Dependency Tree Semantics and Underspecification




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Dependency Tree Semantics and Underspecification


Leonardo Lesmo & Livio Robaldo

Dipartimento di Informatica

Università di Torino

C.so Svizzera, 185 - I-10149 Italy

{lesmo, robaldo}@di.unito.it






Abstract


This paper describes a formalism (DTS: Dependency Tree Semantics) for representing underspecified interpretations of quantifier scope ambiguities and for expressing the various possible disambiguated readings. It is based on a dependency representation of the syntactic structure: the claim is that such a representation is isomorphic to an underspecified semantic representation. The inclusion of arcs that mark semantic dependencies among quantified substructures enable one to express the readings. Moreover, the definition of suitable rules constrain the possible configurations in such a way that the licensed trees correspond to the linguistically admissible readings.


1. Introduction

One of the problems in semantic interpretation concerns the scope of quantifiers. In fact, the choice of the correct scope is notoriously hard, depending on the context, on preferences grounded in the syntactic structure, and on background (world) knowledge. In order to decouple the basic process of semantic interpretation and the disambiguation of quantifier scope, a number of underspecified representations have been proposed (see, for instance, [Alshawi & Crouch 92], [Reyle 93], [Bos 95], [Pinkal 96], [van Deemter & Peters 96], [Niehren et al. 97], [Copestake et al. 99], [Bos 01]). The connections between underspecification, disambiguation and reasoning are discussed in [König and Reyle 99]. An overview of the goals and techniques of underspecification may be found in [Bunt 03].

The basic idea is to devise a formalism such that all different readings associated with the different possible scopes of quantifiers are represented in a single structure. This structure must satisfy a twofold goal: first, it must be usable in applications where the exact determination of the scope is not relevant (e.g. in Machine Translation). Second, it must be able to undergo ‘specification’ procedures which produce the disambiguated readings, i.e. that the underspecified representation can be ‘extended’ (and not reshuffled) by adding further information that express the various different readings.

In this paper, we describe such a formalism. With respect to the other proposals appeared in the literature, it presents the advantage that the underspecified representation largely coincides with the syntactic structure of the sentence. The syntactic structure is represented as a dependency tree ([Mel’cuk 88], [Hudson 90]). Recently, dependency structures have raised interest as a by-product of Lexicalized Tree Adjoining Grammars [Joshi & Schabes 97]. In LTAG, two syntactic trees are relevant: the derived tree, a constituency representation, and the derivation tree, which expresses the way the sentence has been formed using the basic syntactic structures (elementary and auxiliary trees) by means of the TAG operations (substitution and adjoining). It is the derived tree that has been exploited as the syntactic structure on which the semantic interpretation process can be applied. In [Kallmeyer & Joshi 03], it is shown how an underspecified representation can be obtained from a derived tree, and how it is possible to deal, within this framework, with various semantic problems, as, for instance inverse linking [May & Bale, 02]. In the last section we will compare our proposal with some relevant alternatives.

The paper is organized as follows. In the next section, we present the basic dependency trees, which play the role of underspecified semantic representation. In Section 3, we introduce the various readings of our example sentences. In Section 4, we show how it is possible to extend the dependency tree in order to specify the various possible quantifier scopings. In Section 5, we compare our proposal with other approaches. The Conclusion section closes the paper.


2. Dependency trees

A Dependency Tree is a representation of the syntactic structure based on the idea of replacing the explicit representation of the way the words are grouped (constituents) with a representation of the ‘dependencies’ between words. So, instead of specifying that in

  1. A boy ate the cake

‘A boy’ forms a group (NP) and ‘ate the cake’ forms a second group (VP), it is stated that ‘ate’ is the main element of the sentence, and that it ‘governs’ ‘my friend’ and ‘the cake’ (fig.1).




Fig.1 - Constituency and dependency trees for

A boy ate the cake

It is not possible here to discuss the linguistic merits of the two approaches; we only mention that it has often been stated that dependency trees mirror the basic predicate-argument structure. In ex.1, ‘ate’ corresponds to a two-place predicate (to eat) and the two arguments appear explicitly. Note that dependency representations are based on labelled edges, where the labels mark the kind of relationship (e.g. subject) existing between the head and the dependent. For the sake of simplicity, in this paper, we omit arc labels, which are present in our approach to syntax; a description of the arc labels we use may be found in [Bosco & Lombardo 03]. A treebank for Italian based on dependency structures has been described in [Lesmo et al. 03].

