A step towards incremental generation of logical forms
Lu´ısa Coheur
L2F INESC-ID / GRIL
Lisboa, Portugal
Luisa.Coheur@l2f.inesc-id.pt
Nuno Mamede
L2F INESC-ID / IST
Lisboa, Portugal
Nuno.Mamede@inesc-id.pt
Gabriel G. B`es
GRIL / Univ. Blaise-Pascal
Clermont-Ferrand, France
Gabriel.Bes@univ-bpclermont.fr
Abstract
This paper presents AsdeCopas, a module de-
signed to interface syntax and semantics. Asde-
Copas is based on hierarchically organised se-
mantic rules, that output formulas in a flat lan-
guage. In this paper, we show how this system
can be used in the following applications: a) se-
mantic disambiguation; b) logical formulas con-
struction (in Minimal Recursion Semantics); c)
question interpretation.
1 Introduction
We present AsdeCopas, a syntax-semantic in-
terface based on hierarchically organised rules.
AsdeCopas is integrated in a system where
the input text is first transformed into a graph
and then passed to AsdeCopas. AsdeCopas can
be used in several ways.
It can be used to enrich the graph (Figure 1),
for example, by labeling its arrows.
a0 a1 a2 a3 a4 a5 a6 a7 a8 a9 a10
a3 a2
a6
a11
a8 a12
a2 a13 a14 a15 a16
a1 a17 a18
a8 a6
Figure 1: Enriching the graph
It can be used in a desambiguation process
and to generate logical formulas. In this paper
we show how AsdeCopas can be used to choose
between several semantic values of some quan-
tifiers and also how it can generate underspec-
ified formulas in Minimal Recursion Semantics
(MRS) (Copestake et al., 2001). Additionally,
it can be used to add constraints to these under-
specified formulas. As AsdeCopas makes a con-
troled generation of variables, these new formu-
las can be simply added to the previous under-
specified MRS formulas and the rules respon-
sible for generating MRS underspecified struc-
tures remain unchangeable.
a19 a20 a21 a22 a23 a24 a25 a26 a27 a28 a29
a22 a21
a25 a30
a27 a25 a31 a32 a33 a25
a34
a27 a35
a21 a36
a33 a37 a38
a20
a31 a32 a27 a25
Figure 2: Desambiguation
Notice that in all the applications, AsdeCopas
could previously be used to enrich the graph and
the rules used in each task should take it into
consideration.
This paper is organised as follows: we start
with the motivation for this work. Then, in sec-
tion 3, we present our approach. This includes a
description of the semantic rules formalism and
a brief overview of the algorithm behind Asde-
Copas. In section 4 we introduce some applica-
tions. Final remarks and future directions can
be found in section 5.
2 Motivation
In 1992, an exhaustive study of the Portuguese
tourist resources was made by the Direc¸c˜ao
Geral de Turismo (DGT) and afterwards the
Inventory of Tourist Resources (IRT) emerged.
In order to access it, multimedia “kiosks” were
developed and a system called Edite (da Silva,
1997; Reis et al., 1997; da Silva et al., 1997) was
created with the purpose of being integrated in
these “kiosks” and to allow database access us-
ing natural language. Edite had a set of linguis-
tically motivated traditional modules (semantic
rules associated with syntactic rules, bottom-up
parser, and so on) and it soon became saturated:
adding a new syntactic rule caused dramatic
side effects, a new semantic value could dupli-
cate the number of generated formulas, etc. It
was this experiment that made us change our
approach and invest in a more robust method-
ology. We found in the 5P Paradigm (B`es, 1999;
B`es and Hag`ege, 2001; Hag`ege, 2000) the back-
ground we were looking for and the syntax-
semantic interface presented in this paper re-
flects the effort of adapting to a more robust
methodology.
Many systems base their interface in rules,
that according to (A¨ıt-Mokhtar et al., 2002)
“encode hypothetical local interpretations of
sub-strings, yet to be validated by the produc-
tion of a full parse”. This is typically what hap-
pens to syntactic-semantic bottom-up parsers
where each semantic rule is associated with a
syntactic rule. Even if these systems do not
fail when a sub-string interpretation fails, their
parsers need to deal with a combinatory explo-
sion of multiple interpretations of words, even if
syntactic conditions would allow precise values
to be chosen. This is due to the fact that at each
step there is not a whole vision of the (syntac-
tic) context. An additional effect of not having
access to context is that spurious ambiguities
are produced.
