Dependency Tree Semantics
Livio Robaldo, [email protected]
Leonardo Lesmo, [email protected]
Department of Computer Science, University of Turin, Italy.
Tel.: +39 011 6706812, Fax: +39 011 751603
Abstract
This report presents a new way to semantic Underspecification of quantifier scope
ambiguities. The key ideas are to keep the semantic representation as close as possible to the syntactic structure, which is given following the Dependency Paradigm
(whence the name Dependency Tree Semantics (DTS)), and to specify the different
(unambiguous) readings by introducing explicit functional dependencies among sets
of entities. The closeness to the syntactic structure allows for a simple and direct
syntax-semantics interface with respect to the syntactic dependency tree; the explicit representation of functional dependencies exploits the insights coming from
the Skolem theorem. The report explains how incremental disambiguation can be
obtained in DTS; this ability is shared by most current underspecified logics based
on dominance constraints and is important for using partial knowledge of the world
in the disambiguation process. Particular attention is paid to the way the so-called
‘Independent Set (IS) readings’ are represented in DTS, which are not taken into
account in most other approaches to Underspecification. The report discusses their
importance and, consequently, the need to consider them in a logical language aiming at representing Natural Language meaning. It then briefly illustrates the logical
framework in (Robaldo, 2010), which represents the meaning of IS readings via
Second Order Logic formulae augmented with standard Generalized Quantifiers. A
model theory of DTS is provided in terms of an algorithm that translates DTS
well-formed structures into (Robaldo, 2010)’s formulae.
1
Introduction
Generalized Quantifiers (GQs) are standardly accepted as essential logical items needed
to properly represent the semantics of NL sentences; see (Mostowski, 1957), (Lindström,
1966), (Barwise and Cooper, 1981), (Montague, 1988), (Keenan and Westerståhl, 1997),
and (Peters and Westerståhl, 2006) to begin with.
Most NL sentences that involve two or more quantifiers yield different interpretations
depending on how their quantifiers interact to each other. In the standard GQ account,
a typical solution to represent these different readings adopts an operation that we term
in this report as ‘quantifier-embedding’. Some quantifiers are embedded as argument
of other quantifiers. For instance, sentence (1.a) yields two interpretations that, in the
1
standard GQ approach, may be represented as in (1.b-c). In (1.b), the GQ ∀x is embedded
in the second argument (the body) of ∃y . This is the reading of (1.a) where there is a
single sound that has been heard by every man. Conversely, the architecture of (1.c)
features the opposite embedding: ∃y occurs in the scope of ∀x . This is the reading of
(1.a) where every man heard a (potentially different) sound.
(1)
a. Everyx man heard ay mysterious sound.
b. ∃y (mystSound (y), ∀x (man (x), heard (x, y)))
c. ∀x (man (x), ∃y (mystSound (y), heard (x, y)))
Quantifier-embedding enables the so-called ‘linear readings’ only, i.e. those readings
where the quantifiers may be arranged along a linear order. For instance, the linear
order characterizing (1.b) is ‘∃y ∀x ’, while the one characterizing (1.c) is ‘∀x ∃y ’.
On the contrary, it is impossible to represent readings where two or more sets of
entities do not vary to each other, i.e. where they are independent. Those readings are
usually known as ‘Scopeless readings’, but we prefer to term them as ‘Independent Set
(IS) readings’, for reasons that will be clear below (see footnote 4 below). Four kinds of
IS readings have been identified in the literature, starting from (Scha, 1981).
(2)
a. Branching Quantifier readings, e.g. Two students of mine have seen three
drug-dealers in front of the school. (Robaldo, 2009a)
b. Collective readings, e.g. Three boys made a chair yesterday.
(Nakanishi, 2007)
c. Cumulative readings, e.g. Three boys invited four girls. (Landman, 2000)
d. Cover readings, e.g. Three children ate five pizzas. (Kratzer, 2007)
The preferred reading of (2.a) is the one where there are exactly two1 students and exactly
three drug-dealers and each of the students saw each of the drug-dealers. (2.b) may be
true in case three boys cooperated in the construction of a single chair. In the preferred
reading of (2.c), there are three boys and four girls such that each of the boys invited
at least one girl, and each of the girls was invited by at least one boy. Finally, (2.d)2
allows for any sharing of ten pizzas between twenty children. In Cumulative readings,
the single actions are carried out by atomic3 individuals only, while in (2.d) it is likely
that the pizzas are shared among subgroups of children. For instance, (2.d) is satisfied
by the following extension of ate (‘⊕’ is the standard sum operator, taken from (Link,
1983)):
1 In (2.a-d) “two/three/ten/etc.” are interpreted as “exactly two/three/ten/etc.” as in (Scha, 1981).
That is actually a pragmatic implicature, as noted in (Landman, 2000), pp.224-238.
2 The original example by (Kratzer, 2007) is actually Twenty children ate ten pizzas. I changed the
value of the numerals for the sake of simplicity only.
3 In line with (Landman, 2000), pp.129, and (Beck and Sauerland, 2000), def.(3), that explicitly
define Cumulative readings as statements among atomic individuals only. Others, e.g. (Krifka, 1992),
(Sternefeld, 1998), and (Hackl, 2002), proposed definitions of Cumulativity that actually handle the
truth values of Cover readings.
2
(3)
ate M ≡ {c1 ⊕ c2 ⊕ c3 , p1 ⊕ p2 , c2 ⊕ c3 , p3 ⊕ p4 , c3 , p5 }
In (3), children c1 , c2 , and c3 (cut into slices and) share pizzas p1 and p2 , c2 and c3 (cut
into slices and) share p3 and p4 , and c3 also ate pizza p5 on his own.
Branching Quantifier readings have been the more controversial (cf. (Beghelli, BenShalom, and Szabolski, 1997) and (Gierasimczuk and Szymanik, 2009)). Many authors
claim that those readings are always subcases of Cumulative readings, and they often
co-occur with certain adverbs (May, 1989), (Schein, 1993). Collective and Cumulative
readings have been largely studied in the literature; see (Scha, 1981), (Link, 1983), (Beck
and Sauerland, 2000), (Landman, 2000), and (Ben-Avi and Winter, 2003) to begin with.
However, the focus here is on Cover readings. This report assumes – following (van der
Does, 1993), (van der Does and Verkuyl, 1996), (Schwarzschild, 1996), (Kratzer, 2007)
– that they are the Scopeless readings, of which the three kinds exemplified in (2.a-c)
are merely special cases. The name ‘Cover readings’ comes from the fact that their
truth conditions are traditionally captured in terms of Covers, a particular mathematical
structure that will be presented below in section 2.
It is easy to see that quantifier-embedding intrinsically prevents the proper representation of IS readings. By nesting a quantifier in the scope of another one, we implicitly
assume that the sentence is satisfied by models where the set of entities associated with
the former varies depending on each entity quantified by the latter (cf. (Beghelli, BenShalom, and Szabolski, 1997)).
Several alternative accounts have been proposed in the literature to properly handle IS
readings, e.g. (Schein, 1993), (Carpenter, 1997), (Hackl, 2000), (Landman, 2000), (Beck
and Sauerland, 2000) (Ben-Avi and Winter, 2003), and (Kratzer, 2007). However, it is
not always immediatly clear how to integrate the logical frameworks proposed in these
works with theories of Generalized Quantifiers.
A recent logical framework that deals with IS readings explicitly in terms of GQs
is (Robaldo, 2010). The proposal extends two previous attempts, i.e. (Sher, 1997)
and (Robaldo, 2009a), that uniformily represent Branching Quantifiers readings as (2.a)
and standard linear readings as (1.b-c) in terms of Skolem-like functional dependencies4
between the witness sets of the GQs. The extension is achieved by incorporating the Cover
variables defined in (Schwarzschild, 1996) in (Robaldo, 2009a)’s. This move enhances the
expressivity of the latter in order to represent Cover readings, which are assumed to
be the more general cases of IS readings, as in the other ‘Cover approaches’ mentioned
above. The proposal in (Robaldo, 2010) is illustrated in more details in section 2 below.
Nevertheless, (Robaldo, 2010) does not define any syntax-semantics interface, i.e. it
does not explain how to obtain the final formulae from the syntactic structure of the NL
sentence. This is instead the goal of the present report. This report defines a new logical
form acting as an intermediate semantic representation between the syntactic structure
and the fully-unambiguous formulae.
Following moderns theories of syntax-semantic interfaces (see (Bunt, 2003), (Ebert, 2005),
or section 3 below), our logical forms will be underspecified formulae. In Semantic Un4 Since the logical framework in (Robaldo, 2010) is defined upon Skolem-like dependencies rather
than a scope mechanism, we avoid the term ‘Scopeless reading’ in favour of ‘Independent Set reading’ in
order to emphasize the fact that the desired interpretations should be kept conceptually distinct from the
formal instrument used to represent them (functional dependencies rather than quantifier-embeddings).
3
derspecification, quantifier scoping and its relatives are seen as instances of unspecified
meaning that can adopt a more restricted sense depending on preferences grounded on
the contextual world-knowledge, the topic/focus distinction and so forth. Underspecified formulae encapsulate all readings of an NL sentence into a single compact structure,
which is compositionally obtained starting from the syntactic representation. Since an
underspecified formula remains ambiguous, as it refers to multiple interpretations, it is
necessary to provide a mechanism to specify it into one of the latter.
The formalism proposed here is defined with respect to a Dependency Grammar (Tesniere, 1959), (Hudson, 1990), (Mel’cuk, 1988), (Kahane, 2003). In the light of this, we call
our formalism ‘Dependency Tree Semantics (DTS)’. The compositional syntax-semantics
interface of DTS will be illustrated in section 5.
In section 6, we illustrate the DTS disambiguation process, i.e. how it is possible to fullyspecify a DTS underspecified structure. In short, this is done by specifying dependencies
between quantifiers, that resemble Skolem functional dependencies.
Finally, section 8 presents an algorithm that translate a DTS fully-disambiguated structure into the formulae defined in (Robaldo, 2010). Conclusions close the report.
2
Interpreting IS readings in terms of (In)dependence
and Maximality
As pointed out in the Introduction, (Robaldo, 2010) considers Cover readings as the
Independent Set readings of NL sentences, of which Branching quantifiers readings, Cumulative readings, and Collective readings, exemplified in (2.a-c), are mere special cases.
A Cover is a mathematical structure defined with respect to one or more sets. With
respect to two sets S1 and S2 , a Cover C is formally defined as in (4):
(4)
A Cover Cov is a subset of Cov1 × Cov2 , where Cov1 is a set of subsets of
S1 and Cov2 is a set of subsets of S2 , such that:
a. ∀s1 ∈ S1 , ∃cov1 ∈ Cov1 such that s1 ∈ cov1 , and ∀s2 ∈ S2 , ∃cov2 ∈
Cov2 such that s2 ∈ cov2 .
b. ∀cov1 ∈ Cov1 , ∃cov2 ∈ Cov2 such that cov1 , cov2 ∈ Cov.
c. ∀cov2 ∈ Cov2 , ∃cov1 ∈ Cov1 such that cov1 , cov2 ∈ Cov.
In (Schwarzschild, 1996), the Cover approach from which (Robaldo, 2010) mainly draws,
Covers are denoted by 2-order variables called ‘Cover variables’. We may then define a
meta-predicate Cover that, taken a Cover variable C and two unary predicates P1 and
P2 , asserts that the extension of C is a Cover of the extensions of P1 and P2 (subsets are
implemented as plural sums):
4
(5) Cover(C, P1 , P2 ) ⇔
∀X1 X2 [C(X1 , X2 )→∀x1 x2 [((x1 ⊂ X1 ) ∧ (x2 ⊂ X2 ))→(P1 (x1 ) ∧ P2 (x2 ))]] ∧
∀x1 [ P1 (x1 ) → ∃X1 X2 [ (x1 ⊂ X1 ) ∧ C(X1 , X2 ) ] ]
∀x2 [ P2 (x2 ) → ∃X1 X2 [ (x2 ⊂ X2 ) ∧ C(X1 , X2 ) ] ]
The concept of Cover and the formal definition of Cover may be easily extended to tuples
of sets with any cardinality and to n-tuples of unary predicates P1 , . . . , Pn respectively.
The general definition of Cover is reported in (Robaldo, 2010), § 4.
In terms of Cover variables, it is possible to decouple the quantification from the
predications. We introduce relational variables whose extensions specifies the atomic
individuals involved in the sentence’s meaning. The truth values of the GQs are attested
on these relational variables. Then, other relational variables describe how the actions
are actually done. The latter variables are required to cover the former, i.e. they have
all to satisfy the Cover meta-predicate. For instance, in (3), in order to evaluate (2.d)
as true, we may introduce three variables P1 , P2 , and C such that:
(6)
P1 M,g = {c1 , c2 , c3 }
P2 M,g = {p1 , p2 , p3 , p4 , p5 }
CM,g = {c1 ⊕ c2 ⊕ c3 , p1 ⊕ p2 , c2 ⊕ c3 , p3 ⊕ p4 , c3 , p5 }
In (6), children c1 , c2 , and c3 (cut into slices and) share pizzas p1 and p2 , c2 and c3 (cut
into slices and) share p3 and p4 , and c3 also ate pizza p5 on his own.
