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<Paper uid="E91-1008">
  <Title>INDEXING AND REFERENTIAL DEPENDENCIES WITHIN BINDING COMPUTATIONAL FRAMEWORK</Title>
  <Section position="6" start_page="0" end_page="0" type="evalu">
    <SectionTitle>
4 Applications
</SectionTitle>
    <Paragraph position="0"> Two applications of the formalism introduced above are now considered. The discussion will by no means be exhaustive, the purpose being just to show the potentiality of the present proposal.</Paragraph>
    <Section position="1" start_page="0" end_page="0" type="sub_section">
      <SectionTitle>
4.1 Verification
</SectionTitle>
      <Paragraph position="0"> We define the verification problem for BT as follows: let Xw be a phrase structure tree representation for a sentence w and let 5 w be an indexation set for w. We want to know if $ w is compatible with the constraints imposed by BT on Zw. In essence, this is the same problem as that discussed in the last section. A polynomial time algorithmic method that solves it will be briefly discussed. The problem at hand can be reduced to the following one: let R be a set of symbols and GR=(V, E) be a graph whose edges are triples (~1 r ~) where reR; given a regular language LR ~R*, is there any path p in GR with string in LR ? An algorithm can be given, based on a dynamic programming method presented in Aho et al.</Paragraph>
      <Paragraph position="1"> (1974), which takes as input one such graph GR. a finite state non deterministic automaton for LR and computes a IVI x IVI boolean matrix such that its ij-th entry is set to 1 just in case there is a path, from node ni to node nj, with string in LR.</Paragraph>
      <Paragraph position="2"> The verification problem for BT can, then, be solved by the following algorithmic schema: first, compute relations d and b ; then check condition (i) of Theorem 1 for every element in ~,,. If the test is successful, build the directed labelled graph Gv=(V, E) where V={v I veP(N) and either (v r ~)e~qw or (~ r v)~ ~w , for some r in {c, 1, 1(.), s, st.)} } and E=~3~. Now, check conditions (ii) through (v) of Theorem 1, by means of successive runs of the algorithm previously sketched.</Paragraph>
    </Section>
    <Section position="2" start_page="0" end_page="0" type="sub_section">
      <SectionTitle>
4.2 Satisfiability
</SectionTitle>
      <Paragraph position="0"> Satisfiability for BT can be stated as follows: given a sentence w and a phrase structure tree representation for it, Zw, does there exist at least one indexation set which is BT compatible ? Observe that, addressing the problem of BT satisfiability can prove useful in actual parsing systems, since it provides a means to weed out ungrammatical analysis of the input string.</Paragraph>
      <Paragraph position="1"> According to the version of BT considered here, only anaphors need to be considered; in fact, from the point of view of the syntactic theory, it is always possible to assign every R-expression and every pronoun an independent reference so that no interactions arise. In other words, a sentence like She loves her is perfectly grammatical, provided that the two pronouns are neither in the binding nor in the coreference relation, even if uttered without any context from which references can be drawn; in this case the only BT compatible index set is the empty set, i.e. the one that does not specify any mutual dependency between the two elements. On the side of the interpretative processes, this corresponds to (possibly infinitely) many non overlapping reference assignments to the two pronouns. 12 Anaphors make the real difference, though, since Principle A requires them to get their reference from intrasentential items. Our attention will be focused on 9~ w , called the A-restricted binding constraint set and on 3w', called the A-restricted indexation set. ~w' is defined in such a way that (~ r I/t)eg~ w iff either ~p={n}, neA or Ip={m}, meA and r is as in ~w. 3w&amp;quot; is defined in a similar way. The problem, then, is to find out whether an Areduced index set verifying (i), (ii) and (iv) of Theorem 1 exists, for a given pair (w, &amp;quot;rw).</Paragraph>
      <Paragraph position="2"> Theorem 2 - Conditions for BT Satisfiability: Let w be a sentence, ~'w one of its phrase structure tree representations and ~w&amp;quot; its A-restricted binding constraint set; then, w satisfies BT iff for any C/={n}, neA there exists an element, ~={m}, meP, R, such that there is a path, p=(C/ ri ~1) (~1 r2 ~2)... (Om.l rm Ill) in ~' with string wpeb + and (ll/ d ~ra.l)Z~ w.</Paragraph>
      <Paragraph position="3"> An algorithmic solution for the satisfiability problem can be pursued by means of an approach similar to the one sketched above for verification.</Paragraph>
    </Section>
  </Section>
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