It is clear that dependency trees are syntactic representations, so that the nodes appearing in the tree are words, and not semantic elements. However, we assume that, apart from some special cases, there is a one to one correspondence between syntactic nodes and semantic primitives (predicates, constants, etc.). For instance, ‘ate’ corresponds to a two-place predicate, ‘cake’ to one-place predicate, ‘boy’ to a two-place predicate, ‘the’ to the  quantifier; ‘a’ to the existential quantifier. Of course, there are cases where this does not hold (for instance ‘black dog’ should be interpreted by making explicit the property ‘colour’) but this does not affect the ideas behind our approach.

In this paper, we focus on some specific examples, that enable us to face the relevant problems. We start with an extension of the well known example “Two politicians spied on someone from every city” ([Larson 85], [Joshi et al. 03]), i.e.

  1. Two students studied three theorems of four theories.

The basic idea is to first consider the behaviour of numerals, and then treat the other quantifiers as special cases. The dependency representation of (2) is reported in fig.2. The tree has been annotated with semantic primitives, in order to show the correspondence with the semantic formulae we use in the paper.

























Fig.2 - Annotated dependency tree for

Two students read three theorems of four theories


We claim that there exist 13 (thirteen) acceptable readings for (2), and that all of them can be obtained by adding suitable arcs to the dependency tree in fig.2. Note that we can interpret the tree in fig.2 as a flat representation, similar to the one in MRS [Copestake et al. 99] (see Section 6 below). Numerals can be interpreted in the standard way:

2[x] p(x)  x1x2 [x1x2 

x [(x = x1  x = x2)  p(x)]]

Note also that in fig.2 the variable names do matter only insofar they enable us to express the mapping with a standard logical approach, but they are implicit in the DTS representation.


3. Various readings

Before introducing the structures that are added to the dependency tree to express the various readings, it is first necessary to describe these readings. Of course, they arise from the different scopes of the numerical quantifiers. Here, because of space constraints, we report just two of those readings, by providing the corresponding Predicate Calculus formula, and a graphical representation of a model satisfying the formula. All various readings are available in http://www.di.unito.it/~lesmo/DTS/dts-home.html .

A first reading can be paraphrased in this way:

(2’1) There are two students (Mary and Bill), each of which studied the same three theorems (Th1, Th2 and Th3), all of which belong to a well-defined group of four theories (Tr1, Tr2, Tr3, Tr4).

This corresponds to the following formula:

(2f1) x1x2 y1y2y3 z1z2z3z4

[x1x2  y1y2y3  z1z2z3z4 

x [(x=x1  x=x2)  student’(x)]

z [(z=z1  z=z2  z=z3  z=z4)  theory’(z)] 

y [(y=y1  y=y2  y=y3)  theorem’(y)] 

yz [((y=y1 y=y2  y=y3) 

(z=z1 z=z2  z=z3  z=z4))

 belong’(y, z)] 

yx [((y=y1 y=y2  y=y3)  (x=x1 x=x2))

 study’(x,y)]]

A possible model for this reading is the following:




(2m1)


Of course, both the formula and the model reflect an interpretation of the numerals as ‘at least n’ rather than ‘exactly n’. This seems reasonable, since, for instance, (2) does not exclude that Susan did the same as Mary and Bill.


4. Disambiguating the dependency tree

Since the dependency tree corresponds to an underspecified representation, what is needed is a way to express the different (13) readings. This is obtained by introducing new arcs in the tree (which becomes a graph), which roughly correspond to Skolem functions. Since these arcs connect the quantifiers nodes, so that the “content” nodes (e.g. student) are not affected by them, we simplify the figures by keeping only the quantifier nodes.


4.1 Semantic Dependency (semdep) arcs


A disambiguated reading is obtained by specifying the relative scopes of the three quantifiers. This is represented by inserting three semdep (semantic dependency) arcs, shown in the figures as dotted arrows. However, we need a means for expressing the outermost scope; to this aim, a dummy node is inserted in the tree as its root; we call it context, to express the intuition that the elements ‘semantically dependent’ on the context correspond to individual constants that, in some cases, may be contextually determined (as Mary and Bill in the examples of the previous section). So, the tree in fig.2 becomes.