P1 , P2 , and C satisfy Cover(C, P1 , P2 ). In order to correctly evaluate the sentence as
true, we may simply require P1 and P2 to respectively contain three children and five
pizzas, and separately that CM includes ate M . In (Robaldo, 2010), this is done via
the following formula5 :
(7)
Cx : 3!x(child’(x), PxB (x)) ∧ Cy : 5!y(pizza’(y), PyB (y)) ∧
IN : (Cover(C, PxB , PyB ) ∧ ∀xy [C(x, y))→ ate’(x, y)]) ∧
M ax(PxB , PyB , IN )
Cx , Cy , and IN are simply labels, introduced in order to write more compact formulae.
Cx and Cy mark two subformulae called ‘Cardinality conditions’, while IN marks a
subformula called ‘Inclusion condition’. The two cardinality conditions are asserted in
terms of the Generalized Quantifiers 3!x and 5!y . Cx requires PxB ’s extension to include
exactly three children, while Cy requires PyB ’s extension to include exactly five pizzas. In
particular, the extension of PxB and PyB are required to be the witness sets of the GQs,
and so to include atomic individuals only.
On the other hand, the Inclusion Condition IN requires the extension of C to be a Cover
of the extensions of PxB and PyB , and to be included in ate M .
The final clause, i.e. M ax(PxB , PyB , IN ), requires PxB and PyB to be Maximal with
respect to the IN Condition. In particular, M ax(PxB , PyB , IN ) requires the nonexistence of a superset of PxB (or PyB ) for which ate M includes a Cover of P and PyB
(or PxB ) that also includes C. In other words, the following equivalence must hold:
5 Actually, the formula in (7) is a simplified version of the one proposed in (Robaldo, 2010). The
difference is that the latter also introduce two relational variables PxR and PyR that refer to the restrictions
of the two GQs. Full examples are shown below in this section.
5
(8)
M ax(PxB , PyB , IN ) ⇔
∀P1 [(∀x [P1 (x)→P1 (x)] ∧ ∃C [Cover(C , P1 , P2 ) ∧ ∀xy [C(x, y)→ C (x, y)]∧
∀xy [C (x, y)→ate’(x, y)]])→∀x [P1 (x)→P1 (x)] ] ] ∧
∀P2 [(∀y [P2 (y)→P2 (y)] ∧ ∃C [Cover(C , P1 , P2 ) ∧ ∀xy [C(x, y)→ C (x, y)]∧
∀xy [C (x, y)→ate’(x, y)]])→∀y [P2 (y)→P2 (y)] ] ]
Also the definition of M ax is easy to generalize to any tuple of arguments (see (Robaldo,
2010), section 6). Maximality Conditions are needed in order to achieve the proper truth
conditions of IS readings, as extensively explained in (van Benthem, 1986), (Kadmon,
1987), (Sher, 1990), (Sher, 1997), (Spaan, 1996), (Steedman, 2007), (Robaldo, 2009a),
(Robaldo, 2009b), and (Robaldo, 2010), among others. They are not the focus of the
present report, however; therefore, we do not discuss their importance here.
Finally, another important feature of formula (7) is that the relational variables PxB ,
B
Py , and C are pragmatically interpreted. In other words, their value must be provided by
an assignment g rather than being existentially quantified. Accordingly, we indicate their
extensions as PxB M,g , PyB M,g , and CM,g respectively. The need of pragmatically
interpreting the witness sets and the Covers is extensively illustrated and exemplified in
(Schwarzschild, 1996) and (Robaldo, 2010).
The logical framework introduced in (Robaldo, 2010) features an higher degree of
expressivity, uniformity, and scalability with respect to its previous proposals.
The framework is expressive enough to represent IS interpretations on the predicates in
the quantifiers’ restriction. As pointed out above in footnote 5, this is done by inserting a relational variable also in the first argument of the GQ. Such a relational variable
is covered and maximized, possibly with other variables, exactly as it is done with the
variables reifying the witness sets. For instance, the sentence in (9) means that most individuals in the set of all students enrolled in an Italian university arrive. In other words,
the predicate conveyed by “of” must receive a cumulative interpretation: the restriction
set is the cumulation of all students individually enrolled in an Italian university. This
meaning is denoted by the formula below the sentence in (9). The set PyR M,g , i.e. the
restriction of “Most”, is covered and maximized together with PyB M,g , i.e. the body of
the quantifier6 “The”, that includes the set of all Italian universities.
6 In (Robaldo, 2010) and in the present report, definites correspond to GQs, as in the standard
Russellian tradition, e.g. (Barwise and Cooper, 1981), (Beghelli, Ben-Shalom, and Szabolski, 1997), and
(Neale, 1990).
6
(9) Mostx students of they Italian universities arrive.
[ Cx : M ostx (PxR (x), PxB (x)) ∧ Cy : T hey (PyR (y), PyB (y)) ∧
Ry : {Cover(CovPyR , PyR ) ∧∀y [CovPyR (y)→itU niv (y)]} ∧ M ax(PyR , Ry ) ∧
Rx : {Cover(CovPxR PyB , PxR , PyB ) ∧ ∀xy [CovPxR PyB (x, y)→enrolledIn (x, y)]} ∧
M ax(PxR , PyB , Rx ) ∧
IN : {Cover(CovPxB , PxB ) ∧ ∀xy [CovPxB (x)→arrive (x)]} ∧ M ax(PxB , IN ) ]
The logical framework in (Robaldo, 2010) allows to uniformily represent standard linear
readings, besides the IS ones. In that cases, some Cardinality Conditions trigger the
proper Skolem-like dependencies between two or more sets of entities. A Maximality
condition ensures that the functional dependency is Maximal in the same sense explained
above. For instance, in the linear reading of sentence (10) there is a set of two apples for
each individual boy. This meaning is denoted by the formula in (10) below the sentence:
(10) Eachx boy ate twoy apples.
[ Cx : ∀x (PxR (x), PxB (x)) ∧
Rx : {Cover(CovPxR , PxR ) ∧∀x [CovPxR (x)→boy (x)]} ∧ M ax(PxR , Rx ) ∧
Cy : {Cover(CovPxB , PxB ) ∧∀x [CovPxB (x)→2!y (PyR (x, y), PyB (x, y))]} ∧
M ax(PxB , Cy ) ∧
Ry : {Cover(CovPyR , PyR ) ∧∀xy [CovPyR (x, y)→apple (x, y)]} ∧ M ax(PyR , Ry ) ∧
IN : {Cover(CovPyB , PyB ) ∧∀xy [CovPyB (x, y)→ate (x, y)]} ∧ M ax(PyB , IN ) ]
Finally, scalability is achieved in that the logical framework allows to represent any
intermediate cases between those exemplified above, with any number of quantifiers. For
instance, the preferred reading of (11), taken from (Beck and Sauerland, 2000), is an
IS reading where there is a set PxB M,g of many politicians and a set PzB M,g of five
companies such that each politician in the former took a bribe from at least one group
of companies in the latter and each company in the latter gave a bribe to at least one
group of politicians in the former. Therefore, in that reading, PzB M,g is independent
from PzB M,g , while the bribes depend on the entities in both sets. This meaning is
represented by the formula in (11), below the sentence
7
(11) Manyx politicians have taken ay bribe from fivez companies.
[Cx : M anyx (PxR (x), PxB (x)) ∧ Cz : 5!z (PzR (z), PzB (z)) ∧
Rx : {Cover(CovPxR , PxR ) ∧ ∀x [CovPxR (x)→ polit (x)]} ∧ M ax(PxR , Rx ) ∧
Rz : {Cover(CovPzR , PzR ) ∧ ∀z [CovPzR (z)→ comp (z)]} ∧ M ax(PzR , Rz ) ∧
Cy : {Cover(CovPxB PzB , PxB , PzB ) ∧
∀xz [CovPxB PzB (x, z)→ ∃y (PyR (x, z, y), PyB (x, z, y))]} ∧ M ax(PxB , PzB , Cy ) ∧
Ry : {Cover(CovPyR , PyR ) ∧ ∀xzy [CovPyR (x, z, y)→bribe (y)]} ∧ M ax(PyR , Ry ) ∧
IN : {Cover(CovPyB , PyB )∧∀xzy [CovPyB (x, z, y)→took (x, y, z)]} ∧ M ax(PyB , IN ) ]
3
Semantic Underspecification - Background
As pointed out in the Introduction, the present report aims at defining a formalism acting as a bridge between a Dependency Grammar and the logical framework presented
in the previous section. We want our formalism to be underspecified, i.e. such that
quantifier scoping and its relatives are seen as instances of unspecified meaning that
can adopt a more restricted sense depending on preferences grounded on the contextual world-knowledge, the syntactic structure, the topic/focus distinction, and so forth7 .
Underspecified logics must encapsulate all readings of a sentence into a single compact
representation, which, in the following, we call ‘underspecified forms’ (UF). This is done
in order to preserve the one-to-one mapping from syntax to semantics: the UF must be
compositionally obtained starting from the syntactic structure of the sentence.
Since the UF remains ambiguous, at least from a standard logical point of view, one of
its main features must be the ease with which the actual non-ambiguous readings can
be extracted from it afterwards. In other words, an underspecified logical language does
not have an associated model theory, but there are methods for obtaining, out of ambiguous structures, unambiguous formulae in some logical language, as Predicate Logic
or (Robaldo, 2010)’s framework, whose model theory is well defined.
A full overview of goals and techniques recently proposed in Underspecification may be
found in (Bunt, 2003). See also (Ebert, 2005) for a comparison among existing underspecified formalisms, their expressivity, and their computational complexity.
The first approach to the disambiguation of underspecified structures is perhaps the
well-known Hobbs and Shieber’s algorithm (Hobbs and Shieber, 1987), where the UF
is a partial formula Φ, easy to obtain from the syntactic structure of the sentence. An
UF is said to be “partial” as it may contain unsolved terms. In order to obtain the
disambiguated formulae, those terms must be solved. This is done by “pulling out” and
“unstoring” the unsolved terms one by one, respecting certain constraints, until no one of
them appears in the formula any longer. Depending on the order in which unsolved terms
are chosen, different readings are obtained. An example of UF in Hobbs and Shieber’s
7 For instance, the English determiner each tends to favour a wide-scope reading of the NP including
it, while the opposite holds for other universal quantifiers like all. See (Moran, 1988), (Kurtzman and
MacDonald, 1993), and (Villalta, 2003).
8
underspecified logics is shown in (12)
(12) [Everyx representative of [ay company]] saw [mostz samples].
see’(< ∀, x, rep’(x) ∧ of ’(x, < ∃, y, comp’(y) >) >, < M ost, z, samp’(z) >)
This formula contains three unsolved terms, called complex terms, in the form <q, v, r>
where q is a Generalized Quantifier, v an individual variable and r, the restriction of q,
another partial formula. By solving the complex terms in different orders, the algorithm
produces the following readings:
(13)
a. M ostz (samp (z), ∃y (comp (y), ∀x (rep (x)∧of (x, y), see (x, z)))
b. ∃y (comp (y), M ostz (samp (z), ∀x (rep (x)∧of (x, y), see (x, z)))
c. ∀x (∃y (comp (y), rep (x)∧of (x, y)), M ostz (samp (z), see (x, z)))
d. M ostz (samp (z), ∀x (∃y (comp (y), rep (x)∧of (x, y)), see (x, z)))
e. ∃y (comp (y), ∀x (rep (x)∧of (x, y), M ostz (samp (z), see (x, z))))
The ideas behind Hobbs and Shieber’s algorithm have then been imported and extended
in several underspecified frameworks, one of which is Quasi Logical Form (QLF) (Alshawi, 1992) and (Alshawi and Crouch, 1992). The QLF scoping mechanism (see (Moran,
1988)) is basically the same implemented in Hobbs and Shieber’s algorithm. The main
difference is that QLF is able to enumerate the available readings in order of preference.
Furthermore, QLF features a wider linguistic coverage in that it allows to manage the
interaction between quantifiers, modals, opaque operators and other scope bearers and
to underspecify anaphora and implicit relations like ellipsis.
Hobbs and Shieber’s algorithm and QLF clearly resemble some previous strategies based
on overt or covert quantifier movement (e.g. Quantifying-in (Montague, 1988) and
Quantifier-Raising (Heim, 1982), (May, 1985), and (Diesing, 1992)), or similar alternatives like the well-known Cooper Storage (Cooper, 1983), (Larson, 1985), and (Keller,
1988), or Type-raising (Barwise and Cooper, 1981). The added value is the precise definition of an intermediate (underspecified) representation as a way to block the ongoing
derivation and store, in a compact way, all available readings it yields, until the knowledge
needed to determine the contextually relevant one becomes available.
