tern belongs to: 1 (left), r (right), or ~, (,~one); ,n is the ltumber of the subcategory which caused the assertion of this \tern; i is a position nulnber relative to the (re\hi-)chart wi~h type t and number m; p is the ~urnber of an item in another part of the &lt;:hart where the mini-chart is attached to. Por items which have been deriw,'d on the basis of the items given by the input string, t := n, m = 0, and p = 0.</Paragraph>
      <Paragraph position="6"> In figure 4, the extended algorithm is presented.</Paragraph>
      <Paragraph position="7"> From the following discussion, various details and, in particular, a proof of correctness, are omitted due to the laek of space. The rule (compl\s) is roughly equivalent to (complete\) in the basic algorithm: two items which are adjacent in the same piece of the (:hart are combined. The rule (compl\r) allows an il;em at tile beginning of a right mini-chart to cornline witll sorne other item with number al on its left. The function newlast adds the information td~out this new attachment point at the end of the 3In the following, symmetric cases are mosl, ly omitted. right attachment chain of item a2 (see item 6 in figure 2 where the item id 0 is replaced by the id 5 because of'a (compl\ r)-step which combines items 5 and 9). In order to avoid cycles, the intersection o\[' the mini-chart numbers which are fbund by traversing all the tbur attachment chain's has to be empty. The work of the abstraction rule (abstr\) is split into the two sub-tasks: The rule (emit\) pertbrms the assertion of the mini-charts. The rule (di,sc\) (&amp;quot;discharge&amp;quot;) is a specialization of the completer rule for fllnctor categories which are emitters. The conditions in the rule guarantee the following: Both mini-charts which are due to the current emitter must have been used completely in deriving y. The rigi~t index of the attachment point p4 of y (determined by a function ri) must be equal to the left index of the emitter. The value category x must not bear any information about the involvement of the current mini-charts in its derivation. This is achieved by going back to the attachment point of the left mini-chart.</Paragraph>
      <Paragraph position="8"> Since each of the O(n) mini-charts fbr an input string of length n can only be used once in a derivation, and since we currently do not know any reasonable restrictions, we assume that any one of the (c)(n!) permutations of the mini-charts possibly can be used in a specific derivation. For every item in the initial chart, O(n!) completer-steps can be possible.</Paragraph>
      <Paragraph position="9"> This means that the whole parser has a time complexity of O(n ~ x n!). In order to evaluate the result for tim given algorit, hm, one has to be aware of tile fact that. this parsing procedure can handle n-fold extraction from phrases - to speak linguistically.</Paragraph>
      <Paragraph position="10"> If we restrict ourselves to single extraction (this can be implemented by checking the length of the attachment chain before performing (compl\ r)) then the time complexity reduces drastically: The formula n!/(n - k')! tbr tile variations of length k of a set of length 'n equals n tbr k' = 1. Thus, the time cornplexity of the algorithm is O(n:~). If we want to describe two-tbld extraction like in figure 2, i.e. use a &amp;quot;2-restricted version of Lc&amp;quot;, the algorithm needs O(n 4) steps for deriving an input string of length n.</Paragraph>
      <Paragraph position="11"> Conjecture: 1-restricted Lc covers the rules of functional compositio n and type raising, whereas 2rest, rioted LC suffices to derive the rules of predictive combination.</Paragraph>
    </Section>
  </Section>
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