Discourse dependency structures as constrained DAGs
Laurence Danlos
TALANA/ LATTICE
Universit´e Paris 7
Laurence.Danlos@linguist.jussieu.fr
Abstract
I show that the semantic structure for dis-
courses, understood as a dependency represen-
tation, can be mathematically characterized as
DAGs, but these DAGs present heavy structural
constraints. The argumentation is based on a
simple case, i.e. discourses with three clauses
and two discourse connectives. I show that
only four types of DAGs are needed for these
discourses.
1 Introduction
Within a multi-level approach to discourse processing,
this paper focuses on the semantic level. This level re-
flects the discourse structure (how things are said, how
the discourse is rhetorically organized). This structure
plays an important role, e.g., it constrains both anaphora
resolution and the attachment of incoming propositions
in understanding. I assume that the informational content
level (what is said) is based on first order logic.
A nice tool for the semantic level is dependency
graphs. This is what is adopted in RST (rhetorical struc-
tures correspond roughly to dependency structures), but
it is not the case in SDRT1: discourse structures, called
SDRSs, are represented as boxes. Nevertheless, it is easy
to translate the conditions of an SDRS into a dependency
graph (Section 2.1).
Our goal in this paper is to determine to which mathe-
matical object dependency structures for discourses cor-
respond. In RST, it is a basic principle that this object is a
tree. In SDRT, the issue is not discussed. I will show that
this object is an ordered directed acyclic graph (DAG),
1SDRT stands for Segmented Discourse Representation The-
ory (Asher, 1993) (Asher and Lascarides, 2003). It is an ex-
tension of DRT, Discourse Representation Theory (Kamp and
Reyle, 1993). (S)DRS stands for (Segmented) Discourse Repre-
sentation Structure. RST stands for Rhetorical Structure Theory
(Mann and Thompson, 1987).
which may be not tree shaped. Some authors, e.g. (Bate-
man, 1999) and (Blackburn and Gardent, 1998), have al-
ready brought forward discourse structures which are not
tree shaped. However nobody says explicitly that dis-
course dependency structures are DAGs considering seri-
ously all the consequences of this claim2.
Our argumentation is based on one of the simplest
cases of discourses, namely discourses of type S1 Conna
S2 Connb S3 with two discourse connectives (Conna/b)
and three clauses (Si). A discourse connective Conn can
be either a subordinating or coordinating conjunction or
a discourse adverbial. It denotes a discourse relation R,
a predicate with two arguments. I will show (Section 3)
that they are topologically only four types of DAGs for
these discourses. This allows us to state that DAGs for
these discourses are not arbitrary: they satisfy structural
constraints (Section 5). I stipulate that this result can be
extrapolated to discourses in which sentences are sim-
ply juxtaposed without discourse connective. It can also
be foreseen that dependency structures for more complex
discourses (e.g. discourses with more than three clauses)
are also constrained DAGs.
This can be seen as an important result since many au-
thors in the discourse community hang on trees as dis-
course structures, even if it means to use artificial trees
as shown in Section 2.4. They reject DAGs because
they view them as completely unconstrained (except the
acyclicity constraint) and so as unusable in discourse pro-
cessing. This is truly not the case. Semantic dependency
structures for discourses are ordered DAGs but these DAGs
present heavy structural constraints, which can help us
to cut down the number of possibilities when processing
discourses (although this issue is not discussed here).
Before getting to the heart of the matter, let us give
some preliminaries.
2For example, (Blackburn and Gardent, 1998) exhibits an
example the structure of which is a “re-entrant graph”, see (6c).
However, in (Duchier and Gardent, 2001), the semantic repre-
sentations of discourses are always tree shaped.
2 Preliminaries
2.1 Translation of an SDRS into a DAG
Formally, an SDRS is is a couple of sets 〈U,Con〉. U is
a set of labels of DRS or SDRS which may be viewed as
“speech act discourse referents”. Con is a set of condi-
tions 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 dis-
course relation (structuring).
The set of conditions can be translated into a depen-
dency graph by applying the following rules.
• 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 gen-
erally to the “nucleus” in RST), pij its second argu-
ment (it corresponds generally to the “satellite” in
RST).
• 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.
• A condition pi : K in whick K is a DRS is simply
translated as pi.
Figures 1 and 2 give examples of this translation mech-
anism.
2.2 Linear order
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 “canonical” one S1 Conj (,)
S2) or preposed (then the non canonical order is Conj S2,
S1). Following works in MTT3, a trace of the linear or-
der can be recorded in a semantic dependency represen-
tation, 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.
What happens for a sentence with two subordinate
clauses? Establishing the canonical order with only post-
posed 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 =
S2 Conja S1 Conjb S3 or to Y2 = S2 Conjb S3 Conja S1.
3MTT stands for Meaning to Text Theory, a dependency for-
malism for sentences (Mel’cuk, 2001).
In (Danlos, 2003), I have shown, using LTAG as a syn-
tactic 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 se-
mantic structures (see above), X does not yield any other
semantics than Y1 and Y2, i.e. the semantics of X is in-
cluded in the semantics of Y1 and Y2.