Most recent works in Underspecification deal with scope ambiguities via dominance
constraints between subformulae, each of which refers to an unambiguous portion of
meaning conveyed by the sentence. In other words, formalisms of this type are based
on sets of unambiguous pieces of formula that must be linked like a puzzle. In case of
scope ambiguity, pieces can be linked in several ways, leading to different interpretations.
Moreover, it is possible to reduce the number of available readings by adding constraints
9
on allowed links in a monotonic way.
The first relevant proposal along this line is perhaps (Reyle, 1993), (Reyle, 1996) where
UDRT, an underspecified version of Discourse Representation Theory (Kamp and Reyle,
1993) is presented. The ideas lying behind UDRT have been then generalized in Hole
Semantics (Bos, 1996), (Bos, 2004). Hole Semantics is a metalanguage which can be
wrapped around a logical language, called the ‘object language’, in order to create an
underspecified version of it. In (Bos, 1996) this is shown for standard predicate logic,
dynamic predicate logic (Groenendijk and Stokhof, 1991) and DRT but, in principle, the
object language may be any unambiguous logic. For example, taking standard predicate
logic as object language, the underspecified representation of (1.a) in Hole Semantics is
(14)
{h0 , l1 :∀x [h11 →h12 ], l2 :∃y [h21 ∧ h22 ], l3 :man (x), l4 :mystSound (y), l5 :hear (x, y)}
{l1 ≤ h0 , l2 ≤ h0 , l3 ≤ h11 , l4 ≤ h21 , l5 ≤ h12 , l5 ≤ h22 }
This representation consists of two sets: a set of labelled subformulae containing particular variables named holes (e.g. h11 , h12 , etc.) and a set of dominance constraints between
holes and labels (in the form l ≤ h). Disambiguation is made by inserting labels into
holes, according to the dominance constraints, via a plugging function P . A dominance
constraint is in the form l ≤ h and specifies that the piece of formula labelled by l will
fill h either directly or transitively, i.e. h will be filled either by it or by a larger fragment
containing it. Just two pluggings are possible in (14). The first one, which yields reading
(1.b), where a single sound has been heard by every man, is:
(15)
{P (h0 ) = l2 , P (h21 ) = l4 , P (h11 ) = l3 , P (h22 ) = l1 , P (h12 ) = l5 }
∃y [mystSound (y) ∧ ∀x [man (x) → hear (x, y)]]
while (1.c), where every man heard a (potentially different) sound, is:
(16)
{P (h0 ) = l1 , P (h11 ) = l3 , P (h21 ) = l4 , P (h12 ) = l2 , P (h22 ) = l5 }
∀x [man (x) → ∃y [mystSound (y) ∧ hear (x, y)]]
Other formalisms based on dominance constraints are the approach proposed in (Willis,
2000), Constraint Language for Lambda Structures (Egg, Koller, and Niehren, 2001),
Normal Dominance Graphs (Koller, 2004), Minimal Recursion Semantics (Copestake,
Flickinger, and Sag, 2005), and the feature-based approach of (Kallmeyer and Romero,
2008). Their expressive power is compared in (Koller, Niehren, and Thater, 2003),
(Fuchss et al., 2004), and (Ebert, 2005).
The innovation introduced in the constraint-based underspecified formalisms with
respect to their precursors was the possibility to achieve partial disambiguations. This
is needed for a suitable management of partial knowledge. As pointed out above, an
underspecified formula allows to represent all available readings in a compact way until
the knowledge needed to disambiguate becomes available. In real-life scenarios, however,
this knowledge normally does not become available simultaneously. A constraint-based
approach provides a more flexible framework to incrementally perform disambiguation,
10
in that dominance relations can be asserted one by one, independently of each other.
Instead, formalisms like QLF, which aim to (totally or partially) instantiate the final
scoping relations on quantifiers, intrinsically prevent partial disambiguations. This has
been discussed by (Reyle, 1996) in the light of examples as
(17) Everyxbody believed that many ay problem about the environment preoccupied
mostz politicians.
As is well known, quantifiers are (with some exceptions) clause bounded; concerning
sentence (17), this amounts to saying that both M anyAy and M ostz must have narrow
scope with respect to Everyx (in symbols, M anyAy ≤ Everyx and M ostz ≤ Everyx ).
However, the sentence does not provide enough information to establish also the relative
scope between M anyAy and M ostz , i.e. to specify that what is believed by everybody
is either that most politicians are worried about a (potentially different) set of many
problems about the environment (i.e. M anyAy ≤ M ostz ) or that there is a specific
set of many problems about the environment that worries a (potentially different) set of
politicians (i.e. M ostz ≤ M anyAy ).
In QLF it is impossible to specify this partial scoping. Once the relative scope between
two quantifiers is fixed, for example M anyAy ≤ Everyx , it is only possible to specify the
scope of M ostz with respect to both of them, i.e. to establish one of the final orders:
- M ostz ≤ M any ay ≤ Everyx
- M any ay ≤ M ostz ≤ Everyx
Whereas we cannot independently assert only the weaker constraint M ostz ≤ Everyx .
On the contrary, this may be easily handled in Hole Semantics, Minimal Recursion Semantics, or UDRT, since each dominance constraint indicates that a certain piece of
formula must be inserted into another one, but still it leaves unspecified whether this
insertion is achieved directly or rather transitively, via other pieces.
Nevertheless, the possibility to achieve partial disambiguations has been paid in terms
of a minor correspondence with respect to the syntactic architecture, which is instead one
of the peculiarity of QLF. Constraint-based formalisms tend to move material away from
its surface position, because they basically spread the sentence meaning among several
“pieces” of formula and constrain their recomposition. This amounts to saying that the
syntax-semantics interface turns out to be more complex in that it has to manage initial
dominance constraints together with predicate-argument relations. For instance, consider
the constraint-based underspecified representation in (18), built starting from the LTAG
(Joshi and Schabes, 1997) structure of (12) via the syntax-semantics interface proposed
in (Joshi and Kallmeyer, 2003).
(18)
{ l1 :∀x (h11 , h12 ), l2 :∃y (h21 , h22 ), l3 :M ostz (h31 , h32 ), l4 :rep’(x) ∧ of ’(x, y),
l5 :comp’(z), l6 :samp’(z), l7 :saw’(x, z)},
{l4 ≤ h11 , l5 ≤ h21 , l6 ≤ h31 , l7 ≤h12 , l7 ≤h32 , l4 ≤h22 }
In this represention, there are six initial dominance relations. The ones marked in boldface specify the predicates to be included in the bodies of the three quantifiers, while the
11
other ones specify the predicates to be included in their restrictions.
However, only the constraints concerning the restrictions arise from the syntactic structure, while those related with the bodies are just needed to prevent the occurrence of
free variables in the final representation. Furthermore, as noticed in (Joshi, Kallmeyer,
and Romero, 2003), the representation in (18) actually permits more readings than those
that seem necessary, because the sentence contains a syntactic Nested Quantification in
the subject. To avoid the unexpected interpretations, (Joshi, Kallmeyer, and Romero,
2003) introduces further constraints in the syntax-semantics interface.
There is no need to assert similar constraints in QLF, because the prevention of unavailable readings is in charge of the disambiguation process rather than the syntax-semantics
interface. Consequently, the latter turns out to be simpler.
To summarize, the investigation of main underspecified approaches proposed in the
literature highlights a trade-off between ease of definition of the syntax-semantics interface and flexibility of manipulation of the semantic representation. This is mainly due to
the fact that in QLF-like formalisms all required information is embodied in a single UF
(simpler to build but more rigid in structure), while in formalisms based on constraint it
is split in different components (more flexible, but also more distant from the syntactic
structure). It must be pointed out that, in the light of these considerations, some translation processes from formalisms of the first kind to those of the second kind have been
proposed (van Genabith and Crouch, 1996) and (Lev, 2005). Their aim is to combine the
advantages of both by obtaining a QLF-like underspecified formula from the syntactic
representation, then translating it into a constraint-based one.
4
A new way to Underspecification: Dependency Tree
Semantics
We present now a new underspecified formalism, called ‘Dependency Tree Semantics’
(DTS), based on functional (Skolem-like) dependencies among involved sets of entities.
Underspecified structures in DTS are based on a simple graph G that represents the
predicate-argument relations appearing in the sentence. The nodes of G refer either to
predicates or to variables; the variables are called ‘discourse referents’, as in DRT. Each
arc connects a predicate with a discourse referent and is labelled with the argument position. Each discourse referent is also associated with a Generalized Quantifier, according
to a function quant from discourse referents to Generalized Quantifiers, and a restriction,
according to a function restr from discourse referents to subgraphs of G.
While the value of quant is clear, the value of restr involves the syntactic structure of the
sentence, and, consequently, the syntax-semantics mapping. Intuitively, restr(d), where
d is a discourse referent, is the subgraph including all the nodes that contribute to the
identification of the individuals referred to. Note that, from a formal point of view,
restr(d) is just a subgraph of G.
As a first simple example, we show in fig.1 the fully underspecified DTS representation
for sentence (1). As the reader can see, it simply depicts the predicate-argument relations
occurring in (1), the involved quantifiers and the predicate-argument relations needed to
identify the set of individuals over which x and y respectively range.
12
½
Ü
Ü
¾
Ý
½
½
Ý
½
½
Figure 1: DTS underspecified structure for Every man heard a mysterious sound
To disambiguate the representation in fig.1 we must specify the functional dependencies among involved entities. This is done by inserting additional (dotted) arcs, named
SemDep arcs, between discourse referents. Furthermore, to represent that a certain
group of entities does not depend on the entities of any other group, i.e. it “takes scope
everywhere”, the corresponding discourse referent is linked, via a SemDep arc, to an
additional node named Ctx. Ctx refers to the context, that is, the domain of individuals
with respect to which the representation will be evaluated.
For example, in fig.2, on the left, a SemDep arc links y to x: this is the interpretation in
which each individual in the set of all men has heard a potentially different sound. On
the contrary, in fig.2, on the right, y is linked to Ctx; this means that the sound has to
be the same for all men.
½
Ü
½
¾
Ý
½
Ü
½
½
¾
Ý
½
Figure 2: DTS fully disambiguated structures for Every man heard a mysterious sound
It is important to note that SemDep arcs represent a transitive relation and that they
are minimal. This means that:
13
- No arc appears in a figure that can be derived because of the transitivity of the relation (so the semantic dependency of y on Ctx in fig.2.a is not represented explicitly
by means of an arc).
- If two nodes are not connected (either directly or because of transitivity), then the
pair of those nodes does not belong to the relation. So, in fig.2.b, since x is not
connected to y via a SemDep arc (and no such a connection can be inferred via
transitivity), it is assumed that x and y are independent of one another.
Note that the reading in fig.2.b is a (trivial) example of IS reading. In other words, fig.2.b
is not the counterpart of the FOL formula in (1.c) and the Hole Semantics fully-specified
representation in (15), although it has the same truth values. In fact, in those formulae,
∀ is in the scope of ∃, while in fig.2.b, the two discourse referents are independent of each
other, hence none of their corresponding quantifiers is in the scope of the other one.
The DTS counterpart of (1.c) and (15) is shown in fig.3. This reading is equivalent to
the IS one, hence we consider it as redundant. In fact, it describes a model where there
is a set including a single sound and each individual in this set (i.e. that specific sound)
has been heard by a set including all men. Clearly, this is equivalent to take the set of
all men directly from the context and assert that they heard the sound.
½
Ü
¾
Ý
½
½
Figure 3: A third (redundant) reading of Every man heard a mysterious sound
In this work we do not analyse logical redundancy in depth, but we introduce some
constraints to prevent all but one reading in an equivalence class. As said above, with
respect to the current example, the allowed reading will be the IS one, while the one in
fig.3 will be considered redundant and, therefore, forbidden by the constraints.
5
Building underspecified structures in DTS starting
from a Dependency Tree
As the name ‘Dependency Tree Semantics’ suggests, well-formed (fully-underspecified)
structures in DTS are built starting from the Dependency Tree (DT, henceforth) of the
sentence. This section outlines how the syntax-semantics interface DT⇒DTS may be
implemented for the basic examples mentioned in the report.
14
5.1
Dependency Grammars
A Dependency Grammar (DG, henceforth) is a formalism that allows to describe NL
syntax in terms of oriented relations between words, called ‘dependency relations’ or
‘grammatical functions’. In particular, a DG analysis represents a NL sentence by means
of a hierarchical arrangement of words linked via dependency relations.
