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<Paper uid="E93-1041">
  <Title>A Tradeoff between Compositionality and Complexity in the Semantics of Dimensional Adjectives</Title>
  <Section position="6" start_page="355" end_page="356" type="ackno">
    <SectionTitle>
Acknowledgements
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    <Paragraph position="0"> Thanks to Carola Eschenbach, Claudia Maienborn, Andrea Schopp, Heike Tappe and the referees for their comments on earlier versions of this paper.</Paragraph>
    <Paragraph position="1"> Thanks also to Longin Latecki for discussions about constraint propagation, and to Christopher Habel for encouraging me to pursue this work.</Paragraph>
    <Paragraph position="2"> Appendix In the following, the proof of the following theorem (from \[Simmons, 1993\]) is briefly outlined: Theorem 1 If a system of product constraints is consistent and its solution is admissible, then the Waltz algorithm brings it to quiesenee in time O(pS). Recall that a product constraint is of the form ~i Xi = Y, and that a labelling is admissible if it does not assign \[0, 0\] or \[c~, oo\] to any parameter. First we need some terminology defined in \[Davis, 1987, Appendix B\] (recall the definition of refinement operators in section 3.2 above).</Paragraph>
    <Paragraph position="3"> For a refinement operator R, let OUT(R) be the bound affected by R, and let ARGS(R) be the set of bounds other than OUT(R) that enter into the computation of OUT(R). Given a labelling L, R is active on L if it changes L, i.e. if L ~ R(L).</Paragraph>
    <Paragraph position="4"> A series of refinement operators T~ = (RI,..., Rm) is active if each refinement in T~ is active. We say that Ri is an immediate predecessor of Rj in 7~ if i &lt; j, OUT(Ri) E ARGS(Rj), and for all k such that i &lt; k &lt; j, OUT(Rk) # OUT(I~). In other words, some argument of P~ has been set most recently in the series by Rj. We say that Ri depends on Rj if either i = j or Ri depends on Rk and Rj is an immediate predecessor of Rk. Thus the dependence relation is the transitive and reflexive closure of the immediate precedence relation. We say that Ri depends on bound B if for some Rj, Ri depends on Rj and B E ARGS(Rj).</Paragraph>
    <Paragraph position="5"> The series of refinements T~ = (R1,..., R~) is self-dependent if Rn depends on OUT(Rn), its own output bound. In other words, a series is self-dependent if the last bound affected by the series is also an argument to the first refinement in a chain of refinements in the precedence relation, as illustrated below.</Paragraph>
    <Paragraph position="6"> (OUT( Rn ~OUT( R, }~-~OUT( R2 } . . . ~ Davis shows that such self-dependencies are potential infinite loops: Theorem 2 Any infinite sequence of active refinements contains an active, self.dependent subsequence (\[Davis, 1987, Lemma B.15\]}.</Paragraph>
    <Paragraph position="7"> In \[Simmons, 1993\], it is shown that if any self-dependent sequence 7~ is active on the labelling of a system of product constraints, then a certain sub-sequence T~' of ~ will be active infinitely many times. Moreover, on the rn-th execution of each refinement Ri in ~', there is a term 7~n/, where each T/m &gt; T/m-1 &gt; 1, such that OUT(Ri) is multiplied by: (T~) -1, if OUT(e,) is an upper bound sty-, if OUT(R~) is a lower bound It follows that upper bounds are refined so as to become arbitrarily small (asymptotically approaching zero), and that lower bounds become arbitrarily large, up to infinity.</Paragraph>
    <Paragraph position="8"> Thus if there is any constraint Ci in the system that imposes a lowest value greater than zero on an  upper bound that is affected by a refinement operator in ~', that bound will be refined often enough until it becomes inconsistent with Ci. Similarly, if any constraint Cu imposes a largest finite value on a lower bound that is affected by a refinement in 7U, then that bound will be refined until it becomes inconsistent with Cu. In both cases, the system is inconsistent.</Paragraph>
    <Paragraph position="9"> If there are no such constraints, then it is consistent for upper bounds affected by T~' to be asymptotically close to zero and for lower bounds affected by T~' to be arbitrarily large. This can only be consistent if, in the case of upper bounds, the solution assigns \[0, 0\] to the parameter in question, and in the case of lower bounds, the solution assigns \[co, oo\] to its parameter. Hence, the solution is inadmissible.</Paragraph>
    <Paragraph position="10"> But according to Davis' result (Theorem 2), infinite loops must contain an active, self-dependent subsequence such as 7~. It follows that if a system of product constraints is consistent and its solution is admissible, then the Waltz algorithm finds its solution in finite time. The time complexity result is a straightforward extension of Davis' analysis of unit linear inequalities (see \[Simmons, 1993\]).</Paragraph>
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