As an example, consider the Portuguese word
qualquer (roughly, any), which can take several
semantic values (see (M´oia, 1992) for a detailed
discussion about the multiple interpretations of
qualquer):
• In Qualquer c˜ao gosta de ossos (All dogs
like bones) it has an universal value (univ);
• In Ele tem qualquer problema (There is
some problem with him) it has an existen-
tial value (exist);
• In Ele ´e um jornalista qualquer (He is an
insignificant journalist) it is an adjective,
and it means something like with no rele-
vant characteristics in the class denoted by
the noun it qualifies. We will denote this
semantic value as indiscriminate;
• In Ele n˜ao ´e um jornalista qualquer (He is
not an insignificant journalist) it has the
same indiscriminate value.
Let us assume that on the right of a main verb
in the scope of negation, qualquer can only take
the indiscriminate semantic value. Typically,
in a bottom-up parsing (Figure 3) we will not be
able to discard unnecessary values, as in point
(1), when finally we have the whole vision of
the subtree, the semantic rule will not take into
consideration the negation inside V.
Figure 3: Grammar and qualquer example
Another kind of interface can be found in
systems such as ExtrAns (Moll´a et al., 2003),
where the syntax-semantic interface is executed
over dependencies. According to (Moll´a and
Hutchinson, 2002), the current version of Ex-
trAns uses either Link Grammar or the Conexor
FDG parser.
In the first situation, the logical-form is con-
structed by a top-down procedure, starting in
the head of the main dependency and follow-
ing dependencies. The algorithm is prepared
to deal with a certain type of dependencies,
and whenever an unexpected link appears, a
special recovery treatment is applied. When
describing the algorithm, the authors say that
most of these steps “... become very complex,
sometimes involving recursive applications of
the algorithm” and also that “specific partic-
ularities of the dependency structures returned
by Link Grammar add complexity to this pro-
cess” (Moll´a and Hutchinson, 2002).
In the Conexor FDG case, the bottom up
parser used has three stages. In the first one (in-
trospection) possible underspecified predicates
are associated with each word. Object predi-
cates introduce their own arguments, but other
predicates remain incomplete until the second
stage (extrospection). During extrospection,
arguments are filled by examing the relation
between each word and its head. Sometimes
dummy arguments need to be assigned when
the algorithm faces disconnected dependency
structures. A third stage (re-interpretation) is
needed to re-analyse some logical constructs.
According to the authors, the algorithm can-
not produce the correct argument structure for
long distance dependencies.
As we will see, within AsdeCopas:
• rules allow to identify semantic values that
depend on the context;
• the algorithm itself is independent from the
utilised dependency structures. Only se-
mantic rules have to be adapted to the de-
pendency structures;
• there is no need to recursively apply the al-
gorithm or to create dummy arguments due
to disconnected dependency structures: in
these situations, default rules are triggered;
• long distance dependencies cause no prob-
lem as rules are sensitive for the (possibly
non-local) syntactic context;
• all words, independently from their cate-
gory, are mapped into formulas in one step:
since rules are self-contained, they contain
all the necessary information to calculate
the corresponding formula.
3 Our approach
3.1 Brief overview
A¨ıt-Mokhtar (A¨ıt-Mokhtar et al., 2002) defines
an incremental rule as “a self-contained oper-
ation, whose result depends on the set of con-
textual restrictions stated in the rule itself. [...]
If a sub-string matches the contextual restric-
tions, the corresponding operation applies with-
out later backtracking” . This is the gold prop-
erty we achieved for our semantic rules.
Now the question is: how are we going to de-
fine an incremental rule if in our output we have
predicates sharing variables, scope relations to
define, and so on? We propose a solution based
on the following:
• we split each rule in three parts: a) the ele-
ment or elements to transform (notice that
each rule can transform more than one el-
ement); b) the context of the elements to
transform (it can be seen as a set of condi-
tions that, being verified, indicate that the
rule can be applied); c) the output (spec-
ified by a set of functions that will trans-
form the elements according to the chosen
representation).
• we assume that there is a set of fixed vari-
ables associated with each word. Each vari-
able has the position the word occupies in
the text as index. As a result, if two ele-
ments are connected (directly or not) they
know each other variables, and they can be
used to build their formulas.