As pointed out above, dependency relations are oriented; therefore, for each pair of linked
words, we can identify a head (the origin of the link) and a dependent (the destination
of the link). The dependent plays the role of “completion” of the head, i.e. it provides a
sort of “parameter” to the latter, instantiating, in this way, the head meaning on the dependent meaning. For this reason, Dependency Grammars are traditionally expressed in
terms of valency8 . Dependency relations are usually labeled in order to make explicit the
function played by a dependent with respect to the head. Although the precise inventory
of labels varies from theory to theory, many grammatical functions, such as ‘subject’,
‘object’, etc., are commonly accepted in the literature and may be found in all Dependency Grammars. Morevoer, it is important to note that, in a dependency relation, not
only the dependent, but the whole subtree having the dependent as root can contribute
to the “completion” of the head.
Modern Dependency Grammars are attributed to (Tesniere, 1959)9. Tesniere’s Structural Syntax analyzes sentences in terms of functions held by the words, which trigger
subordinative relations called connections. Structural Syntax has been formalized in a
computational model in Functional Dependency Grammar (Tapanainen, 1999).
Other important works belonging to the DG approach are Hudson’s Word Grammar
(Hudson, 1990), a Dependency Grammar structured in an “is–a” taxonomy that allows
for linguistic generalizations and enables one to deal with default cases/exceptions, and
Meaning⇔Text Theory ((Mel’cuk, 1988), (Kahane, 2003)), a formal framework structured along seven sequential levels of description, which allows for the parallel representation of various kinds of linguistic knowledge.
One of the most recent proposals in the field of Dependency Grammar is Extensible Dependency Grammar (XDG, henceforth) (Debusmann, 2006), a modular framework able to
handle different aspects of Natural Language, as Meaning⇔Text Theory does. However,
contrary to the latter, in XDG the various levels of description, called dimensions, are
more integrated (besides being more fine-grained). In particular, while in Meaning⇔Text
Theory the levels of description are strictly sequential, i.e. the rules to interface a level
to another one are defined for adjacent levels only, in XDG such rules can be in principle
defined for each pair of dimensions.
The Dependency Structures we refer to in the ongoing research on DTS are being used
at the University of Torino in a number of projects, including the development of a Treebank for Italian (Bosco, 2004) and the implementation of a rule-based dependency parser
(Lesmo and Lombardo, 2002). The dependency arcs are labelled according to a dependency scheme that encodes the surface relations between words. The scheme is based
8 The word ‘valency’ is taken from chemistry: the valency number of an atom is the number of electrons
that it is able to take from (or give to) another atom in order to form a chemical compound.
9 But insights about dependencies can be found in the work of the Indian grammarian Panini, who
created a formal description of Sanskrit in the 5th (or 7th, according to some scholars) Century B.C.
(Bharati, Chaitanya, and Sangal, 1996).
15
on a twofold distinction between Functional and Non-functional dependents. The latter
are dependents not having domain-based semantic import, and are furtherly classified
into Aux (auxiliaries), Contin (continuations, in idioms), Coordinator (arcs related with
conjunctions), Separator (most punctuation marks), Visitors (e.g. in raising structures),
Interjections, and some particles void of semantic contents (as the Italian “accorger-si”
- remark –, where the “si” reflexive pronoun is lexicalized into a pseudo-reflexive verb).
The Function class is split into Arguments and Modifiers, corresponding to the standard
distinction between syntactic actants and syntactic circumstantials (see the discussion in
(Mel’cuk, 2004)). Finally, the modifiers can be Rmod (restrictive modifiers) or Appositions. Two simple examples are reported in fig.4. It is worth noting that determiners
(and quantifiers) are the head of nominal subtrees, i.e. they govern the associated noun.
This is in agreement with the solution proposed in Word Grammar (Hudson, 1990).
verb-subj
det-arg
verb-obj
verb-subj
det-arg
det-arg
verb-obj
ØÛÓ
det-arg
verb-rmod
prep-rmod
prep-arg
verb-subj
det-arg
Figure 4: Syntactic dependency trees associated with the sentences Every representative
of a company saw most samples and The boy who arrived eats two apples.
5.2
DTS syntax-semantics interface
The DG⇒DTS interface allows to associate a Dependency Tree with a fully-underspecified
well-formed structure in DTS. It is important to remind that this interface is compositional, i.e. a Dependency Tree is related to only one DTS well-formed structure.
The DG⇒DTS interface is a generalization of XDG. The main difference between the two
is that the former relates structures with different sets of nodes while, in the latter, the
various dimensions share the same set of nodes. This choice has been made to keep apart
the nodes in the DT (words like saw, every, of, etc.) from the nodes in the DTS graph
(predicates10 like see , of , ιxJohn(x), etc. or discourse referents like x, y, z, etc.).
The mapping from words in a DG to nodes in DTS is implemented by means of the
functions Sem and SV ar (Lex and LV ar implement the inverse mapping).
Sem relates verbs, common nouns, and other content words in DG to a predicate, while
SV ar relates determiners, proper names, and relative pronouns in DG to a discourse
10 The ι-operator has the standard semantics: if α is a constant, ιxα(x) is a unary predicate which is
true only for the individual denoted by α.
16
referent.
In this report, we consider a very simple Dependency Grammar for the sake of example. The domain objects of this grammar are words of English vocabulary classified into
seven Parts Of Speech (POS): IV (intransitive verbs), TV (transitive verbs), PN (proper
names), CN (common nouns), PREP (prepositions), DET (determiners), and RP (relative pronouns). The grammatical functions are subject (verb-subj), object (verb-obj),
argument of a preposition (prep-arg), argument of a determiner (det-arg), prepositional
modifier (prep-rmod), and verbal modifier (verb-rmod) needed for relative clauses.
It is now possible to state how to obtain a DTS well-formed structure from a DT in this
toy grammar. The mapping is defined by a set of if-then rules, called LinkDG,DT S , between “pieces” of structures. An example of such rules is presented in fig.5. In this report,
we adopt argument numbering for the semantic representation (1, 2, etc.). In principle,
the approach applies also in case the “semantic” arcs are labelled with thematic relations, in which case the governing verb must be suitably subcategorized. However, this
is outside the scope of the report.
ÁÎ
ÌÎ
Ú Ú
½ ¾
Ú
½
ÌË
½
½
ÌË
Figure 5: LinkDG,DT S rules for verb subjects and arguments of determiners
The two rules specify how the subjects of verbs and the arguments of determiners are interpreted: the rule on the left asserts that the subject of an intransitive (transitive) verb
corresponds to the first argument of the unary (binary) predicate associated with the verb
by the function Sem. Clearly, the arguments of these predicates are the discourse referents associated with the dependent of the verb by the function SV ar. Analogous rules
constrain the semantic realization of the direct object and the argument of a preposition:
ÌÎ
Ú Ú
¾
¾
½
Ú
ÌË
¾
ÌË
Figure 6: LinkDG,DT S rules for verb object and arguments of preposition
For dealing with adjuncts, it is necessary to introduce rules with an additional level of
complexity. The two optional dependency relations allowed in our toy grammar, i.e.
prepositional and verbal modifiers, link a common noun (associated with a predicate p1 )
to a preposition or a verb (associated with another predicate p2 ); clearly, p1 and p2 have
to be applied to the same discourse referent. For example, in “the man with the hat”,
assuming that “with” is associated with the predicate wear’, the argument of man’ (the
entity which is a man) and the first one of wear’ (the entity which wears the hat) refer to
the same individual. Thus, we introduce in the rules a new variable d, referring to this
17
common discourse referent; however, contrary to the variables v and u, in the rules there
is no constraint on d’s counterpart (i.e. on LV ar(d)). These rules are shown in fig.7.
½
½
Ú
¾
¾
Ú Ú
ÌË
ÁÎ
½ ¾
½
¾
ÌÎ
ÌË
Figure 7: LinkDG,DT S rules for prepositional and verbal (relative clauses) modifiers
The last ingredient needed to build the underspecified DTS representation is a criterion
to define restr(x), for each discourse referent x in the main graph11 . restr(x) is simply set
to the subgraph of all predicate-argument relations P (x1 , . . . , xn ) s.t.:
- P (x1 , . . . , xn ) arises from the subtree having the determiner associated with x as root.
- x is one of the x1 , . . . , xn .
This seems to be consistent with the ideas lying behind the architecture of a Dependency
Tree. In fact, as pointed out above, in a dependency relation between a word head and
a word dependent, the latter does not circumscribe/restrict only the head meaning, but
the whole subtree having the dependent as root.
We close this section with a final example, showing how the different rules interact in
the transduction process. Let us apply the rules to the DT of “Every representative of a
company saw most samples” shown in fig.4, on the left. Most steps of the transduction
are trivial, i.e. the application of the rules in fig.5 to the DT substructures at the top of
fig.8, which produce the DTS substructures at its bottom.
½
¾
½
½
Ü
Ý
Ý
Ü
½
½
Ý
Þ
Figure 8: Some LinkDG,DT S matches on the dependency tree of the sentence Every
representative of a company saw most samples, shown in fig.4.
One tricky point concerns the prepositional modifier. In this case, the applied rule is the
left one in fig. 7. The variable d appearing in the rule is unified with the discourse object
x. The application of the prep-rmod rule leads to the inclusion of the arc from of ’ to x,
as shown in fig.9.
11 The
value of quant depends only on the lexical meaning of the associated NL quantifier.
18
½
¾
Ü
Figure 9: LinkDG,DT S match for the adjunct relation on the dependency tree of the
sentence Every representative of a company saw most samples, shown in fig.4.
In other words in the case of prep-rmod (as for all modifiers), the semantic effect is
to add further arcs to already existing discourse referents. This sounds rather natural,
since the involved discourse referents had to be created because of their presence in a
connected structure (e.g. as actants of verbs) and the modifiers just add them some more
specifications. Of course, the latter are also represented as participants in other relations
(e.g. the of ’ relation, which could be expanded in “work for”).
The final result of the application of the DG⇒DTS syntax-semantic interface to the DT
in fig.4 on the left is reported in fig.10.
verb-subj
det-arg
verb-obj
1
Ý
det-arg
prep-arg
Ý
det-arg
1
2
1
2
Ü
1
Þ
Ý
1
1
1
prep-rmod
2
Ü
1
Þ
1
Figure 10: DG⇒DTS Interface application for the sentence Every representative of a
company saw most samples.
6
Monotonic semantic disambiguation in DTS
As pointed out in section 3, in order to perform monotonic disambiguation, the underspecified logic must enable partial scope orderings. Whenever such orderings are dominance relations between some kind of scope-bearers and some kind of scope-arguments,
as in Hole Semantics, or dependencies between involved groups of entities, as in DTS,
incremental disambiguations may be achieved rather easily. However, the SemDep arcs
introduced above are not enough for carrying out an incremental disambiguation. To see
why, consider again sentence (17), repeated below as (19):
19
(19) Everyxbody believed that many ay problem about the environment preoccupied
mostz politicians.
In DTS, in order to represent that both ManyAy and Mostz have narrow scope with
respect to Everyx , it seems that SemDep arcs should be instantiated as in fig.11.a.
Nevertheless, fig.11.a already describes a final ordering for (17): it specifies that ManyAy
and Mostz must be independent of one another.
Ü
Ý
Þ
Ü
Ý
Þ
Ü
Ý
Þ
Figure 11: Three readings for sentence (17) in DTS.
In other words, in DTS each of the three representations in fig.11 is a possible final
configurations having ManyAy and Mostz with narrow scope with respect to Everyx . In
fact, the dependencies represented via SemDep arcs are assumed to be transitive, but
“minimal” (see section 3). In the case of the graph in fig.11.a, this means that neither y
depends on z, nor z depends on y.
On the other hand, during incremental disambiguation, the two dependencies cannot
be excluded (i.e. negated), but only left unknown. We need a way to underspecify the
three readings, but clearly we cannot use one of them to this end.
Therefore, it is necessary to distinguish between partial and fully specified representations. In order to obtain this result, we introduce another type of arc, which we call
SkDep, for which the minimality assumption is not enforced. With this distinction, we
can now use the SkDep in fig.12 to underspecify the three SemDep configurations in
fig.11 (SkDep’s are graphically shown by means of thinner dotted arcs).
