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<Paper uid="W04-2324">
  <Title>Discourse dependency structures as constrained DAGs</Title>
  <Section position="3" start_page="0" end_page="0" type="intro">
    <SectionTitle>
2 Preliminaries
</SectionTitle>
    <Paragraph position="0"/>
    <Section position="1" start_page="0" end_page="0" type="sub_section">
      <SectionTitle>
2.1 Translation of an SDRS into a DAG
</SectionTitle>
      <Paragraph position="0"> Formally, an SDRS is is a couple of sets &lt;U,Con&gt; . U is a set of labels of DRS or SDRS which may be viewed as &amp;quot;speech act discourse referents&amp;quot;. Con is a set of conditions on labels of the form: * pi : K, where pi is a label from U and K is a (S)DRS (labelling); * R(pii,pij), where pii and pij are labels and R a discourse relation (structuring).</Paragraph>
      <Paragraph position="1"> The set of conditions can be translated into a dependency graph by applying the following rules.</Paragraph>
      <Paragraph position="2"> * A condition R(pii,pij) is translated as a binary tree, the root of which is R, the ordered leaves are pii and pij. pii is the first argument of R (it corresponds generally to the &amp;quot;nucleus&amp;quot; in RST), pij its second argument (it corresponds generally to the &amp;quot;satellite&amp;quot; in RST).</Paragraph>
      <Paragraph position="3"> * A condition pi : K in which K is a SDRS leads to a sub-graph obtained by translating recursively the conditions in K, this sub-graph is labeled pi.</Paragraph>
      <Paragraph position="4"> * A condition pi : K in whick K is a DRS is simply translated as pi.</Paragraph>
      <Paragraph position="5"> Figures 1 and 2 give examples of this translation mechanism. null</Paragraph>
    </Section>
    <Section position="2" start_page="0" end_page="0" type="sub_section">
      <SectionTitle>
2.2 Linear order
</SectionTitle>
      <Paragraph position="0"> Subordinate conjunctions (noted as Conj) are the only discourse connectives which allow us to invert the order of the sentences: a subordinate clause can be postposed (the linear order is then the &amp;quot;canonical&amp;quot; one S1 Conj (,) S2) or preposed (then the non canonical order is Conj S2, S1). Following works in MTT3, a trace of the linear order can be recorded in a semantic dependency representation, however it should not affect its structure. From this principle, the position of subordinate clauses should not affect semantic structures. That is to say that S1 Conj S2 and Conj S2, S1 are both represented as R(pi1,pi2) in which pii is the semantic representation of Si.</Paragraph>
      <Paragraph position="1"> What happens for a sentence with two subordinate clauses? Establishing the canonical order with only postposed subordinate clauses may generate ambiguities: for example, a sentence X of the type Conja S1, S2 Conjb S3, with a preposed subordinate clause and a postposed one, corresponds, in the canonical order, either to Y1 =  malism for sentences (Mel'cuk, 2001).</Paragraph>
      <Paragraph position="2"> In (Danlos, 2003), I have shown, using LTAG as a syntactic formalism, that X receives two syntactic analyses which allow us to compute Y1 and Y2. From the principle that the position of subordinate clauses does not affect semantic structures (see above), X does not yield any other semantics than Y1 and Y2, i.e. the semantics of X is included in the semantics of Y1 and Y2.</Paragraph>
      <Paragraph position="3"> As a consequence, our study on the semantics of sentences with two subordinate clauses can be limited to the study of such sentences in the canonical order. Since subordinate conjunctions are the only discourse connectives which allow us to invert the order of the sentences, our study on the semantics of discourses with three clauses and two discourse connectives can be limited to discourses which satisfy the linear order S1 Conna S2 Connb S3.</Paragraph>
    </Section>
    <Section position="3" start_page="0" end_page="0" type="sub_section">
      <SectionTitle>
2.3 Compositionality principle
</SectionTitle>