Fig.3 - Simplified dependency tree for

Two students read three theorems of four theories


As a first example, we report in fig.4 the disambiguated tree corresponding to the first reading presented in section 3. It is possible to see that, in that reading (see (2fi)), no existential is outscoped by a universal. In the given paraphrase, as well as in the model, the various elements are associated with some ‘contextually defined’ constants. So, the resulting representation is:




Fig.4 - The reading (2’1), (2f1), (2m1)


In a second reading, on the contrary, the existentials associated with the theories (zi variables) occur in the scope of the universal associated with the theorems (yi variables), so that the particular set of theories may ‘depend on’ the specific theorem under consideration (in a skolemized representation, we would have that the z’s are substituted by f(y), where f is a Skolem function):

(2f2) x1x2 y1y2y3

[x1x2  y1y2y3 

x [(x = x1  x = x2)  student’(x)] 

y [(y=y1 y=y2  y=y3)  theorem’(y)] 

y [(y=y1 y=y2  y=y3) 

z1z2z3z4 [z1z2z3z4 

(z [(z=z1 z=z2  z=z3  z=z4) 

theory’(z)] 

z[(z=z1 z=z2  z=z3  z=z4) 

belong’(y,z)])] 

xy [((x=x1 x=x2)  (y=y1 y=y2  y=y3)) 

study’(x, y)]]


Correspondingly, the disambiguated reading is expressed as in fig.5.





Fig.5 - A second reading of

Two students read three theorems of four theories


4.2 Constraints on semdep arcs


It is clear that there are many combinations of semdep links that are not valid, in the sense that they produce unacceptable readings. The simplest example is the case where two nodes are linked in both directions: this would mean that each quantifier occurs in the scope of the other, which is clearly impossible. So, what must be done is to devise a set of constraints that enable the insertion of all and only the arcs that correspond to the thirteen acceptable readings. The readings we considered as acceptable are produced by enforcing the two following constraints.

  1. There are no cycles

  2. There are no multiple dependencies, i.e. the same pair of nodes cannot be linked via two different paths.

  3. A node n1 cannot be linked to a node n2 if they do not share a predicate unless:

a. There is another node n3, which shares a predicate with n1 and is linked to n2.

b. There is another node n3, which shares a predicate with n2 and there is a path of links from n3 to n1.

The first constraint is rather obvious: a two-step cycle corresponds to a quantifier q1 which is included in the scope of another quantifier q2, which is included in the scope of q1, which is clearly impossible. Similarly, for n-step cycles. The absence of cycles imposes that at least one of the nodes is linked to context, i.e. at least one of them must have outermost scope.

The second constraint specifies that there must be no redundancy (so that no two different DTS have the same model). This constraint is important in connection with Multiple Semdep arcs (see §4.3).

The third constraint states that two nodes may have a semantic dependency just in case such a dependency is licensed by the syntactic structure of the sentence; in (2), two and three share the predicate study’ (it is not represented in the figures, but, in general this corresponds to having a common mother, see fig.2) and three ad four share the predicate belong’ (i.e. one syntactically depends on the other). On the contrary, no predicate is shared by two and four; this means that, since no direct relation exist between the students and the theories, no direct semantic dependency can exist between them. However, the ‘unless’ clause in 3 partially relaxes this constraint: there can be an indirect semantic dependency in two ways. Subclause 3.a says that the quantifier associated with n1 can occur in the scope of the quantifier associated with n2 in case there is already another quantifier (the one associated with n3) that is in that scope and shares a predicate with n1. This corresponds to the reading where the three and four quantifiers are, at the same level, within the scope of two. This can be paraphrased as:


(2’2) There are two students (Mary and Bill) each of which chose three theorems (ThM1, ThM2 and ThM3 for Mary and ThB1, ThB2 and ThB3 for Bill), and four theories (TrM1, TrM2, TrM3 and TrM4 for Mary and TrB1, TrB2, TrB3 and TrB4 for Bill). Bill and Mary studied their theorems, and each theorem of Mary must belong to all the theories chosen by Mary; the same for Bill.

It seems that this reading can easily be perceived as an acceptable one.

Subclause 3.b says that the quantifier associated with n1 can occur in the scope of the quantifier associated with n2 in case there is another quantifier n3 which shares a predicate with n2 and which occurs in the scope of n1. Since this case is rather complex, we report in fig.6 an example graph.