Ü
Ý
Þ
Figure 12: A partially specified reading for sentence (17)
Now, the interpretation process proceeds as follows: first, we initalize the graph by linking each discourse referent to Ctx and then we add SkDep arcs in the representation
according to the constraints coming from the context; each new SkDep arc provides a
new piece of information, that further constrains the scope of the quantifiers and the set
of possible models satisfying the graph. When the disambiguation is complete, the set
of SkDep arcs is “frozen” and converted into a set of SemDep arcs. Now, the modeltheoretic interpretation of the graph can be given, according to the semantics provided
below in section 7. The freezing process consists in copying all SkDep arcs as SemDep
arcs, except the ones that are deducible on the basis of transitivity. For instance, after
20
the SkDep arcs in fig.12 have been introduced, what remains to be done is to decide if
z is functionally dependent on y or viceversa, or they are independent of each other (IS
reading). If the context provides no answer, then no freezing is possible (unless one of
the three possibilities is chosen at random). If z does depend on y, then a SkDep arc is
added, leaving z and entering y. The one from z to x becomes transitive, and so it is no
longer drawn. The case where y depends on x is exactly symmetric, while, if the context
specifies that y and z are independent of each other, no new SkDep arc is added and the
structure in fig.12 is frozen as such, yielding fig.11.a.
Before proceeding, it must be pointed out that SemDep and SkDep arcs are clearly
just a graphical way to characterize binary relations among discourse referents and Ctx.
This implies that the actual relations, as they are described below in section 7, include
more pairs than the ones appearing in the figures. In particular, both the the SemDep
and the SkDep relations are transitive, while the set of arcs do not include the arcs that
may be inferred because of transitivity. However, what keeps apart SemDep and SkDep
arcs is that the former is minimal, in the sense explained above, while the latter is not
so. This minimality is a feature of the drawings, introduced to underline what appear to
be the most relevant dependencies, but has no counterpart in the formal definitions.
The availability of an underspecified representation based on the predicate-argument
structure makes it particularly easy to identify the questions that must be asked to the
semantic KB (or “the context”) to choose the correct dependencies. For most other underspecification formalisms it is somewhat harder to find out the relations to verify. This
is due to the fact that predicates and variables are scattered around in the representation. For instance, in Hole Semantics, the interaction between labels and holes makes it
difficult to determine the local effects (and, consequently, the decision criteria) of a given
choice. In particular, in Hole Semantics, each quantifier is associated with two holes, one
for the body and one for the restriction, so that different choices on the dominance relations lead to different restrictions. Conversely, in DTS, the predicate-argument relations
constituting the quantifier restrictions, i.e. the subgraphs as values of the restr function,
that are needed to determine which discourse referents depend on which ones, are fixed
regardless of the chosen interpretation. Therefore, the exploitation of world knowledge
seems in principle harder in Hole Semantics (cf. (Robaldo and Di Carlo, 2009), (Robaldo
and Di Carlo, 2010)), since it requires additional reasoning steps to identify the predicate
argument relations involved in the checks or to determine if a label has to be dominated
by the hole associated with the restriction rather than the body of a certain quantifier.
6.1
Constraints on admissible readings
As explained above, SkDep/SemDep arcs have to define a partial order between discourse
referents and Ctx, with Ctx as its maximal element. However, not all partial orders can
be allowed. For example, the dependencies depicted in (20) describe an unreasonable
interpretation of the associated sentence. In fact, it is awkward to take this sentence
as describing a situation in which each representative (of the universal, contextually
determined, set) saw most samples and such that there is a different company to which
s/he belongs for each sample s/he saw. This kind of constraints usually arises in sentences
featuring syntactically nested quantification, i.e. a quantifier occurring in the restriction
of another quantifier, as in (20), in which “a” occurs in the restriction of “every”.
21
(20) [Everyx representative of [ay company]] saw [mostz samples]
1
Ü
Ý
Þ
1
1
2
Ü
2
1
Þ
1
1
1
1
Ý
1
2
Ý
These readings have been avoided in current approaches to underspecification by introducing additional constraints in the basic framework. For example, (Hobbs and Shieber,
1987) requires, at each disambiguation step, to choose a complex term among those not
included in any other complex term. On the other hand, (Larson, 1985), (Park, 1996),
(Willis, 2000), (Joshi, Kallmeyer, and Romero, 2003) claim that if a NL quantifier occurs
in the syntactic restriction of another NL quantifier, no other quantifier can ‘intercalate’
between their corresponding GQs in the scope order. Accordingly, their logical framework forbid both the linear order ∀x M ostz ∃y , corresponding to the structure in (20), and
∃y M ostz ∀x , which is instead accepted in Hobbs and Shieber’s (see (13.b) above).
In DTS, similar constraints are defined. Here, the availability of the restr function puts at
disposal all that is needed to express the constraints. In fact, though restr is part of the
logical formalism, its values are determined by the syntax-semantics interface by taking
into account the syntactic structure of the sentence. So, the fact that a node d1 occurs
in restr(d2 ) mirrors the fact that the word that in the sentence introduces d1 , governs the
word that in the sentence introduces d2 .
Of course, the constraints that are introduced below extend the more basic constraints on
the well-formedness of the graph (as the absence of cycles). The constraints are given in
terms of SkDep arcs, since SemDep is just a “freezing” of SkDep, in the sense explained
above, so that all constraints on SkDep transfer directly to their SemDep counterparts.
The set of constraints is the following:
Definition 1[Constraints on SkDep]
Let d be a discourse referent, and let R(d) i.e. the smallest set such that12 :
- d ∈ R(d)
- if d ∈ R(d) and d ∈ restr(d ) then d ∈ R(d)
Then, it must hold that:
- If d1 ∈ R(d), d2 ∈ R(d), and d1 depends13 on d2 , then also d depends on d2 .
- If d1 ∈ R(d), d2 ∈ R(d), and d2 depends on d1 , then also d depends on d1 .
22
¾
¾
¾
½
½
¾
½
½
Figure 13: Allowed dependencies in DTS
Fig.13 shows a graphical representation of the constraints.
The first constraint specifies that if a discourse referent d1 , which contributes to the
identification of d, semantically depends on d2 , then also d depends on d2 . In other
words, the discourse referent d with which R is associated “goes together” with the
referents that contribute to its description in depending on other referents (d2 ). But
d may depend on d2 in two ways: directly (as in fig.13.a) or indirectly via d1 (as in
fig.13.b). In the first case we obtain an IS reading, whereas in the second case, we obtain
the so-called Inverse Linking reading (May and Bale, 2005).
Analogously, the second constraint specifies that if d2 semantically depends on d1 , then
also d depends on d1 . Again, this can be achieved in two ways (triggering an IS reading or
an Inverse Linking reading respectively): directly (fig.13.c) or indirectly via d2 (fig.13.d).
An example of Inverse Linking (enabled by the pattern in fig.13.b) is reported in (21.a).
(21) [Twox students] must write [any essay about [threez poets]]
Ü
Ý
Ü
Þ
Ý
Ü
Þ
Þ
Ü
Ý
Ü
Þ
Ý
Þ
12 R(d)
13 I.e.
Ý
is a kind of transitive closure of the restriction of a node.
The structure includes a SkDep arc from d1 to d2 .
23
(21.a) describes a reading of (21) in which there are two particular boys, e.g. John and
Jack, each of which must write about a possibly different set of three poets, and for each
poet (transitively, for each student) there may be a different essay (so, we have up to
six poets and six essays). (21.b) features both an Inverse Linking and an Independence
between the sets of entities denoted by x and by z; in (21.b), the set of poets is the same
for both boys, and the essays depend both on boys and on poets (three poets and up to
six essays). (21.c) is like (21.b), but instead of an Inverse Linking, it features a semantic
dependency of the essays on the students; in this case, each of the two students must
write a single essay talking about three poets.
What the two constraints exclude is a set of dependencies as the one in (21.d) and (21.e),
i.e. we assume that the sentence cannot be used to describe a situation where the poets
can be different for different boys, but there is a single essay involved, nor a situation
where there is just an essay, three poets, and up to six students.
Finally, it seems worth spending some more words about the problem of logical redundancy, a problem not new in Underspecification (see (Vestre, 1991), (Chaves, 2005),
and (Koller and Thater, 2006)). Redundancy may arise when universal and existential
quantifiers occur in the sentence. Universals, since they range on the whole domain of
individuals, cannot exhibit any dependency on another quantifier, i.e. their inclusion
within the scope of another quantifier does not affect the selection of individuals they
range on, so that the SkDep arc exiting them can be assumed to be fixed and to enter
Ctx. For analogous reasons, no arc can enter an existential quantifier. Anyway, there
is an exception to the rule about universals. It concerns an universal having a modifier
(which includes another quantifier) in its restriction. For example, in the case of (22),
there are two possibilities, according to the intention of referring to ‘all friends of any
participant’ or to ‘all friends of all participants’. In the first case, we enable the upper
universal to be linked via a SkDep arc to its modifier as in (22.a). Clearly, the standard
link to Ctx is possible, so also the second case is accounted for (22.b).
(22) Every friend of every participant arrives.
Ü
Ü
Ý
Ý
There are other cases where a universal quantifiers can be ‘restricted’ in this way. An
example could be ‘Two lecturers failed every student’. However, this sentence could
involve some kind of ellipsis (i.e ‘every student of him’). In this case, it could be up to
the syntax to hypothesize this type of ‘overt’ change: with the inclusion of this modifier,
this and similar examples are covered by the redundancy rule introduced above, but it is
clear that this requires further investigation.
With respect to the sentence (12), the constraints introduced in this section allows for
the six disambiguations shown in fig.14. Fig.14 also shows the correspondence with the
five formulae generated by the Hobbs and Shieber’s algorithm, reported above in (13).
24
Ü
Þ
Ü
Þ
Ý
Ý
Ü
Ü
Þ
Ý
Þ
Ü
Þ
Ý
Ý
Ü
Þ
Ý
Figure 14: The six DTS disambiguations for the sentence Everyx representative of ay
company saw mostz samples. The correspondence with the five disambiguations in (13)
is indicated under the figures.
The additional reading, i.e. the third one from the left in fig.14, involves a fixed set of most
samples such that every representative of a (potentially different) company saw each of
them. It is not possible to represent such a reading via quantifier-embedding. Therefore
neither Hobbs and Shieber’s algorithm nor Hole Semantics and the other constraint-based
underspecified formalisms seen above can engender it. Nevertheless, even if we do believe
that this interpretation is actually available, it is definitely not the preferred one for the
sentence under examination. Consider instead the following alternative context: suppose
there is a university that organizes a conference series. Its dean, in order to make the
students of the final year interested in research, decides to allow them to attend most
conferences free of charge. In such a context, among the preferred readings of the sentence:
(23) Everyx student of ay graduating class will attend mostz conferences free of charge.
there is obviously the one where the set of conferences that may be attended free of charge
is chosen in advance by the dean, so that it does not vary from student to student. It is
instead likely that the students of the final year can freely access all but some conferences,
e.g. the ones with very important invited speakers and a limited number of seats, to which
no discount applies to anyone. On the other hand, the graduating class does clearly vary
from student to student.
Finally note that another exception to the constraints on logical redundancy can
be obtained also because of the existence of the constraints on Nested Quantification.
For instance, the first constraint on logical redundancy imposes every discourse referent
associated with a universal quantifier, as x in (23), to enter14 Ctx. Accordingly, in fig.14,
the first and the fifth disambiguations from the left should be blocked.
But since y may depend on z (the company varies depending on the sample under
consideration), also x must, in those cases, depend on z (i.e. all representatives of that
company saw that sample). There could be a single company for each sample (as in
(13.a), where the company does not depend on the representative), or more than one (as
in (13.d)), if they can vary both with the sample and with the representative.
14 Or to depend on another discourse referent in its restriction; but in (23) the only available one is y,
which is associated to an existential quantifier and so no discourse referent may depend on it, according
to the second constraint against logical redundancy.
25
7
Syntax of DTS
This section defines DTS well-formed structures. A semantics of such structures is then
given in the next section, in terms of an algorithm for translating DTS disambiguated
graphs into (Robaldo, 2010)’s formulae, presented above in section 2.
In DTS, there are two kinds of well-formed structures. The first one is called Scoped
Dependency Graph (SDG), which is a non-ambiguous representation corresponding to
one reading of a sentence, while the second one is the Underspecified Scoped Dependency
Graph (USDG), which is the underspecified version of the SDG. The difference between
the two structures however, is simply that the latter includes SkDep arcs only, while the
former SemDep arcs only (cf. section 6 above). An USDG can be converted into one of
the SDGs it refers to by means of a “freezing” operation, which simply substitutes all
SkDep arcs with corresponding SemDep arcs.
Both structures are defined in terms of a third structure, the Flat Dependency Graph
(FDG), that simply contains the predicate-argument relations occurring in the sentence.
As in standard semantics, the definitions below are stated in terms of three sets: a set of
predicates pred, a set of constants name and a set D of individual variables15 . Moreover,
we write P 1 ⊆pred for the unary predicates, P 2 ⊆pred for the binary ones and ιname ⊆P 1
for those obtained by applying the ι-operator to a constant in name
Definition 2 [Flat Dependency Graphs (FDG)]
A Flat Dependency Graph is a tuple N , L, A, Dom, f s.t.:
- N is a set of nodes {n1 , n2 , . . . , nk }.
- L is a set of labels {l1 , l2 , . . ., lm }16 .