Moreover, in order to incrementally add new
information to our system without having to
rewrite more general rules, semantic rules are
organised in a subsumption hierarchy. As a re-
sult, if a set of rules can be applied, only the
rules that do not subsume other rules are trig-
gered.
3.2 Semantic rules
3.2.1 Syntax
Let W be a set of words, C a set of category
labels, D a set of dependency labels and is
used to represent an underspecified value.
Element: elem(w, c) is an element, where:
• w ∈ { } ∪ W;
• c ∈ { } ∪ C.
Arrow: arrow(c1, c2, d, l) is a dependency, and
no arrow(c1, c2, d, l) a non existing depen-
dency where:
• c1, c2 ∈ C (c1 and c2 are, respectively,
the source and the target);
• d ∈ { } ∪ {L, R} (d is the dependency
orientation: L if it goes from right to
left, R from left to right);
• l ∈ { } ∪ D (l is a possibly undefined
dependency label).
Semantic Rule: [Ri] Σ : Θ mapsto→ Γ is a semantic
rule where:
• Σ is a possibly empty set of elements
(the elements to operate on);
• Θ is a possible empty set of exist-
ing and non existing dependencies (the
rule’s context);
• Γ is a set of functions, that vary ac-
cording to the chosen representation
language.
Extra constraints over semantic rules syntax
can be found in (Coheur et al., 2003b; Coheur,
2004).
3.2.2 Hierarchy of rules
In the following we define the subsumption re-
lation between semantic rules. This relation es-
tablishes the hierarchy of rules and it is based
on the subsumption relation between categories.
Although we use labels to represent categories,
each category is a set of attribute/value pairs
organized in a subsumption hierarchy.
Element subsumption: Given
e1 = elem(w1, c1) and e2 = elem(w2, c2)
from Σ, e1 subsumes e2 (e1 subsetsqequale e2) iff:
• c1 subsetsqequal c2;
• (w1 negationslash= ) ⇒ (w2 = w1).
Dependency subsumption: Given a1 = ar-
row(c1, c2, d1, l1) and a2 = ar-
row(c3, c4, d2, l2) from Θ, a1 subsumes a2
(a1 subsetsqequala a2) iff:
• c1 subsetsqequal c3 ∧ c2 subsetsqequal c4;
• (d1 negationslash= ) ⇒ (d2 = d1);
• (l1 negationslash= ) ⇒ (l2 = l1).
Subsumption of non existing dependencies:
Given a1 = no arrow(c1, c2, d1, l1) and
a2 = no arrow(c3, c4, d2, l2) from Θ, a1
subsumes a2 (a1 subsetsqequala a2) iff:
• c1 subsetsqequal c3 ∧ c2 subsetsqequal c4;
• (d1 negationslash= ) ⇒ (d2 = d1);
• (l1 negationslash= ) ⇒ (l2 = l1).
Rule subsumption: Given two semantic rules
R1 = (Σ1, Θ1, Γ1) and R2 = (Σ2, Θ2, Γ2),
R1 subsumes R2 (R1 subsetsqequalr R2) iff:
• (∀ e1 ∈ Σ1)(∃ e2 ∈ Σ2) (e1 subsetsqequale e2);
• (∀ a1 ∈ Θ1)(∃ a2 ∈ Θ2)(a1 subsetsqequala a2).
Finally, if R1 subsumes R2, R2 is said to be
more specific than R1. If both rules can apply,
only the most specific rule does so.
3.3 AsdeCopas
AsdeCopas is integrated in a system called
Javali (Coheur et al., 2003a), where a module
called Ogre (Coheur, 2004) generates a graph,
which is AsdeCopas’ input. Given the ques-
tion Qual a maior praia do Algarve(Which is
the largest beach in Algarve?), the following fig-
ure shows the graph generated by Ogre:
Each graph node is a triple, representing: a) a
word; b) its associated category; c) its position
(in the text). Each graph arrow is also a triple,
Figure 4: Ogre’s output.
maintaining information about: a) the position
associated with the source node; b) the position
associated with the target node; c) the arrow
label (possibly undefined)1.
AsdeCopas is implemented in Prolog. It goes
through each graph node and:
• identifies the rules that can be applied;
• chooses the most specific rules;
• triggers the most specific rules.
Then it continuous to the next node. Notice
that since rules are self-contained, the way it
goes through the graph and the order of rule’s
application is not relevant, and results remain
the same. Notice also, that at each step more
than one rule can be triggered.