As a consequence, our study on the semantics of sen-
tences with two subordinate clauses can be limited to the
study of such sentences in the canonical order. Since sub-
ordinate 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 dis-
courses which satisfy the linear order S1 Conna S2 Connb
S3.
2.3 Compositionality principle
Let Dn be a DAG with n leaves representing the depen-
dency 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 < p < 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.
2.4 Interpretation of dependency relations in trees
Two different ways can be used to interpret dependency
relations in trees: the standard one used in mathematics
and computer science, and the “nuclearity principle” put
forward in RST (Marcu, 1996). Let us illustrate them
with the tree in Figure 3. With the standard interpreta-
tion, the first argument (nucleus) of Rc is its left daughter
(the tree rooted at Ra), while with the nuclearity prin-
ciple, it is pi1 (the leaf which is the first argument (nu-
cleus) 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 princi-
ple, it is pi2 (the leaf which is the first argument (nucleus)
of Rb). To put it in a nutshell, the arguments of a dis-
course relation can be intermediary nodes or leaves with
the standard interpretation, while they can only be leaves
with the nuclearity interpretation.
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).
(1) a. Although John is very generous -
b. if you need some money,
c. you only have to ask him for it -
d. he’s very hard to find.
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 “concession” in their tree are a and d.
These predicate-argument dependencies are given by the
nuclearity interpretation.
(2) a. Although John is very generous,
d. he’s very hard to find.
From (1), (3) can also be inferred. This is evidence that
the arguments of “elaboration” in their tree are a and the
tree rooted at “condition”. These dependencies are given
by the standard interpretation.
(3) a’. John is very generous -
b. if you need some money,
c. you only have to ask him for it.
Nevertheless, one should not feel free to use trees rely-
ing 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 Sec-
tion 4, I will make an attempt to lay down rules on the
choice of one of these two interpretations according to
the “coordinating or subordinating” type of discourse re-
lations. 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 ex-
ample, 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
”a” in the DAG.
3 DAGs for S1 Conna S2 Connb S3
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
4I thank an anonymous reviewer for drawing my attention
on this point.
5A text unit (noted as U) is either a clause or, recursively, a
non discontinuous sequence Ui Conn Uj.
a weaker assumption, that I call “left1-right2 principle”
which states the following: the first (resp. second) ar-
gument of a discourse relation expressed through a dis-
course 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.
A consequence of the left1-right2 principle in dis-
courses 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 “wide scope” 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.
• 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).
Similarly, the second argument of Rb is compulsorily
pi3, the only text unit on the right of Connb, but depend-
ing on the scope of Connb, its first argument may a pri-
ori be Ra(pi1,pi2), see (B) in Figure 6, or pi2 in (B1) =
Rb(pi2,pi3) or pi1 in (B2) = Rb(pi1,pi3).
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 dis-
courses S1 Conna S2 Connb S3 from those which do not
(i.e. which cannot be linguistically realized).
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
(4d) has also wide scope.
6In this figure, as well as in other subsequent figures, the
label for the sub-graph is omitted.
7To indicate that it is stressed when spoken, the word while
is written in capital letters in (4).
8When while is not stressed, the question in (4b) may be
given as answer only Because S2. The interpretation of (4a)
corresponds then to DAG (C) in Figure 6.
(4) a. Mary is in a bad mood because Fred played tuba
WHILE she was taking a nap.
b. - Why is Mary in a bad mood?
c. - Because Fred played tuba WHILE she was tak-
ing a nap.
d. Fred wanted to bother Mary. Therefore, he
played tuba WHILE she was taking a nap.
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 composi-
tionality principle, (B) includes the sub-graph Ra(pi1,pi2)
and S1 Connb S2 can be inferred from (5a). The adver-
bial 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 tak-
ing a nap?
c. - In order to bother her.
d. Fred played tuba WHILE Mary was taking a nap.
Therefore, she is in a bad mood.
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 “factorized” 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).
(6) a. Mary is in a bad mood because her son is ill.
Specifically, he has an attack of bronchitis.
b. Fred played tuba. Next he prepared a pizza to
please Mary.
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 “factorized” 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.
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 “re-entrant graph”.
(7) a. Fred prepared a pizza to please Mary. Next, he
took a nap.
b. Fred prepared a pizza, while it was raining, be-
fore taking a walk.
c. Fred is ill. More specifically, he has an attack of
bronchitis. Therefore, Mary is in a bad mood.
In discourses analyzed as (D), S3 is linked to S1 (which
is not adjacent) and not to S2 (which is adjacent). There-
fore, these discourses are counter-examples to the adja-
cency principle adopted in RST.
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 “discourse structure itself does not ad-
mit crossing structural dependencies”11.
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 subordinat-
ing 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.
So far, we have examined only cases where a discourse
relation has two arguments. It remains to examine what is
called “multi satellite or nucleus cases” in RST, in which
a discourse relation is supposed to have more than two
arguments.
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) dis-
tinguish “structural connectives” (e.g. subordinating conjunc-
tions) 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 connec-
tives at the semantic level, but I emphasize that (7b) comprises
only structural connectives (subordinating conjunctions) and its
structure exhibits crossing structural dependencies.