- Dom ≡ pred ∪ D is a set of domain objects: predicates and discourse referents.
- f is a function f : N → Dom, specifying the node referent, i.e. the domain object
with which the node is associated.
- A is a set of arcs. An arc is a triple ns , nd , l, where ns , nd ∈ N , ns is of type
pred, nd is of type D and l ∈ L.
In the following, if f (n) ∈ pred, we say that node n is of type pred, otherwise we say
it is of type D. Moreover, without going into further details, we stipulate that a FDG
Gf is a connected acyclic graph such that each node of type pred has one exiting arc for
each of its argument places. Note that there can be two different nodes nu and nv s.t.
f (nu )=f (nv ), i.e. the nodes in N can be seen as occurrences of symbols from Dom.
15 Those variables refers to entities taken from the discourse context. Therefore, we call them discourse
referents as in DRT (Kamp and Reyle, 1993).
16 In all figures above, L≡{1, 2}.
26
Definition 3[Underspecified Scoped Dependency Graph (USDG)]
An Underspecified Scoped Dependency Graph is a tuple Gf , Ctx, quant, restr, SkDep such
that:
- Gf = N, L, A, Dom, f is an FDG.
- Ctx is a special element called “the context”.
- quant is a total function ND → Q, where ND ⊆ N are the nodes of type D and Q
is a set of 2-place Mostowskian quantifiers (F orAll, M ost, T wo, . . .).
- restr is a total function assigning to each d ∈ ND its restriction, which is a subgraph
of Gf .
- SkDep is a total transitive function ND →℘(ND ∪{Ctx})\℘(ND ). SkDep satisfies
the constraints formally defined in def.1
Definition 4 [Scoped Dependency Graph (SDG)]
A SDG Gs is a tuple Gf , Ctx, quant, restr, SemDep such that:
- Gf , Ctx, quant and restr are defined as in def. 3
- SemDep is a total transitive function ND →℘(ND ∪{Ctx})\℘(ND ). SemDep satisfies
the constraints formally defined in def.1
As pointed out above, defs.3-4 are almost identical, the only difference being that in the
former there appears the SkDep function, while in the latter there is SemDep.
The next definition concerns the monotonic semantic disambiguation as described above.
It allows to “transform” an USDG into a corresponding SDG.
Definition 5[Freezing]
A SDG G = SGf , Ctx1 , quant1 , restr1 , SemDep is a freezing of the USDG G = U SGf ,
Ctx2 , quant2 , restr2 , SkDep iff:
- SGf =U SGf , Ctx1 =Ctx2 , quant1 =quant2 , restr1 =restr2
- ∀(n ∈ ND )SemDep(n) = SkDep(n).
So, a graph G is a freezing of a graph G when the SemDep function of the former
is identical to the SkDep function of the latter. Again, the two graphs are trivially
isomorphic, but they express very different constraints on the dependencies.
27
8
Translating Scoped Dependency Graphs into
(Robaldo, 2010)’s formulae.
In this section, we outline an algorithm that translates Scoped Dependency Graphs into
(Robaldo, 2010)’s Second Order Logic (SOL) formulae. This algorithm establishes an
indirect model-theoretic interpretation of DTS disambiguated structures.
Although DTS structures and SOL formulae are different types of objects, we adopt in
the following a terminology that enables us to simplify the description. As we have seen,
the domain Dom is partitioned into two subsets: pred and D. The first of them includes
predicates that correspond exactly to the predicates of a standard logic (as SOL), while
the second includes discourse referents that have the same role that variables have in SOL.
Although discourse referents and variables are different types of objects, we occasionally
say that a variable is equal to or is the same as a given discourse referent. This is intended
to mean that we use the same name for them, in order to keep explicit the correspondence
between the DTS structure and the corresponding SOL formula. In the same vein, we will
use “the restriction” and “the body” of a discourse referent to refer to the restriction and
the body of the Generalized Quantifier that, in the resulting SOL formula, is associated
with the variable corresponding to that discourse referent.
Basically, our object formulae are conjunctions of two kinds of conditions: Maximal
Inclusion conditions (mIc) and Maximal/non-Maximal Quantifier conditions (Qc).
For instance, with respect to example (10), repeated in (24), we have that each mIc’s or
Qc’s correspond to the pair of conjuncts (the second of which is an application of the Max
operator) appearing in the same line. In case no implication appears in the corresponding
condition, the Max operation does not occur. This may happen only for Qc. Therefore,
in (24), clause Cx is a non-Maximal Quantifier condition, Cy a Maximal one, and Rx ,
Ry , and IN , three Maximal Inclusion conditions.
(24) Eachx boy ate twoy apples. = (10)
[ Cx : ∀x (PxR (x), PxB (x)) ∧
Rx : {Cover(CovPxR , PxR ) ∧∀x [CovPxR (x)→boy (x)]} ∧ M ax(PxR , Rx ) ∧
Cy : {Cover(CovPxB , PxB ) ∧∀x [CovPxB (x)→2!y (PyR (x, y), PyB (x, y))]} ∧
M ax(PxB , Cy ) ∧
Ry : {Cover(CovPyR , PyR ) ∧∀xy [CovPyR (x, y)→apple (x, y)]} ∧ M ax(PyR , Ry ) ∧
IN : {Cover(CovPyB , PyB ) ∧∀xy [CovPyB (x, y)→ate (x, y)]} ∧ M ax(PyB , IN ) ]
In spite of their different role, both types of conditions are in the form
(25) C: {Cover(CovPd1... Pdn , Pd1 , . . . , Pdn ) ∧ ∀x1 ...xi [CovPd1... Pdn (x1 , . . . , xi ) → Φ ]} ∧
M ax(Pd1 , . . . , Pdn , C)
The general formalization of the meta-predicates Cover and M ax are given in (Robaldo,
2010). In the following, we do not specify them for the sake of compactness.
The first line of (25) is the Inclusion or Quantifier Condition, while the second one is
the associated Maximality condition. As stated above, the Maximality condition can be
28
absent for some/all of the Quantifier Conditions, in case no implication appears in the
first conjunct. In this case, the whole condition has the simple form Φ.
Pd1 . . . Pdn are relational variables associated with the discourse referents d1 . . . dn .
{x1 . . . xi } is the set of all the variables appearing as arguments of one or more of those
predicates. A Cover is then a set of tuples X1 . . . Xi where X1 , . . . Xi may be plural
individuals. The implication in (25) asserts that the subformula Φ holds for every tuple
in the pragmatically relevant Cover of Pd1 , . . ., Pdn .
In case of mIc, Φ is a conjunction of FOL predicates (each of which is applied to some of
the variables in x1 . . . xi ), whereas in case of Qc, Φ is a conjunction of 2-place Mostowskian
Generalized Quantifiers each of which has the form:
(26)
quant(ndk )(PdR (yk1 , . . . , ykjk , dk ), PdB (yk1 , . . . , ykjk , dk ))
k
k
Where the discourse referents yki are the ones on which dk depends.
For instance, in (24), the expression:
Cy : {Cover(CovPxB , PxB ) ∧∀x [CovPxB (x)→2!y (PyR (x, y), PyB (x, y))]} ∧
M ax(PxB , Cy )
is built to force that for all members of the extension of the discourse referent x in
the DTS graph there are exactly two members in the extension of y that are apples
(PyR ) and are eaten by x (PyB ). Of course, only a single value for the Cover CovPxB
seems to be acceptable: the set of all individual boys, i.e. CovPxB M,g ≡ PxB M,g .
M ax(PxB , Cy ) forces the extension of x to be maximal with respect to dependency
from y to x: PxB M,g must be the set of all boys who ate each exactly two apples.
In general, for building any type of condition, two pieces of information are needed.
The first piece of information concerns the variables to which a given predicate must
be applied. These predicates are either the ones associated with items in the sentence
(e.g. apple , ate , . . . ) or the ones associated with the restriction and the body of the
quantifiers resulting from the translation of discourse referents (e.g PxB , PzR ). In the first
case, the involved variables are determined on the basis of the flat dependency graph
(apple applies to y if y is an apple, while ate applies to x and y if x ate y). In the
second case, the involved variables are the discourse referent itself plus the ones on which
it depends.
The second piece of information concerns the antecedents of the implications occurring
in the first conjunct (e.g. in the first line of (25)).
Since the determination of the correct expression is a subtle matter, it will be deferred
to the presentation of the algorithm for building mIc’s and Qc’s.
8.1
Building maximal Inclusion Conditions
Maximal Inclusion conditions are built by a function buildM IC that takes as input a
Scoped Dependency Graph and returns the conjunction of all Maximal Inclusion conditions associated with it. Before presenting buildM IC, it is convenient to introduce the
following notations:
29
G
- If Gf = N, L, A, Dom, f is a Flat Dependency Graph, NP f ⊆ N is the set of
G
Predicative Nodes, and ND f ⊆ N is the set of Discourse Referent Nodes. It holds
Gf
Gf
G
G
that NP ∪ ND = N and NP f ∩ ND f = Ø.
- If G is any subgraph of a Flat Dependency Graph, such that for each predicative
node in G, G also includes all of its arguments, G∗ is the SOL formula
∧
f (npi )(f (ndi1 ), f (ndi2 ), ..., f (ndiki ))
G
npi ∈ NP
where ndi1 , . . ., ndiki are all the nodes in Gf linked to npi in A. For instance,
for the G that coincides with the Flat Dependency Graph in fig.1, we have that
G∗ = man (x) ∧ mSound (y) ∧ hear (x, y).
- If Gs = Gf , Ctx, quant, restr, SemDep is a Scoped Dependency Graph, body(Gs )
is the subgraph of Gf that contains all and only the predicate-argument relations
G
that do not belong to restr(nd ) for any node nd ∈ ND f . In all examples presented
in this paper, this subgraph is made up just by the predicate denoted by the main
verb, with its arguments.
In order to make clearer the presentation of the algorithm, we interleave its description
with its application to a specific example. This example refers to the analysis of the
complex sentence discussed in (Robaldo, 2010):
(27) Mostx biologists of twoy labs analyzed thez results of everyw experimental therapy
lasting more than fivek months.
The DTS fully-underspecified representation of sentence (27) appears in fig.15.
Ü
1
Ü
1
1
2
1
1
2
2
1
Û
Ý
2
Ý
1
1
2
1
1
1
Þ
Û
Þ
1
2
2
1
1
Û
Ý
1
1
1
Figure 15: DTS representation for the sentence Most biologists of two labs analyzed the
results of every experimental therapy lasting more than five months.
30
The interpretation of sentence (27) that we discuss here, and which is presumably its
preferred reading, features the dependencies described by fig.16. In this reading there
are two particular labs such that the majority of biologists working in either lab analyzed
the result of each experimental therapy that lasted more than five months. Of course,
each therapy has a different set of results (Inverse Linking), and a different duration, i.e.
a different set of associated (more than five) months17 .
Ü
Û
Ý
Þ
Figure 16: SemDep arcs corresponding to a suitable reading of sentence (27).
G
The overall organization of the algorithm is: first introduce 2 ∗ |ND f | relational variables, then apply the two steps of building mIc and Qc on the set of variables involved
in the conditions. According to the discussion above, the set of relational variables is
partitioned into two sets: the first set refers to the restrictions of the quantified variables,
while the second one refers to the bodies.
G
The construction of the mIc is obtained via |ND f | + 1 applications of the function
buildOneM C(P, Φ) that takes as input a set P of some of the relational variables and a
formula Φ, which can be either body ∗ (Gs ) or restr∗ (nj ).
buildOneM C(P, Φ) introduces a Cover variable, and requires its extension to both cover
the variables in P and to satisfy the formula Φ. Finally, it adds a Maximality condition
that requires the variables in P to be a Maximal tuple of variables satisfying the inclusion
condition so built.
The set P is determined in the following way. If Φ = body ∗ (Gs ), then P includes all
those PkB such that nk is a node of body(Gs ) (i.e. an argument of the main predicate) on
which no other node in body(Gs ) depends. Also if Φ = restr∗ (nj ), then P includes a Pk
for any node nk in restr(nj ) on which no other node depends. But for the main node of
the restriction (nj , i.e. the one that is being restricted) it will be PdRj ; for all other nodes
it will be PdBk as in the previous case. The function buildM IC is defined as following:
17 Note that these months are presumably continuous. However, this is actually an Implicature that is
mirrored in the final formula by pragmatically interpreting the relational variables via an assignment g.