AsdeCopas is responsible for variable genera-
tion. Thus, instead of randomly generating vari-
ables, each variable is indexed by the position
that the related word occupies in the text. Al-
though apparently naive, this is an important
feature of our system which allows different se-
mantic processes to run at different times and
results to be merged at the end.
4 Case studies
We present three applications. First we show
how AsdeCopas can be used in a disambigua-
tion process. Then we use it to build formulas
in MRS (Copestake et al., 2001). Finally, we
present an application where AsdeCopas gener-
ates logical forms from questions. Quantifica-
tion is ignored in this last task.
Notice, however, that in order to have a se-
rious evaluation of AsdeCopas capabilities, it
needs to be applied to more demanding tasks.
4.1 Disambiguation process
Consider again the quantifier qualquer. As we
saw, it can take several semantic values. Some-
times the syntactic context allows to limit these
1Within our applications, dependencies are unla-
belled, and go from dependents to the head. The motiva-
tion behind these structures came from the 5P Paradigm.
possibilities. In some situations, one semantic
value can be chosen, allowing a full desambigua-
tion.
Let us assume that all is an underspeci-
fied value (Poesio, 1994) representing all of the
semantic values. If no desambiguation takes
place, this is the value that will represent this
word’s semantics. Alternatively, we could opt
for a default value. For example, the universal
value since it is the most common.
Let us opt for the universal default value.
We can write a default rule, as the following:
[R1] {elem(qualquer, qt)} : ∅
mapsto→ {sem(qt) = univ}
Assuming again, as we did in section 2,
that on the right of the main verb in the
scope of negation, qualquer takes the value
indiscriminate the following rule allows to
choose the correct value for qualquer in that
context:2
[R2] {elem(qualquer, qt)}
: {arrow(qt, n, L, ),
arrow(n, v, L, ),
arrow(neg, v, R, )}
mapsto→ {sem(qt) = indiscriminate}
R2 is more specific than rule R1, thus it is
applied in these particular conditions. In order
to disambiguate, or at least to limit semantic
values, other semantic rules would have to be
added.
Consider now the Portuguese quantifier
algum. When it appears on the left side of a
noun (n), it means “some” (some). On the right
side it means “none” (none), unless it is in the
scope of negation. In this particular situation
it has an universal value. The following rules
allow the right values to be chosen – in this
particular situations – for this quantifier (notice
that rule R5 is more specific than rule R4):
[R3] {elem(algum, qt)}
: {arrow(qt, n, R, )}
mapsto→ {sem(qt) = some}
2We assume that the object with category n arrowing
an object with category v is the same object with cat-
egory n that receives an arrow from a qt. An index is
used when we need to distinguish two different objects
with the same category.
[R4] {elem(algum, qt)}
: {arrow(qt, n, L, )}
mapsto→ {sem(qt) = none}
[R5] {elem(algum, qt)}
: {arrow(qt, n, L, )
arrow(n, v, L, )
arrow(neg, v, , )}
mapsto→ {sem(qt) = every}3
A precise study of the disambiguation of the
word qualquer can be found in (Coheur, 2003)
and (Coheur, 2004), where we try to go as far
as possible in the disambiguation process of this
word (an some paraphrases of it), by using its
syntactic context. Obviously, there are limits
to this task, as in some situations information
from semantics and pragmatics should also be
taken into account to find the correct semantic
value.
4.2 Logical forms generation
4.2.1 Minimal Recursion Semantics
Linking syntax with semantics is not an easy
task. As Allen says in (Allen, 1995) there seems
to be a structural inconsistency between syn-
tactic structure and the structure of the logical
form.
We can ease this process by using an ad-
equate representation language. In fact, al-
though the concept is not new (Hobbs, 1983),
state of the art frameworks such as (Moll´a et
al., 2003; Baldridge and Kruijff, 2002) are using
flat semantic representations, taht is formulas
with no embedded structures (see (Moll´a, 2000)
for details about flatness), which simplify the
syntactic-semantic interface. At the same time,
and because it is not reasonable to generate all
the possible interpretations of a sentence, many
frameworks are using representation languages
that leave underspecified semantic interpreta-
tions (also an old concept (Woods, 1978)).
MRS (Copestake et al., 2001) uses a flat rep-
resentation with explicit pointers (called han-
dles) to encode scope effects, corresponding to
recursive structures in more conventional formal
semantic representations.