12Recall that I feel entitled to make this claim because I have
studied in detail the syntactic analyses of sentences of the type
S1 Conja S2 Conjb S3 in (Danlos, 2003).
both S1 Conna S2 and S1 Conna S3 can be inferred from
(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.
The following question arises: is Rb in a dependency rela-
tion with Ra? It is hard to give an answer for (8a). How-
ever the answer seems positive for (8b), which could also
be analyzed as a multi-satellite case with Ra = Explana-
tion. 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 rep-
resented as DAG (A) with Ra = Explanation and Rb =
Disjunction.
(8) a. Guy experienced a lovely evening last night.
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.
c. Mary is in a bad mood because she had’nt slept
well or it is raining.
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 more-
over that (4a) with wide scope of Conna is also repre-
sented as (A). All in all, (A) happens to be a semantic
structure which is shared by discourses whose informa-
tional content shows quite different relations between the
eventualities at stake. Is it a problem? I would say no, be-
cause, 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:
e1 ∧e2 ∧e3 ∧precede(e2,e3)
∧subevent(e1,e2) ∧subevent(e1,e3)
• for (8b) with Ra = Explanation and Rb = Joint: e1 ∧
e2 ∧e3 ∧cause(e1,and(e2,e3))
→ e1 ∧e2 ∧e3 ∧cause(e1,e2) ∧cause(e1,e3)
13This discourse is a modified version (including discourse
connectives) of an example taken in (Asher and Lascarides,
2003).
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 con-
sidered as the “topic” (common theme) for pi2 and pi3.
• for (8c) with Ra = Explanation and Rb = Disjunc-
tion: e1 ∧e2 ∧e3 ∧cause(e1,or(e2,e3))
→ e1 ∧e2 ∧e3 ∧ (cause(e1,e2) ∨cause(e1,e3))
• for (4a) with Ra = Explanation and Rb = Circum-
stances: 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 ex-
tent 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 se-
mantic level trough a predicate - a discourse relation -
with more than two arguments. The multi-satelitte anal-
ysis 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.
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 repre-
sented as (B).
(9) Fred washed the dishes and Guy cleaned up the bath-
room, while Mary was taking a nap.
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.
J) Graphs (A2), (B1) and (B2): Along the same lines,
the fusion of these graphs lead to a DAG for which I can-
not find any instance.
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.
Before commenting on this result, let us come back to
the interpretation of dependency relations in trees.
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 princi-
ple amounts to interpreting (A) as (C), and (B) as (D)15.
But then, cases with wide scope are not taken into ac-
count, which is unacceptable. Therefore, the standard in-
terpretation 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 stan-
dard 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
“sometimes”. The only sound answer consists in stat-
ing that there exist two types of discourse relations: the
dependency relations are computed with the standard in-
terpretation for the first type, and computed with the nu-
clearity interpretation for the second one. The only types
of discourse relations which have been put forward up
to now are the “coordinating and subordinating” types
(Hobbs, 1979), (Asher and Lascarides, 2003), (Asher and
Vieu, 2003). Laurence Delort in (Delort, 2004) has ex-
amined, 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 pos-
sible 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 inter-
pretation when Ra is subordinating.
However, let us examine the situation for the tree (A).
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 stan-
dard 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.
16For the other dependency relations in (B), both interpreta-
tions give the same result.
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.
(10) a. Fred has made no domestic chore this morning.
However, this afternoon, he wed up the dishes.
Next he ran the vacuum cleaner.
b. Fred has made no domestic chore this morn-
ing. However, this afternoon, he washed up the
dishes. Next he went to see a movie.
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 conse-
quence, since the standard interpretation is needed for
wide scope cases, the nuclearity principle should be dis-
carded.
5 Analysis of the result and conclusion
The result I arrived at does not take into account the dis-
course 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 con-
junction, then (B) with wide scope of Rb should be ex-
cluded. On the top of part of speech considerations, the
lexical value of each connective may exclude some of
these DAGs. Finally, the distinction between coordinat-
ing 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 cor-
responds to the 2star2 = 4 possibilities given by the combi-
natory Ra/b coordinating or subordinating discourse rela-
tion.
To put it in a nutshell, there is a maximum of four or-
dered DAGs representing the semantic structures of dis-
courses S1 Conna S2 Connb S3. I stipulate that this result
can be extrapolated to cases where sentences are simply
juxtaposed without discourse connective.
It can be considered that there is only a few DAGs cor-
responding to coherent discourses with three clauses17.
First, recall that the left1-right2 principle (Section 3) dis-
cards 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 sup-
posed 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
“left-dominate” 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.
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 connec-
tives 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.
I have just half-opened the door towards an extension
of this study to discourses with more than three clauses.
I stipulate that the conclusion of this forthcoming study
will be the same. Namely, semantic dependency struc-
tures for discourses are ordered DAGs which satisfy heavy
structural constraints, which can help us to cut down the
number of possibilities when processing discourses.
Acknowledgements
I want to thank Laura Kallmeyer for her many valuable
comments.

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