31
Definition 6 (buildM IC) 18
buildM IC(Gs ), where Gs = Gf , Ctx, quant, restr, SemDep is a SDG, returns a conjunction of Maximal Inclusion Conditions:
buildM IC(Gs ) =
- set M axInclConds=true
- set sGraphs = {Ctx, body(Gs ), nd1 , restr(nd1 ), . . . , ndn , restr(ndn )},
G
where {nd1 , . . . , ndn } ≡ ND f
(Note that sGraphs includes all the subgraphs for which a mIc must be defined)
- for each nd , sG ∈ sGraphs do
sG
- set nT oM ax = ND
\(
∪
sG
n∈ND
SemDep(n) )
(Note that nT oM ax are all discourse referent nodes in sG on which no other
node in sG depends)
- if nd ∈ nT oM ax then set P = {PfR(nd ) } else set P = {}
- for each n∈(nT oM ax \ nd ) do set P = P ∪ {PfB(n) }
- set InclConds = InclConds ∧ buildOneM C(P, sG*)
- return InclConds
With respect to the SDG in figg.15-16 , the buildM IC function returns the following
conjunction of Maximal Inclusion Conditions:
Ry : {Cover(CovPyR , PyR ) ∧∀y [CovPyR (y)→lab (y)]} ∧ M ax(PyR , Ry ) ∧
Rx : {Cover(CovPxR PyB , PxR, PyB ) ∧∀xy [CovPxR PyB (x, y)→(biologist (x) ∧ of (x, y))]} ∧
M ax(PxR , PyB , Rx ) ∧
Rk : {Cover(CovPkR , PkR ) ∧∀wk [CovPkR (w, k)→month (k)]} ∧ M ax(PkR , Rk ) ∧
Rw : {Cover(CovPkB , PkB ) ∧∀wk [CovPkB (w, k)→(expAnalysis (w) ∧ last (w, k))]} ∧
M ax(PkB , Rw ) ∧
Rz : {Cover(CovPzR , PzR ) ∧∀wz [CovPzR (w, z)→(result (z) ∧ of (z, w))]} ∧
M ax(PzR , Rz ) ∧
IN : {Cover(CovPxB PzB , PxB , PzB ) ∧∀xwz [CovPxB PzB (x, w, z)→analyze (x, z)]} ∧
M ax(PxB , PzB , IN )
18 In
this and in the following algorithms, we included some comments enclosed in parentheses.
32
8.2
Building Quantifier conditions
As said above, a formula in (Robaldo, 2010) includes both Maximal and non-Maximal
Quantifier Conditions. Non-Maximal Quantifier Conditions are in the form (26). One
such condition is built for each node having the SemDep’s value equal to {Ctx}.
Conversely, Maximal Quantifier Conditions are asserted between sets of discourse referents whose nodes are related via the relation of ‘immediate dependence’ (cf. (Sher,
1997)). To this end, the set of nodes of type D is partitioned into clusters. A cluster is
a class of equivalence under the relation of common ‘immediate’ dependence. In order
to obtain the correct clusters, a function called buildClusters is invoked. This function
takes as input a Scoped Dependency Graph GS and returns the set of all pairs C1 , C2 ,
where C1 and C2 are two (non-empty) sets such that the nodes in C2 immediately depend
on those in C1 . We omit details of buildClusters(GS ).
For the SDG in figg.15-16 there are only two clusters; therefore, in this case19 , the function buildClusters(SemDep) returns the following singleton:
{ {x, y, w}, {z, k} }
From each pair of clusters C1 , C2 , some quantifier conditions are built. Each discourse
referent in C2 depends on some discourse referents in C1 . However, not all discourse
referents in C1 necessarily have, among their dependents, a discourse referent in C2 . For
instance, in the example under examination, x and y do not have any dependent among
the discourse referents in C2 = {z, k}.
If a discourse referent d∈C2 belongs to restr(d ), such that d ∈C1 , or to SemDep(d ),
such that d ∈C1 and d ∈restr(d ), then the quantifier condition associated with d must
be asserted on PdR . Otherwise, it must be asserted on PdB .
This is achieved by the function buildQC, formally defined as follows:
19 To give another example, consider (21.a). In this case, there are three clusters, each containing a
single discourse referent. The function buildClusters returns the set of pairs {{x}, {z}, {z}, {y}}.
33
Definition 7 (buildQC)
buildQC(Gs ), where Gs = Gf , Ctx, quant, restr, SemDep is a Scoped Dependency
Graph, returns a conjunction of quantifier conditions:
buildQC(Gs ) =
- set N G = {d such that SemDep(nd ) = {Ctx}}
(Note that this instruction inserts in NG all the nodes having wide scope)
∧
- set quantConds =n
quant(nd )(Pf (nd )R (f (nd )), Pf (nd )B (f (nd )))
d ∈ NG
- set Clusters = buildClusters(GS )
- for each C1 , C2 ∈ Clusters do
- set ExtRestr = Ø
- for each nd ∈ C1 do
- set ExtRestr = ExtRestr ∪ (nd , C2 ∩(ND (restr(nd ))∪
∪
SemDep(n)))
n∈ND (restr(nd ))
(Note that ExtRestr associates with each node in C1 the C2 nodes that
either are in the restriction of nd or are among the ones on which a node
in the restriction depends)
- set AllRestrs =
∪
nRestrd
nd , nRestrd ∈ExtRestr
(This instruction adds to ExtRestr a dummy pair, that includes all C2 nodes
not appearing in the Extended Restriction of some other node)
- set ExtRestr= ExtRestr ∪ {dummy, C2 \ Allrestrs}
(We use the extended restrictions to build the quantifier conditions)
- for each nd , nRestrd ∈ ExtRestr do
- if nRestrd = Ø do
- set allDeps = (
∪
immGov(n, SemDep))\{nd }
n∈nRestrd
- if nd = dummy then set P = {} else set P = {PfR(nd ) }
- for each m ∈ allDeps do P = P ∪ {PfB(m) }
- set Φ = true
- for each n ∈ nRestrd do
set Φ =Φ ∧ quant(n)(PfR(n)(y1, . . . , yk , f (n)), PfB(n) (y1, . . . , yk , f (n)))
where {y1 , . . . , yk } ≡ SemDep(n).
- set quantConds = quantConds ∧ buildOneM C(P, Φ)
- return quantConds
34
In the first two instruction of buildQC, the algorithm builds the non-Maximal Quantifier
Conditions, i.e. the ones associated with the discourse referents that enters Ctx only.
After their execution, quantConds holds the following value:
Cx : M ostx (PxR (x), PxB (x)) ∧ Cy : 2!y (PyR (y), PyB (y)) ∧ Cw : ∀w (PwR (w), PwB (w))
Then, for each pair of clusters C1 , C2 , buildQC builds the extended restriction, which is
a set of pairs of sets of discourse referents needed to find out if some discourse referents
in C2 contribute to the restriction of some discourse referents in C1 . If there is any,
ExtRestr contains a pair that describes the correspondence. The discourse referents in
C2 that do not contribute to the restriction of any discourse referent in C1 are inserted
in ExtRestr as well, but they are paired up with a dummy node called dummy. Thus,
at the beginning of the last “for” cycle of buildQC, ExtRestr holds the following value:
ExtRestr = { {x}, Ø, {y}, Ø, {w}, {k}, dummy, {z} }
In the final “for” cycle, the first two pairs of ExtRestr are ignored. The pair {w}, {k}
leads to P = {PwR } while dummy, {z} to P = {PwB }. Thus, the two following Maximal
Quantifier Conditions are added to the three previous non-Maximal ones:
Ck : {Cover(CovPwR , PwR ) ∧∀w [CovPwR (w)→ M T 5k (PkR (w, k), PkB (w, k))]} ∧
M ax(PwR , Ck ) ∧
Cz : {Cover(CovPwB , PwB ) ∧∀w [CovPwB (w)→ T hez (PzR (w, z), PzB (w, z))]} ∧
M ax(PwB , Cz )
To summarize, the translation of an SDG is obtained in the following way:
Definition 8 (DTS to SOL)
A Scoped Dependency Graph Gs = Gf , Ctx, quant, restr, SemDep) is translated in the
Second Order Logic formula:
[ buildM IC(Gs ) ∧ buildQC(Gs ) ]
where {f (d1 ) . . . f (dn )} ≡ D.
As explained above, the formula obtained via this translation needs to be interpreted
with respect to a model M and an assignment g, which assigns a value, determined on
pragmatic grounds, to every relational variable occurring in the formula.
Therefore, the SOL formula corresponding to the SDG in figg.15-16 is simply given by
the conjunction of all Inclusion and Quantifier Conditions built by the two algorithms
presented above. The whole formula is finally reported in (28):
35
(28)
[ Cx : M ostx (PxR (x), PxB (x)) ∧ Cy : 2!y (PyR (y), PyB (y)) ∧ Cw : ∀w (PwR (w), PwB (w)) ∧
Ry : {Cover(CovPyR , PyR ) ∧∀y [CovPyR (y)→lab (y)]} ∧ M ax(PyR , Ry )∧
Rx : {Cover(CovPxR PyB , PxR, PyB ) ∧∀xy [CovPxR PyB (x, y)→(biol (x) ∧ workIn (x, y))]}∧
M ax(PxR , PyB , Rx ) ∧
Ck : {Cover(CovPwR , PwR ) ∧∀w [CovPwR (w)→ M T 5k (PkR (w, k), PkB (w, k))]} ∧
M ax(PwR , Ck ) ∧
Rk : {Cover(CovPkR , PkR ) ∧∀wk [CovPkR (w, k)→month (k)]} ∧ M ax(PkR , Rk ) ∧
Rw : {Cover(CovPkB , PkB ) ∧∀wk [CovPkB (w, k)→(expAnalysis (w) ∧ last (w, k))]} ∧
M ax(PkB , Rw ) ∧
Cz : {Cover(CovPwB , PwB ) ∧∀w [CovPwB (w)→ T hez (PzR (w, z), PzB (w, z))]} ∧
M ax(PwB , Cz ) ∧
Rz : {Cover(CovPzR , PzR ) ∧∀wz [CovPzR (w, z)→resultOf (z, w)]} ∧ M ax(PzR , Rz ) ∧
IN : {Cover(CovPxB PzB , PxB , PzB ) ∧∀xwz [CovPxB PzB (x, w, z)→analyze(x, z)]} ∧
M ax(PxB , PzB , IN ) ]
9
Conclusions
In this report, we present a new semantic underspecified formalism called Dependency
Tree Semantics (DTS). DTS acts as a bridge between a Dependency Grammar, e.g. the
one developed for the Turin University Dependency Parser (Lesmo and Lombardo, 2002)
and Treebank (Bosco, 2004), and the logical framework defined in (Robaldo, 2010), which
uniformily represents both standard linear readings and Independent Set readings, with
any number of quantifiers, any monotonicity, and any partial order between them.
Semantic underspecification provided us with an elegant solution to the problem of
preserving a direct mapping from syntax to semantics. In this way, the intuition about
the existence of pure semantic ambiguities can be supported by the management of such
ambiguities within the realm of semantics, without involving syntactic structures very
far from the surface form of the sentences.
Of course, other underspecified formalisms have been proposed (and briefly described in
the report). However, their underlying logical language uses quantifier-embedding operations for specifying the available readings. Quantifier-embedding intrinsically prevents
the proper representation of IS readings. DTS is instead able to achieve this result by
underspecifying Skolem-like functional dependencies rather than quantifier-embeddings.
The adoption of Skolem terms has the effect of making the specification of referring terms
more local, getting rid of the notion of “scope”, usual in standard logics. Scope is replaced
36
by functional dependency, so that each Skolem term contains inside itself the specification
of the elements that contribute to the determination of its referents. As a side effect, the
constraints coming from the need to specify a linear order among quantifiers are avoided,
thus being able to face partial orders.
Furthermore, it must be noted that some underspecified formalisms proposed in the
literature favour the closeness to syntactic structures, others tend to prefer the flexibility of management of those structures. In DTS, we focus on the similarity between
dependency-based syntactic structures and semantic representations, so that a simple
syntax-semantic interface may be defined without preventing flexible disambiguation.
However, the flexibility is preserved by a suitable way for incrementally adding the functional dependencies.
Finally, an algorithm for translating DTS fully-specified structures into (Robaldo, 2010)’s
formulae has been presented. The algorithm defines an indirect model theory of Dependency Tree Semantics.
A final remark concerns the treatment of other scope ambiguities in DTS, as the ones
arising from the multiple interactions between boolean connectives, modals, adverbs,
intensional verbs, and their relatives. In the other underspecified formalisms they are
again dealt with by means of quantifier-embedding operations. The present report does
not discuss how they may be incorporated in DTS, but it is clear that it is not easy
to treat them in terms of functional dependencies. We do not have a specific answer
for incorporating their management in the formalism, and we do not advocate here any
alternative account with respect to the solution proposed in the other underspecified
constraint-based approaches. In other words, they should be dealt in DTS simply by
introducing additional constructs (possibly) similar to holes\labels in Hole Semantics, a
solution that does not seem difficult to implement, but that deserves further study. The
main point of this report is instead claiming that quantifier-scope ambiguities deserve a
separate account. The motivation of such an alternative account is of course the need
of representing Independent Set readings, whose occurrence in NL is rather widespread,
especially with respect to Cumulative and Collective readings, as remarked in the Introduction. For these readings, it seems that Dependency Tree Semantics is an elegant
solution that enables to cover all relevant aspects that must be faced.