We have chosen this language because it has
three fundamental characteristics: a) it is a flat
language; b) it allows the treatment of quan-
tification; c) it allows underspecification. Un-
3Notice, that by choosing the universal value, in the
final formula this quantifier will no longer be in the scope
of negation.
derspecified MRS structures can be converted
into scope-resolved structures that, according to
(Copestake et al., 1997), “correspond to those
obeyed by a conventionally written bracketed
structure”.
As an example, MRS represents Qualquer
menino adora algum c˜ao(Every boy adores
some dog) in the following underspecified struc-
ture (the =q constraint stands for the equality
modulo quantifiers and relates a handle in an
argument position to a label (Copestake et al.,
2001)):
top p4
h1:every(x, r1, n), h3:menino(x),
r1 =q h3, h7:c~ao(y),
h5:some(y, r5, m), r5 =q h7,
h4:adora(e, x, y)
where h1 outscopes h3 and h5 outscopes h7.
Then, by means of a set of constraints, such
that an MRS structure must be a tree, there
should be a unique top-level handle, etc., the
following readings are obtained:
p=h1 (wide scope “every”)
h1:every(x, h3, h5), h3:menino(x),
h5:some(y, h7, h4), h7:c~ao(y),
h4:adora(e, x, y)
p=h5 (wide scope “some”),
h5:some(y, h7, h1), h7:c~ao(y),
h1:every(x, h3, h4), h3:menino(x),
h4:adora(e, x, y)
In the next section we will show how to reach
these formulas.
4.2.2 Toy example
We will show how to reach an underspeci-
fied MRS representation for constructions as
Qualquer67 menino68 adora69 a70 Maria715
and Qualquer678 menino679 adora680 algum681
c˜ao682. Notice that, for expository reasons,
we are simplifying the process. Actual rules
use fine grained categories for quantifiers,
and scope restrictions are imposed differently
(Coheur, 2004).
In order to perform this task we use the fol-
lowing functions:
4p is the variable over the top.
5Every boy adores Maria
• sem returns a (default) predicate
ex: sem(Maria) = Maria6;
• var returns a variable
ex: var(Maria) = x71;
• handle returns a variable for an handle
ex: handle(Maria) = h71;
• restrictor returns a variable for a restrictor
ex: restrictor(Maria) = r71;
• scope returns a scope variable
ex: scope(Maria) = s71.
The following rule applies to nouns, either
common nouns (nc) or proper nouns (npr),
everytime it finds one (because the arrow set is
empty).
[R1]{elem( , n)}
: ∅
mapsto→ {handle(n): sem(n)(var(n)}
If only this rule is defined, the first sentence
is translated into:
h68:menino(x68)
h71:Maria(x71)
and the second sentence into:
h679:menino(x679)
h682:c~ao(x682)
Nonetheless, h71:Maria(x71) is not the
representation we want for Maria. Instead
we use the predicate NAME. Thus, we define
R2, subsumed by R1 (because n subsetsqequal npr), and
consequently more specific.
[R2]{elem( , npr)}
: ∅
mapsto→ {handle(npr):NAME(var(npr), sem(npr))}
Rule R2 is triggered instead of R1 and we
obtain for the first sentence
h71:NAME(x71, Maria)
instead of
h71:Maria(x71).
6Although these values can be obtained in a lexi-
con, in this application they are generated from sentence
words.
Notice that a new rule needs to be de-
fined for the situations where the npr arrows
an nc and not a v, since we want to trans-
late m˜ae807 Maria81 into m~ae(x80), NAME(x80,
Maria) and not into m~ae(x80), NAME(x81,
Maria). In order to do this, we need only to
add a rule for npr (like the previous rule) to be
applied when a npr arrows an nc. This rule, be-
ing more specific than rule R2, is applied in this
particular situation. As the npr is connected
with the nc, it “knows” its variable, which can
be used is the associated formula.
The next rule is applied to a verb (v) when
the verb has an n arrowing from left (typically
the subject) and an n arrowing from right
(typically the direct object), and no preposition
arrows these nouns.
[R3]{elem( , v)}
: {arrow(ni, v, R, ),
arrow(nj, v, L, ),
no arrow(prep, ni, R),
no arrow(prep, nj, R)}
mapsto→ {handle(v):sem(v)(var(v),var(ni), var(nj))}
As a result, in the first sentence, adora is
translated into:
h69:adora(x69, x68, x71)
and, in the second one, it is translated into:
h680:adora(x680, x679, x682).
