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<?xml version="1.0" standalone="yes"?> <Paper uid="W04-2324"> <Title>Discourse dependency structures as constrained DAGs</Title> <Section position="4" start_page="0" end_page="0" type="metho"> <SectionTitle> 3 DAGs for S1 Conna S2 Connb S3 </SectionTitle> <Paragraph position="0"> It is standardly assumed that the arguments of a discourse relation expressed through a discourse connective are given by text units5 which are adjacent to the discourse connective (Mann and Thompson, 1987), (Duchier and Gardent, 2001). However, there exist counter-examples to this adjacency principle, see (7) below. So I make non discontinuous sequence Ui Conn Uj.</Paragraph> <Paragraph position="1"> a weaker assumption, that I call &quot;left1-right2 principle&quot; which states the following: the first (resp. second) argument of a discourse relation expressed through a discourse connective is given by a text unit which occurs on the left (resp. right) of the discourse connective. This principle makes sense only for discourses in the canonical order. Recall (Section 2.2) that our study can be limited to discourses which satisfy the canonical linear order S1 Conna S2 Connb S3.</Paragraph> <Paragraph position="2"> A consequence of the left1-right2 principle in discourses of the type S1 Conna S2 Connb S3, is that the first argument of Ra is compulsorily pi1, the only text unit which occurs on the left of Conna. On the other hand, its second argument may vary depending on scope. More specifically, it may a priori be: * either the representation of the whole right hand side of Conna, i.e. the semantic representation of the text unit S2 Connb S3. I call this case &quot;wide scope&quot; of Conna or Ra. It leads to DAG (A) in Figure 66. The dependency relations in (A), which is tree shaped, must be interpreted in the standard way: the second argument of Ra is its right daughter, i.e. the tree rooted at Rb.</Paragraph> <Paragraph position="3"> * or the representation of one of the two clauses on the right of Conna. This case leads either to tree (A1) = Ra(pi1,pi2) or to tree (A2) = Ra(pi1,pi3).</Paragraph> <Paragraph position="4"> Similarly, the second argument of Rb is compulsorily pi3, the only text unit on the right of Connb, but depending on the scope of Connb, its first argument may a priori be Ra(pi1,pi2), see (B) in Figure 6, or pi2 in (B1) = Rb(pi2,pi3) or pi1 in (B2) = Rb(pi1,pi3).</Paragraph> <Paragraph position="5"> We are now ready to study the combinatory coming from the fusion of DAGs (Ai) and (Bj). The goal is to distinguish the DAGs which correspond to coherent discourses S1 Conna S2 Connb S3 from those which do not (i.e. which cannot be linguistically realized).</Paragraph> <Paragraph position="6"> A) Graph (A): This graph is linguistically realized in (4a)7. The wide scope of Conna = because can be seen in the dialogue in (4b-c) in which the answer is Because S2 Connb S38. In conformity with our compositionality principle, (A) includes the sub-graph Rb(pi2,pi3) and S2 Connb S3 can be inferred: if (4a) is true, then it is true that Fred played tuba while Mary was taking a nap. The reader will check that the adverbial Conna = therefore in given as answer only Because S2. The interpretation of (4a) corresponds then to DAG (C) in Figure 6.</Paragraph> <Paragraph position="7"> (4) a. Mary is in a bad mood because Fred played tuba WHILE she was taking a nap.</Paragraph> <Paragraph position="8"> b. - Why is Mary in a bad mood? c. - Because Fred played tuba WHILE she was taking a nap.</Paragraph> <Paragraph position="9"> d. Fred wanted to bother Mary. Therefore, he played tuba WHILE she was taking a nap.</Paragraph> <Paragraph position="10"> B) Graph (B): This graph is linguistically realized in (5a). The wide scope of Connb = in order that/to can be seen in the dialogue in (5b-c) in which the question is Why S1 Connb S2? In conformity with our compositionality principle, (B) includes the sub-graph Ra(pi1,pi2) and S1 Connb S2 can be inferred from (5a). The adverbial Connb = therefore in (5d) has also wide scope. (5) a. Fred played tuba WHILE Mary was taking a nap in order to bother her.9 b. - Why did Fred play tuba WHILE Mary was taking a nap? c. - In order to bother her.</Paragraph> <Paragraph position="11"> d. Fred played tuba WHILE Mary was taking a nap.</Paragraph> <Paragraph position="12"> Therefore, she is in a bad mood.