References
Alshawi, H., editor. 1992. The Core Language Engine. Mit Press, Cambridge, MA.
Alshawi, H. and R. Crouch. 1992. Monotonic semantic interpretation. In Proc. of the
30th Annual Meeting of the ACL, pages 32–39. MIT Press.
Barwise, J. and R. Cooper. 1981. Generalized quantifiers and natural language. Linguistics and Philosophy, 4(2):159–219.
Beck, S. and U. Sauerland. 2000. Cumulation is needed: A reply to winter (2000).
Natural Language Semantics, 8(4):349–371.
37
Beghelli, F., D. Ben-Shalom, and A. Szabolski. 1997. Variation, distributivity, and
the illusion of branching. In Ways of Scope Taking. Kluwer Academic Publishers,
Dordrecht, pages 29–69.
Ben-Avi, G. and Y. Winter. 2003. Monotonicity and collective quantification. Journal
of Logic, Language and Information, 12:127–151.
Bharati, A., V. Chaitanya, and R. Sangal, editors. 1996. Natural Language Processing A Paninian Perspective. Mit Press, Prentice Hall of India, New Delhi.
Bos, Johan. 1996. Predicate logic unplugged. In Proc. Of the 10th Amsterdam Colloquium, pages 133–142, Amsterdam.
Bos, Johan. 2004. Computational semantics in discourse: Underspecification, resolution,
and inference. Journal of Logic, Language and Information, (13):139–157.
Bosco, C. 2004. A grammatical relation system for treebank annotation. Ph.D. thesis,
University of Turin, Italy.
Bunt, H. 2003. Underspecification in semantic representations: Which technique for
what purpose? In Proc. of the 5th Workshop on Computational Semantics (IWCS5), pages 37–54, Tilburg, The Netherlands.
Carpenter, B. 1997. Type-Logical Semantics. The MIT Press, Cambridge, Massachusetts.
Chaves, R. P. 2005. Non-redundant scope disambiguation in underspecified semantics.
Proceedings of the 6th Workshop on Computational Semantics (IWCS-6).
Cooper, R. 1983. Quantification and Syntactic Theory. Dordrecht.
Copestake, A., D. Flickinger, and I.A. Sag. 2005. Minimal recursion semantics. an
introduction. 23(3):281–332.
Debusmann, R. 2006. Extensible Dependency Grammar: A Modular Grammar Formalism Based On Multigraph Description. Ph.D. thesis, Saarland University, Germany.
Diesing, M. 1992. Indefinites. Cambridge, MA: MIT Press.
Ebert, C. 2005. Formal Investigations of Underspecified Representations. Ph.D. thesis,
Department of Computer Science, King’s College London.
Egg, Markus, Alexander Koller, and Joachim Niehren. 2001. The constraint language
for lambda structures. Journal of Logic, Language and Information, 10(4):457–485.
Fuchss, R., A. Koller, J. Niehren, and J. Thater. 2004. Minimal recursion semantics as
dominance constraints: Translation, evaluation, and analysis. In Proc. of the the 42th
annual meeting on Association for Computational Linguistics, Buffalo, New York.
Gierasimczuk, N. and J. Szymanik. 2009. Branching quantification vs. two-way quantification. The Journal of Semantics. In press.
38
Groenendijk, J. and M. Stokhof. 1991. Dynamic predicate logic. Linguistics and Philosophy, 14:39–100.
Hackl, M. 2000. Comparative quantifiers. Ph.D. thesis, Massachusetts Institute of
Technology.
Hackl, M. 2002. The ingredients of essentially plural predicates. In Proc. of the 32nd
North East Linguistic Society conference, pages 171–182, University of Massachusetts.
Heim, I. 1982. The Semantics of Definite and Indefinite Noun Phrases. Ph.D. thesis,
University of Massachusetts at Amherst.
Hobbs, J.R. and S. Shieber. 1987. An algorithm for generating quantifier scoping. Computational Linguistics, 13:47–63.
Hudson, R. 1990. English word grammar. Oxford and Cambridge: Basil Blackwell.
Joshi, A. and L. Kallmeyer. 2003. Factoring predicate argument and scope semantics:
Underspecified semantics with ltag. Research on Language and Computation, 1.
Joshi, A.K., L. Kallmeyer, and M. Romero. 2003. Flexible composition in ltag: Quantifier
scope and inverse linking. In Computing Meaning. Kluwer.
Joshi, A.K. and Y. Schabes. 1997. Tree-adjoining grammars. In Handbook of Formal
Languages. Springer Verlag, Berlin.
Kadmon, N. 1987. On unique and non-unique reference and asymmetric quantification.
Ph.D. thesis, University of Massachusetts, Amherst.
Kahane, S. 2003. The meaning-text theory. In Dependency and Valency: an International Handbook of Contemporary Research. Berlin, De Gruyter.
Kallmeyer, L. and M. Romero. 2008. Scope and situation binding in ltag using semantic
unification. Research on Language and Computation, 6(1).
Kamp, H. and U. Reyle. 1993. From Discourse to Logic: an introduction to modeltheoretic semantics, formal logic and Discourse Representation Theory. Kluwer Academic Publishers, Dordrecht.
Keenan, E. and D. Westerståhl. 1997. Generalized quantifiers in linguistics and logic. In
Handbook of Logic and Language. Cambridge: MIT Press, pages 837–893.
Keller, W. 1988. Nested cooper storage: The proper treatment of quantification in
ordinary noun phrases. In Natural Language Parsing and Linguistic Theories. Reidel,
Dordrecht.
Koller, A. 2004. Constrained-Based and Graph-Based Resolution of Ambiguities in Natural Language. Ph.D. thesis, Universitat des Saarlandes, Germany.
Koller, A., J. Niehren, and S. Thater. 2003. Bridging the gap between underspecification
formalisms: Hole semantics as dominance constraints. In In Proc. of the European
Chapter of the Association of Computational Linguistics.
39
Koller, A. and S. Thater. 2006. Towards a redundancy elimination algorithm for underspecified descriptions. Proceedings of the 5th Intl. Workshop on Inference in Computational Semantics (ICoS-5).
Kratzer, A. 2007. On the plurality of verbs. In Event Structures in Linguistic Form and
Interpretation. Mouton de Gruyter, Berlin.
Krifka, Manfred. 1992. Thematic relations as links between nominal reference and temporal constitution. In Lexical Matters. CSLI Publications, pages 29–53.
Kurtzman, H.S. and M.C. MacDonald. 1993. Resolution of quantifier scope ambiguities.
Cognition, (48):243–279.
Landman, F. 2000. Events and Plurality: The Jerusalem Lectures. Kluwer Academic
Publishers.
Larson, R.K. 1985. Quantifying into np. Manuscript, University of Wisconsin.
Lesmo, L. and V. Lombardo. 2002. Transformed subcategorization frames in chunk
parsing. In In Proc. of the 3rd Int. Conf. on Language Resources and Evaluation
(LREC 2002), pages 512–519, Las Palmas.
Lev, Iddo. 2005. Decoupling scope resolution from semantic composition. In In Proc.
6th Workshop on Computational Semantics (IWCS-6), pages 139–150, Tilburg.
Lindström, P. 1966. First order predicate logic with generalized quantifiers. Theoria,
32:186–195.
Link, G. 1983. The logical analysis of plurals and mass terms. In Meaning, Use, and
Interpretation in Language. de Gruyter, Berlin, pages 302–323.
May, R. 1985. Logical Form in Natural Language. Cambridge, MA: MIT Press.
May, R. 1989. Interpreting logical form. Linguistics and Philosophy, 12(4):387–437.
May, R. and A. Bale. 2005. Inverse linking. In The Blackwell Companion to Syntax,
Volume II. Wiley-Blackwell, Malden.
Mel’cuk, I. 1988. Dependency syntax: theory and practice. New York: SUNY University
Press.
Mel’cuk, I. 2004. Actants in semantics and syntax. ii: actants in syntax. Linguistics,
(42):247–291.
Montague, R. 1988. The proper treatment of quantification in ordinary english. In
Philosophy, Language, and Artificial Intelligence: Resources for Processing Natural
Language. Kluwer, Boston, pages 141–162.
Moran, D.B. 1988. Quantifier scoping in the sri core language engine. In Proc. of the 26th
annual meeting on Association for Computational Linguistics, Buffalo, New York.
40
Mostowski, A. 1957. On a generalization of quantifiers. Fundamenta Mathematicae,
44:12–36.
Nakanishi, K. 2007. Event quantification and distributivity. In Event Structures in
Linguistic Form and Interpretation. Mouton de Gruyter.
Neale, S. 1990. Descriptions. Cambridge, MIT Press.
Park, J. 1996. A Lexical Theory of Quantification in Ambiguous Query Interpretation.
Ph.D. thesis, University of Pennsylvania.
Peters, S. and D. Westerståhl. 2006. Quantifiers in Language and Logic. Oxford University Press, Oxford.
Reyle, U. 1993. Dealing with ambiguities by underspecification: Construction, representation and deduction. Journal of Semantics, 13:123–179.
Reyle, U. 1996. Co-indexing labelled drss to represent and reason with ambiguities. In
Semantic Ambiguity and Underspecification. CSLI Lecture Notes, Stanford.
Robaldo, L. 2009a. Independent set readings and generalized quantifiers. The Journal
of Philosophical Logic, 39(1):23–58.
Robaldo, L. 2009b. Interpretation and inference with maximal referential terms. The
Journal of Computer and System Sciences. In press.
Robaldo, L.
2010.
Distributivity, collectivity, and cumulativity in terms of
(in)dependence and maximality.
Manuscript accepted for publication by the
Journal of Logic, Language, and Information. The final version is available at
http://www.di.unito.it/∼robaldo/Draft-JLLI.pdf.
Robaldo, L. and J. Di Carlo. 2009. Disambiguating quantifier scope in dependency tree
semantics. In In Proc. of 8th International Workshop on Computational Semantics
(IWCS-8), Tilburg, The Netherlands.
Robaldo, L. and J. Di Carlo. 2010. Flexible disambiguation in dependency tree semantics. In In Proc. of 11th International Conference on Intelligent Text Processing and
Computational Linguistics (CicLing2010), Iasi, Romania.
Scha, R. 1981. Distributive, collective and cumulative quantification. In Formal Methods
in the Study of Language, Part 2. Mathematisch Centrum, Amsterdam.
Schein, B. 1993. Plurals and Events. MIT Press, Cambridge, MA, USA.
Schwarzschild, R. 1996. Pluralities. Kluwer, Dordrecht.
Sher, G. 1990. Ways of branching quantifiers. Linguistics and Philosophy, 13:393–422.
Sher, G. 1997. Partially-ordered (branching) generalized quantifiers: a general definition.
The Journal of Philosophical Logic, 26:1–43.
41
Spaan, M. 1996. Parallel quantification. In Quantifiers, Logic, and Language, volume 54.
CSLI Publications, Stanford, pages 281–309.
Steedman, M.
2007.
The Grammar of Scope.
forthcoming. See
’Surface-Compositional
Scope-Alternation
Without
Existential
Quantifiers’. Draft 5.2, Sept 2007. Retrieved September 25, 2007 from
ftp://ftp.cogsci.ed.ac.uk/pub/steedman/quantifiers/journal6.pdf.
Sternefeld, W. 1998. Reciprocity and cumulative predication. Natural Language Semantics, 6:303–337.
Tapanainen, P. 1999. Parsing in two frameworks: finite-state and functional dependency
grammar. Ph.D. thesis, University of Helsinki, Finland.
Tesniere, L. 1959. Eléments de Syntaxe Structurale. Librairie C. Klincksieck, Paris.
van Benthem, J. 1986. Essays in logical semantics. Dordrecht, Reidel.
van der Does, J. 1993. Sums and quantifiers. Linguistics and Philosophy, 16:509–550.
van der Does, J. and H. Verkuyl. 1996. The semantics of plural noun phrases. In
Quantifiers, Logic and Language. CSLI.
van Genabith, J. and R. Crouch. 1996. F-structures, qlfs and udrss. In Proc. of the the
1st Int. Conference on LFG.
Vestre, E. J. 1991. An algorithm for generating non-redundant quantifier scopings.
Proceedings of the 5th conference on European chapter of the Association for Computational Linguistics, pages 251–256.
Villalta, E. 2003. The role of context in the resolution of quantifier scope ambiguities.
Journal of Semantics, 20.
Willis, A. 2000. An Efficient Treatment of Quantification in Underspecified Semantic
Representations. Ph.D. thesis, University of York.
42