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<Paper uid="P98-1057">
  <Title>Group Theory and Linguistic Processing*</Title>
  <Section position="6" start_page="350" end_page="351" type="concl">
    <SectionTitle>
5 Parsing
</SectionTitle>
    <Paragraph position="0"> To the compatible preorder ~ on F(V) there corresponds a &amp;quot;reverse&amp;quot; compatible preorder ---, defined as a ---, b iff b ~ a, or, equivalently, a- 1 __+ b- 1. The normal submonoid M' in F(V) associated with ---, is the inverse monoid of the normal submonoid M associated with ~, that is, M' contains a iff M contains a- 1.</Paragraph>
    <Paragraph position="1"> It is then clear that one can present the relations: j john-i--+ 1 A-Ir(A) ran -I-+ 1 sm(N,X,P\[X\]) P\[X\]-I~-IX N-isom e-l-+ etc.</Paragraph>
    <Paragraph position="2"> in the equivalent way: john j -1._., 1 ran r (A) -IA ---7 1 some N x-lo ' P\[X\] etc.</Paragraph>
    <Paragraph position="3"> sm(N,X,P\[X\])-1~-1-v 1 Long-distance movement and quantifiers The second and third examples are parallel to each other and show the derivation of the same string ever}' man saw some woman from two different logical forms. The penultimate and last steps of each example are the most interesting. In the penultimate step of the second example,/3 is instanciated to saw -1 x -1 . This has the effect of &amp;quot;moving&amp;quot; as a whole the expression some woman y-~ to the position just before y, and therefore to allow for the cancellation of y- * and y. The net effect is thus to &amp;quot;replace&amp;quot; the identifier y by the string some woman; in the last step c~ is instanciated to the neutral element 1, which has the effect of replacing x by ever}' man. In the penultimate step of the third example, a. is instanciated to the neutral element, which has the effect of replacing x by every man; then fl is instanciated to saw-1man-levery-1, which has the effect of replacing y by some woman. Remark. In all cases in which an expression similar to a al ... am a-1 appears (with the ai arbitrary vocabulary elements), it is easily seen that, by giving a an appropriate value in F(V), the al ... am can move arbitrarily to the left or to the right, but only together in solidarity; they can also freely permute cyclically, that is, by giving an appropriate value to a, the expression a al ... am a -l can take on the value ak ak+l ..a,,, al * *, ak-1 (other permutations are in general not possible). The values given to the or, fl, etc., in the examples of this paper can be understood intuitively in terms of these two properties.</Paragraph>
    <Paragraph position="4"> We see that, by this mechanism of concerted move- null Suppose now that we move to the right of the --7 arrow all elements appearing on the left of it, but for the single phonological element of each relator. We obtain the rules of Fig. 4, which we call the &amp;quot;parsing-oriented&amp;quot; rules associated with the G-grammar.</Paragraph>
    <Paragraph position="5"> By the same reasoning as in the generation case, it is easy to show that any derivation using these rules and leading to the relation PS --, LF, where PS is a phonological string and LF a logical form, corresponds to a public result LF PS -1 in the G-grammar.</Paragraph>
    <Paragraph position="6"> A few parsing examples are given in Fig. 5; they are the converses of the generation examples given earlier. In the first example, we first rewrite each of the phonological elements into the expression appearing on  ~-*yM-lw the right-hand side of the rules (and where the meta-variables have been renamed in the standard way to avoid name clashes). The rewriting has taken place in parallel, which is of course permitted (we could have obtained the same result by rewriting the words one by one). We then perform certain unifications: A is unified with j, C with p; then B is unified to 1. 5 Finally E is unified with s ( j, i ), and we obtain the logical form +- ( s ( j, 3. ), p ). In this last step, it might seem feasible to unify v. to +- (E, p) instead, but that is in fact forbidden for it would mean that the logical form -i ( E, p) is not a finite tree, as we do require. This condition prevents &amp;quot;self-cancellation&amp;quot; of a logical form with a logical form that it strictly contains.</Paragraph>
    <Paragraph position="7"> Quantifier scoping In the second example, we start by unifying m with N and w with M; then we &amp;quot;move&amp;quot; P\[x\] -1 next to s (A,B) by taking a = xA-1; 6 then again we &amp;quot;move&amp;quot; Q \[y\] -1 next to s (A, B) by taking fl = B sm (w, y, Q \[y\] ) -1; x is then unified with A and y with B. This leads to the expression: ev(m,x, P\[x\] ) P\[x\]-ls (x, y)Q\[y\]-lsm(w, y,Q\[y\] ) where we now have a choice. We can either unify s(x,y) with Q\[y\], or with P\[x\]. In the 5Another possibility at this point would be to unify 1 with E rather than with E. This would lead to the construction of the logical form i ( 1, p ), and, after unification of E with that logical form, would conduct to the output s ( j, i ( 1, p) ). If one wants to prevent this output, several approaches are possible. The first one consists in typing the logical form with syntactic categories. The second one is to have some notion of logical-form well-formedness (or perhaps interpretability) disallowing the logical forms i ( 1, p) \[louise in paris\] or i ( t (w), p) \[(the woman) in paris\], although it might allow the form t (i (w, p) ) \[the (woman in paris)\].</Paragraph>
    <Paragraph position="8"> t'We have assumed that the meta-variables corresponding to identifiers in P and Q have been instanciated to arbitrary, but different, values x and y. See (Dy,netman, 1998) for a discussion of this point. first case, we continue by now unifying P Ix\] with sm(w,y,s(x,y) ), leading to the output ev(m,x, sm(w,y,s(x,y))). In the second case, we continue by now unifying Q\[y\] with ev(m,x,s(x,y) ), leading to the output sm(w,y, ev(m,x,s(x,y)). The two possible quantifier scopings for the input string are thus obtained, each corresponding to a certain order of performing the unifications.</Paragraph>
  </Section>
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