</Paragraph> <Paragraph position="13"> C) Graphs (A1) and (B1): The fusion of (A1) and (B1) leads to DAG (C) in Figure 6. This DAG is not tree shaped: pi2 has two parents. It is linguistically realized in (6a), in which S2 is said to be &quot;factorized&quot; since both S1 Conna S2 = Mary is in a bad mood because her son is ill and S2 Connb S3 = Her son is ill. Specifically, he has an attack of bronchitis can be inferred from (6a), which is in conformity with our compositionality principle since (C) includes both (A1) = Ra(pi1,pi2) and (B1) = Rb(pi2,pi3). A similar situation is observed in (6b) and (6c).</Paragraph> <Paragraph position="14"> (6) a. Mary is in a bad mood because her son is ill.</Paragraph> <Paragraph position="15"> Specifically, he has an attack of bronchitis.</Paragraph> <Paragraph position="16"> b. Fred played tuba. Next he prepared a pizza to please Mary.</Paragraph> <Paragraph position="17"> c. Fred was in a foul humor because he hadn't slept well that night because his electric blanket hadn't worked.10 D) Graphs (A1) and (B2): The fusion of (A1) and (B2) leads to DAG (D) in Figure 6. This DAG is not tree shaped: pi1 has two parents. It is linguistically realized in (7a), in which S1 is said to be &quot;factorized&quot; since both S1 Conna S2 = Fred prepared a pizza to please Mary and S1 Connb S3 = Fred prepared a pizza. Next he took a nap can be inferred, in conformity with our compositionality principle. A similar situation is observed in (7b) and (7c). 9When while is not stressed, the interpretation of (5a) may correspond to DAG (D) in Figure 6.</Paragraph> <Paragraph position="18"> 10This discourse is a modified version (including discourse connectives) of an example taken in (Blackburn and Gardent, 1998). These authors acknowledged that the structure of this discourse is a &quot;re-entrant graph&quot;.</Paragraph> <Paragraph position="19"> (7) a. Fred prepared a pizza to please Mary. Next, he took a nap.</Paragraph> <Paragraph position="20"> b. Fred prepared a pizza, while it was raining, before taking a walk.</Paragraph> <Paragraph position="21"> c. Fred is ill. More specifically, he has an attack of bronchitis. Therefore, Mary is in a bad mood.</Paragraph> <Paragraph position="22"> In discourses analyzed as (D), S3 is linked to S1 (which is not adjacent) and not to S2 (which is adjacent). Therefore, these discourses are counter-examples to the adjacency principle adopted in RST.</Paragraph> <Paragraph position="23"> The DAG (D) exhibits crossing dependencies and it does correspond to coherent discourses. (D) is thus a counter-example to the stipulation made by (Webber et al., 2003), namely &quot;discourse structure itself does not admit crossing structural dependencies&quot;11.</Paragraph> <Paragraph position="24"> E) Graphs (A2) and (B1): The fusion of (A2) and (B1) leads to DAG (E) in Figure 7, in which pi3 has two parents. I cannot find any discourse corresponding to (E), i.e. with S3 factorized, although I wrote down all possible examples I could think of. Laurence Delort, who works on (French) corpus neither. I cannot prove that something does not exist, I can just stipulate it. However there is some evidence, coming from syntax, which supports my stipulation when Conna and Connb are both subordinating conjunctions (Conj). Namely, no standard syntactic analysis of sentences of the type S1 Conja S2 Conjb S3 can lead, in a compositional way, to an interpretation in which S3 is factorized12. As I see no reason to make a difference between subordinating conjunctions and other discourse connectives at the semantic level (see note 11), I extrapolate this result to other discourse connectives. F) Graphs (A2) and (B2): The fusion of (A2) and (B2) leads to DAG (F) in Figure 7. This graph cannot represent a discourse S1 Conna S2 Connb S3 since it does not include pi2.</Paragraph> <Paragraph position="25"> So far, we have examined only cases where a discourse relation has two arguments. It remains to examine what is called &quot;multi satellite or nucleus cases&quot; in RST, in which a discourse relation is supposed to have more than two arguments.</Paragraph> <Paragraph position="26"> G) Graphs (A1), (A2) and (B2): The fusion of (A1), (A2) and (B2) leads to DAG (G) in Figure 7. This DAG could be said to be linguistically realized in (8a): since 11Among discourse connectives, (Webber et al., 2003) distinguish &quot;structural connectives&quot; (e.g. subordinating conjunctions) from discourse adverbials including then, also, otherwise, . . . . They argue that discourse adverbials do admit crossing of predicate-argument dependencies, while structural connectives do not. I don't make any distinction between discourse connectives at the semantic level, but I emphasize that (7b) comprises only structural connectives (subordinating conjunctions) and its structure exhibits crossing structural dependencies.