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<Paper uid="P98-2156">
  <Title>An alternative LR algorithm for TAGs</Title>
  <Section position="6" start_page="950" end_page="951" type="evalu">
    <SectionTitle>
6 Extensions
</SectionTitle>
    <Paragraph position="0"> The recognizer can be turned into a parser by attaching information to the stack elements from .~4. At reductions, such information is gathered and combined, and the resulting data is attached to the new element from Iv\[ that is pushed onto the stack. This can be used for computation of derived trees or derivation trees, and for computation of features. Since this technique is almost identical to that for the context-free case, it suffices to refer to existing literature, e.g. Aho et al. (1986, Section 5.3).</Paragraph>
    <Paragraph position="1"> We have treated a classical type of TAG, which has adjunction as the only operation for composing trees. Many modern types of TAG also allow tree substitution next to adjunction. Our algorithm can be straightforwardly extended to handle tree substitution. The main changes that are required lie in the closure function, which needs an extra case (much like the corresponding operation in context-free LR parsing), in adding a third type of goto function, and in adding a fourth step, consisting of reduction of initial trees, which is almost identical to the reduction of auxiliary trees. The main difference is that all Xj are elements from Af; the X that is pushed can be a substitution node or a nonterminal (see also Section 7).</Paragraph>
    <Paragraph position="2"> Up to now we have assumed that the grammar does not assign the empty string as label to any of the leaves of the elementary trees.</Paragraph>
    <Paragraph position="3"> The problem introduced by allowing the empty string is that it does not leave any trace on the stack, and therefore CS(Rt) and CS+(N) are no longer suffix-closed. We have solved this by extending items with a third component E, which is a set of nodes labelled with C/ that have been traversed by the closure function. Upon encountering a completed item IT, N --+ ~ *, E\], a reduction is performed according to the sets CS(Rt, E) or CS+(N, E), which are subsets of CS(Rt) and CS+(N), respectively, containing only those cross-sections in which the nodes la- null belled with E are exactly those in E. An automaton for such a set is deterministic and has one final state, without outgoing transitions.</Paragraph>
  </Section>
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