Notice that, at this point, although we don’t
have rules for all the elements within the
example sentences, we already have a partial
representation.
Consider now, a generic rule for quantifiers
(qt):
[R4] {elem( ,qt)}
: {arrow(qt, nc, , )}
mapsto→{handle(qt): sem(qt)(var(nc), restrictor(qt),
scope(qt)), restrictor(qt) =q handle(nc)}
Now, the results depend on previous process-
ing: if the disambiguation task described in the
previous section was performed, sem(qualquer)
= every and sem(algum) = some. Otherwise,
underspecified values are used.
7mother.
Let us consider that the disambiguation
stage took place before. Thus, this rule adds to
the first sentence:
h67: every(x68, r67, s67),
r67 =q h68
and to the second sentence:
h678: every(x679, r678, s678),
r678 =q h679
and
h681: some(x682, r681, s681),
r681 =q h682.
Notice that we reach the underspecified for-
mula from 4.2.1 for the first sentence.
We will conclude now this example. It should
be clear that additional rules could impose ex-
tra constraints to the formula, avoiding spurious
ambiguities.
4.3 Question interpretation
From system Edite we inherited a corpus of
680 questions about tourist resources and we
made a preliminar evaluation over 30 questions.
There was no pre-processing of these questions,
no compound terms were detected, no mistakes
were corrected.
Let “correct representation” be a set of for-
mulas representing a question, where the exact
number of expected predicates are produced,
and variables are in the correct places.
Let “system representation” be the set of for-
mulas that the system suggests as the question
representation. Each question can have more
than one “system representation”. Moreover,
a correct “system representation” is a “system
representation” that is equal to the “correct rep-
resentation”.
A general evaluation (of the whole system)
results in a precision of 0,19 (number of correct
“system representations”/number of total “sys-
tem representations”) and a recall of 0,77 (num-
ber of correct “system representations”/30). If
we eliminate two particularly bad results (one
associated 42 “system representations” to a
question and the other 21), we have a precision
of 0,37.
The low precision results from previous
stages, as several graphs are associated with
each question. In fact, with the actual set of
semantic rules only one representation is asso-
ciated with each graph.
Nevertheless, the analysis is not over. The
majority of “system representations” produced
by AsdeCopas are just incomplete and result
from unknown words. For example, the state-
ment Quais os roteiros pedestres sinalizados
em Lisboa?(Which are the signalised footways
in Lisbon?), originated the following formula,
where AM is the predicate for adjectival modifi-
cation and ? indicates the focus of the question:
?x759
roteiros(x759)
AM(x759, x760), pedestres(x760)
em(x759, x763)
NAME(x763, Lisboa)
As the word sinalizados was not recognised,
the “system representation” is not correct, be-
cause a predicate associated with this word is
missing. Nevertheless, most of the information
contained in the question is retrieved.
Within AsdeCopas framework a special ef-
fort was made with paraphrastic relations. As
an example, both phrases Quais os hot´eis com
piscina?(Which are the hotels with a swimming
pool?) and Em que hot´eis h´a piscina?(In which
hotels is there a swimming pool?), result in the
following formulas:
?x22
hot´eis(x22)
com(x22, x24)
piscina(x24)
Note that in order to reach this result, we had
just to look into the particular syntactic condi-
tions that make verb haver (to have) behave as
the preposition com (with).
5 Conclusions
We presented AsdeCopas, a syntax-semantics
interface based on hierarchically organized se-
mantic rules. AsdeCopas is integrated in a sys-
tem called Javali and it has been applied to sev-
eral tasks. Apart from some adjustments, Asde-
Copas should beable to process any dependency
structure.
In the near future, we will have to study co-
ordination properly. We also indent to extend
our work to English.
6 Acknowledgements
We are greatful for the corrections of Dia-
mantino Caseiro, David Matos and S´ergio
Paulo. We also acknowledge Ricardo Ribeiro
and Rui Chaves. Finally, we thank for the use-
ful comments of this paper reviewers. As usual,
the responsability for the contents of this paper
lies with the authors alone.
This paper was supported by FCT (Funda¸c˜ao
para a Ciˆencia e Tecnologia) and by Project
POSI/PLP/41319/2001 (FEDER).

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