</Paragraph> <Paragraph position="27"> 12Recall that I feel entitled to make this claim because I have studied in detail the syntactic analyses of sentences of the type</Paragraph> <Paragraph position="29"> (8a), one may be willing to lay down both Ra(pi1,pi2) and Ra(pi1,pi3), i.e. to consider (8a) as a multi-satellite case with Ra = Elaboration. Rb = Narration links pi2 and pi3.</Paragraph> <Paragraph position="30"> The following question arises: is Rb in a dependency relation with Ra? It is hard to give an answer for (8a). However the answer seems positive for (8b), which could also be analyzed as a multi-satellite case with Ra = Explanation. Rb = Joint links pi2 and pi3. This leads to DAG (G') in Figure 7. However, consider (8c) which differs from (8b) only by the use of or instead of and. Graphs (G) or (G') would not do justice to (8c): neither Ra(pi1,pi2) nor Ra(pi1,pi3) can be laid down. (8c) can only be represented as DAG (A) with Ra = Explanation and Rb = Disjunction.</Paragraph> <Paragraph position="31"> (8) a. Guy experienced a lovely evening last night.</Paragraph> <Paragraph position="32"> More specifically, he had a fantastic meal. Next he won a dancing competition.13 b. Mary is in a bad mood because she had'nt slept well and it is raining.</Paragraph> <Paragraph position="33"> c. Mary is in a bad mood because she had'nt slept well or it is raining.</Paragraph> <Paragraph position="34"> It seems clear that (8b) and (8c) should be represented at the semantic level as the very same graph. This graph can only be (A), which is the only possibility for (8c). For the sake of homogeneity and compatibility with SDRT, (8a) should also be represented as (A)14. Recall moreover that (4a) with wide scope of Conna is also represented as (A). All in all, (A) happens to be a semantic structure which is shared by discourses whose informational content shows quite different relations between the eventualities at stake. Is it a problem? I would say no, because, from (A), semantic to content rules, based on the values of Ra and Rb, can make the difference: they can compute the following (simplified) logical forms, which show that the discourses in (8) and (4a) do not have the same type of informational content as far as the relations between eventualities are concerned, althoug they share the same (dependency) semantic structure: * for (8a) with Ra = Elaboration and Rb = Narration:</Paragraph> <Paragraph position="36"> 13This discourse is a modified version (including discourse connectives) of an example taken in (Asher and Lascarides, 2003).</Paragraph> <Paragraph position="37"> 14The (A) analysis is the translation of the SDRS proposed by (Asher and Lascarides, 2003) for (8a), namely the SDRS in Figure 1 with Ra = Elaboration and Rb = Narration. pi1 is considered as the &quot;topic&quot; (common theme) for pi2 and pi3. * for (8c) with Ra = Explanation and Rb = Disjunction: e1 [?]e2 [?]e3 [?]cause(e1,or(e2,e3)) - e1 [?]e2 [?]e3 [?] (cause(e1,e2) [?]cause(e1,e3)) * for (4a) with Ra = Explanation and Rb = Circumstances: e1 [?]e2 [?]e3 [?]overlap(e2,e3) [?]cause(e1,overlap(e2,e3)) We have touched here a crucial question in discourse processing (within a multi-level approach): to what extent should the semantic (dependency) level (how things are said) echo the informational content level (what is said)? I don't pretend to give a general answer to this fundamental question. However we have seen that the same semantic dependency structure (or SDRS) can lead to quite different informational contents according to the values of the discourse relations at stake. What is called multi-satellite case in RST, e.g. (8a) or (8b), leads to a logical form in which the same eventuality variable, here e1, occurs conjunctively multi-times as the argument of the same predicate, e.g. preda(e1,e2) [?] preda(e1,e3) with preda = subevent in (8a) and preda = cause in (8b). It is unnecessary to represent such a case at the semantic level trough a predicate - a discourse relation with more than two arguments. The multi-satelitte analysis in RST comes from the following principle: if a sub-discourse Dp can be inferred from a discouse Dn, with 1 < p < n, then the graph Dp must be a sub-graph of Dn. This principle is simply wrong. On the other hand, the converse implication is true.