      <Paragraph position="0"> Let Dn be a DAG with n leaves representing the dependency structure of a discourse Dn. It will be shown that the following principle is true: if Dp is a sub-graph of Dn with p leaves, 1 &lt; p &lt; n, then the discourse Dp corresponding to Dp can be inferred from the discourse Dn. On the other hand, it will be shown that the converse principle is not always true, i.e. if a sub-discourse Dp can be inferred from Dn, it does not always mean that the graph Dp is a sub-graph of Dn.</Paragraph>
    </Section>
    <Section position="4" start_page="0" end_page="0" type="sub_section">
      <SectionTitle>
2.4 Interpretation of dependency relations in trees
</SectionTitle>
      <Paragraph position="0"> Two different ways can be used to interpret dependency relations in trees: the standard one used in mathematics and computer science, and the &amp;quot;nuclearity principle&amp;quot; put forward in RST (Marcu, 1996). Let us illustrate them with the tree in Figure 3. With the standard interpretation, the first argument (nucleus) of Rc is its left daughter (the tree rooted at Ra), while with the nuclearity principle, it is pi1 (the leaf which is the first argument (nucleus) of Ra). Similarly, with the standard interpretation, the second argument (satellite) of Ra is its right daughter (the tree rooted at Rb), while with the nuclearity principle, it is pi2 (the leaf which is the first argument (nucleus) of Rb). To put it in a nutshell, the arguments of a discourse relation can be intermediary nodes or leaves with the standard interpretation, while they can only be leaves with the nuclearity interpretation.</Paragraph>
      <Paragraph position="1"> I will show (Section 4) that the standard interpretation should be adopted. The point I want to make now is that one could argue that the nuclearity interpretation should be adopted instead, but one should not feel free to use both interpretations for the same tree. This is however what is done by some authors. For example, in (Webber et al., 2003), the tree in Figure 4 is the discourse structure associated with (1).</Paragraph>
      <Paragraph position="2">  (1) a. Although John is very generous b. if you need some money, c. you only have to ask him for it - null d. he's very hard to find.</Paragraph>
      <Paragraph position="3"> Let us show that some predicate-argument relations are given by the nuclearity interpretation and other ones by the standard interpretation in their tree. From (1), (2) can be inferred. This is evidence that the arguments of the discourse relation &amp;quot;concession&amp;quot; in their tree are a and d. These predicate-argument dependencies are given by the nuclearity interpretation.</Paragraph>
      <Paragraph position="4"> (2) a. Although John is very generous, d. he's very hard to find.</Paragraph>
      <Paragraph position="5"> From (1), (3) can also be inferred. This is evidence that the arguments of &amp;quot;elaboration&amp;quot; in their tree are a and the tree rooted at &amp;quot;condition&amp;quot;. These dependencies are given by the standard interpretation.</Paragraph>
      <Paragraph position="6">  (3) a'. John is very generous b. if you need some money, c. you only have to ask him for it.</Paragraph>
      <Paragraph position="7">  Nevertheless, one should not feel free to use trees relying on a mixed interpretation (the standard and nuclearity ones), except if the conditions governing the use of one or the other interpretation are formally defined4 . In Section 4, I will make an attempt to lay down rules on the choice of one of these two interpretations according to the &amp;quot;coordinating or subordinating&amp;quot; type of discourse relations. However, this enterprise leads to a failure: no general rule can be laid down. Mixed interpretation for trees should thus be discarded. As a consequence, one has to admit that discourse structures are DAGs, for example, the DAG in Figure 5 for (1). This DAG is conform to our compositionality principle: it can be viewed as the fusion of the dependency graphs for (2) and (3), while the discourse in (1) can be viewed as the fusion of the discourses in (2) and (3), with the factorization of John is very generous which corresponds to the factorization of &amp;quot;a&amp;quot; in the DAG.</Paragraph>
    </Section>
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