</Paragraph> <Paragraph position="38"> H) Graphs (A1), (B1) and (B2): The fusion of (A1), (B1) and (B2) leads to a DAG which could be said to be linguistically realized in (9). This discourse allows us to infer both S1 Connb S3 and S2 Connb S3. So it would be classified as a multi-nucleus case in RST. However, by the same argumentation as previously, it should be represented as (B).</Paragraph> <Paragraph position="39"> (9) Fred washed the dishes and Guy cleaned up the bathroom, while Mary was taking a nap.</Paragraph> <Paragraph position="40"> I) Graphs (A1), (A2) and (B2): The fusion of these graphs lead to DAG (I) in Figure 8. I cannot find any example corresponding to this DAG.</Paragraph> <Paragraph position="41"> J) Graphs (A2), (B1) and (B2): Along the same lines, the fusion of these graphs lead to a DAG for which I cannot find any instance.</Paragraph> <Paragraph position="42"> No other fusion of graphs (Ai) and (Bj) leads to a DAG which corresponds to a coherent discourse. So we have arrived at the following result: The dependency structure of a discourse S1 Conna S2 Connb S3 is one of the four DAGs (A), (B), (C) and (D). (A) and (B), which are tree shaped, cover wide scope cases (and multi satellite or nucleus cases in RST). (C) and (D), which are not tree shaped, cover multi parent cases (factorization of a sentence). (D) exhibits crossing dependencies.</Paragraph> <Paragraph position="43"> Before commenting on this result, let us come back to the interpretation of dependency relations in trees.</Paragraph> <Paragraph position="44"> 4 Interpretation of dependency relations in trees (concluding episode) First, let us underline the following point. Interpreting tree shaped graphs (A) and (B) with the nuclearity principle amounts to interpreting (A) as (C), and (B) as (D)15. But then, cases with wide scope are not taken into account, which is unacceptable. Therefore, the standard interpretation of dependency relations in a tree is needed. Next, the following question arises: is it possible to state that the dependency relations in a tree should be computed sometimes by the standard interpretation and some other times by the nuclearity one? In the tree (B), this question is instantiated in the following way: should the first argument of Rb be given sometimes by the standard interpretation (it is then the tree rooted at Ra) and some other times by the nuclearity principle (it is then pi1, and (B) is equivalent to (D))16? An answer to this question is sound only if it is possible to define formally &quot;sometimes&quot;. The only sound answer consists in stating that there exist two types of discourse relations: the dependency relations are computed with the standard interpretation for the first type, and computed with the nuclearity interpretation for the second one. The only types of discourse relations which have been put forward up to now are the &quot;coordinating and subordinating&quot; types (Hobbs, 1979), (Asher and Lascarides, 2003), (Asher and Vieu, 2003). Laurence Delort in (Delort, 2004) has examined, in the framework of SDRT, my DAGs (A)-(D) in studying for each relation Ra or Rb if it could be of the coordinating and/or subordinating type. Her results are summarized in Table 1. This table shows that (B) is possible only when Ra is coordinating and (D) only when Ra is subordinating (in both cases, Rb can be equally co-ordinating or subordinating). Therefore, it is possible to lay down the following rule: the dependency relations in the tree (B) are computed with the standard interpretation when Ra is coordinating, and with the nuclearity interpretation when Ra is subordinating.</Paragraph> <Paragraph position="45"> However, let us examine the situation for the tree (A).</Paragraph> <Paragraph position="46"> From Table 1, the reader can check that no rule can be laid down for the dependency relations in (A) when Rb is coordinating: they can be computed with either the standard or the nuclearity interpretation. These two cases are 15With the nuclearity principle, the second argument of Ra in (A) is pi2, and the first argument of Rb in (B) is pi1.</Paragraph> <Paragraph position="47"> 16For the other dependency relations in (B), both interpretations give the same result.</Paragraph> <Paragraph position="48"> illustrated in (10) with Ra = Contrast and Rb = Narration: (10a) should be analyzed with the standard interpretation of (A) with wide scope of Conna, while (10b) should be analyzed with the nuclearity interpretation of (A), i.e. as (C) with S2 factorized.</Paragraph> <Paragraph position="49"> (10) a. Fred has made no domestic chore this morning.</Paragraph> <Paragraph position="50"> However, this afternoon, he wed up the dishes.</Paragraph> <Paragraph position="51"> Next he ran the vacuum cleaner.</Paragraph> <Paragraph position="52"> b. Fred has made no domestic chore this morning. However, this afternoon, he washed up the dishes. Next he went to see a movie.</Paragraph> <Paragraph position="53"> In conclusion, a mixed interpretation for trees must be discarded: the coordinating or subordinating type of discourse relations does not allow us to choose between the standard and nuclearity interpretations. As a consequence, since the standard interpretation is needed for wide scope cases, the nuclearity principle should be discarded. null 5 Analysis of the result and conclusion The result I arrived at does not take into account the discourse connectives / relations at stake. However, for a given pair of connectives, it may happen that only some of the DAGs among (A)-(D) are observed. For example, if Conna is an adverbial and Connb a subordinate conjunction, then (B) with wide scope of Rb should be excluded. On the top of part of speech considerations, the lexical value of each connective may exclude some of these DAGs. Finally, the distinction between coordinating and subordinating discourse relations must be taken into account. Table 1 from (Delort, 2004) presented as in Table 2 shows that a given DAG among (A)-(D) never corresponds to the 2star2 = 4 possibilities given by the combinatory Ra/b coordinating or subordinating discourse relation. null To put it in a nutshell, there is a maximum of four ordered DAGs representing the semantic structures of discourses S1 Conna S2 Connb S3. I stipulate that this result can be extrapolated to cases where sentences are simply juxtaposed without discourse connective.</Paragraph> <Paragraph position="54"> It can be considered that there is only a few DAGs corresponding to coherent discourses with three clauses17. First, recall that the left1-right2 principle (Section 3) discards right away a number of DAGs, for example (K) in Figure 8 (in (K), Ra is not the mother of pi1). Secondly, among the DAGs which satisfy the left1-right2 principle, some are not instantiated, e.g. (E), and also (F). A look 17In RST, there are only 2 trees (2 is the number of binary trees with 3 leaves), namely trees (A) and (B), which are supposed to be interpreted with the nuclearity principle (being so interpreted as (B) and (D) respectively). We have seen that this is too restrictive: wide scope cases are not taken into account. on the topology of the ordered DAGs (A)-(D) allows us to bring forward this other structural constraint: Ra must &quot;left-dominate&quot; pi2. The definition of left-dominance in a tree is the following (Danlos, 2003): a node X left-dominates a node Y iff Y is a daughter of X (immediate dominance) or there exists a daughter Z of X such that Y belongs to the left-frontier of the tree rooted at Z. For example, Ra left-dominates pi1, Rb and pi2 in (A), while Rb left-dominates Ra, pi1 and pi3 in (B)18.</Paragraph> <Paragraph position="55"> Let us here examine the consequences of this left-dominance constraint in non formal terms. Ra must be the mother of pi1 and must left-dominate pi2. This means that Ra establishes some semantic link between S1 and S219. This result may sound trivial on psycho-linguisitics grounds: what would be a discourse in which the second clause is not linked at all to the first one?20 It has the following consequence: the semantic representation of a discourse with four clauses and three discourse connectives cannot be DAG (L) in Figure 8. In (L), Ra does not left-dominate pi2, or informally, there is no link between S1 and S2. (L) includes two crossing dependencies.</Paragraph> <Paragraph position="56"> I have just half-opened the door towards an extension of this study to discourses with more than three clauses.</Paragraph> <Paragraph position="57"> I stipulate that the conclusion of this forthcoming study will be the same. Namely, semantic dependency structures for discourses are ordered DAGs which satisfy heavy structural constraints, which can help us to cut down the number of possibilities when processing discourses.</Paragraph> </Section> class="xml-element"></Paper>