Fig.6 - A representation licensed by clause 3.b


Here, four (theories) could not be linked to two (students), but since three (theorems) share a predicate (study’) with the students, three can ‘carry with itself’ four within the scope of two. So, the two students chose the theories to which the theorems they will study belong (possibly three different theorems for each theory). Note that constraint 3 is effective, i.e. it actually rules out some readings (4 configurations), two of which are reported in fig.7.




Fig.7 - Two representations excluded by constraint 3.


4.3 Multiple semdep arcs

It is possible that a given quantifier occurs in the scope of more than one other quantifier. Although a chain of semdep arcs account for this in standard cases, there are situations that are not covered by standard arcs. This happens in case two (or more) quantifiers have parallel scope. For instance, in the partially specified DTS shown in fig.8a, both two and four have wide scope, so that a semdep arc linking three to either of them specifies that three occurs only in the scope of it. In order to represent the missing reading, a multiple arc is required, having one tail (three) and two heads (two and four), as shown in fig.8b.




(a) (b)

Fig.8 - A multiple semdep arc


Actually, no extension is required to the basic algorithm for introducing semdep arcs, if multiple arcs are allowed. In fact, the arc in 8b respects the constraints 1, 2 and 3, and it is the only one, so that the representation in 8b provides the thirteenth reading of (2).


5. Universal and existential quantifiers

As stated above, existential and universal quantifiers are treated as special cases of the numerical quantifiers. Universals, since they range on the whole domain of individuals, cannot exhibit any semantic 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 semdep arc exiting them can be assumed to be fixed and to enter the context.1 For analogous reasons, no semdep arc can enter an existential quantifier.

5.1 Universals

Let’s modify example (2) into:

  1. Two students studied three theorems of all theories.

If we assume that there exist (or are contextually relevant) just four theories, the model of the first reading (2m1) does not change. On the contrary, the logical formula changes as follows (we repeat below also (2f1) for making easier the comparison).

(3f1) x1x2 y1y2y3

[x1x2  y1y2y3 

x [(x = x1  x = x2)  student’(x)]

y [(y=y1 y=y2  y=y3)  theorem’(y)] 

yz [((y=y1 y=y2  y=y3)  theory’(z)

belong’(y, z)] 

yx [((y=y1 y=y2  y=y3)  (x=x1 x=x2)) 

study’(x,y)]]

(2f1) x1x2 y1y2y3 z1z2z3z4

[x1x2  y1y2y3  z1z2z3z4

x [(x = x1  x = x2)  student’(x)]

z [(z=z1 z=z2 z=z3 z=z4)theory’(z)]

y [(y=y1 y=y2  y=y3)  theorem’(y)] 

yz [((y=y1 y=y2  y=y3) 

(z=z1 z=z2 z=z3 z=z4))  belong’(y, z)] 

yx [((y=y1 y=y2  y=y3)  (x=x1 x=x2)) 

study’(x,y)]]

It is clear that (3f1) is obtained from (2f1) by removing the existential part referring to the theories (variables zi, in boldface) and substituting the restriction on the universal (variable z) with the theory’ predicate (in boldface italics). This produces the conflation of some formulas, which corresponds to the application of the constraint imposing that an arc exiting a universal enter the context. Similar changes occur if we substitute the other numerical quantifiers with universals, although in some cases there occurs the usual alternation between conjunction and implication. It may be shown that the 13 readings reduce for (3) to just 8 readings.

5.2 An exception to the rule on universals

As stated above, there is an exception to the rule that a semdep arc exiting a universal can only enter the context. This exception concerns a universal acting as a modifier for another universal, as in (4):

  1. Two students studied all theorems of all theories.

In this case, there are two possibilities, according to the intention of the speaker of referring to ‘all theorems that belong to any theory’ or to ‘all theorems that belong to every theory’. Of course, the same situation arises in case the modifier is a numeral (‘all theorems of four theories’). In such a case, we enable the upper universal to by linked via a semdep arc to its modifier. If this is the case, the associated reading is the ‘any’ one, since the particular set of ‘all’ theorems is determined on the basis of the particular theory (all the theorems of that theory). Clearly, the standard link to the context is possible, so that the reading where the theorem must belong to all theories is accounted for.


6. Comparison with other approaches

In order to compare the predictions of the DTS with ones made by other approaches, we can now test the behaviour of the DTS on a well known from the literature (see fig.9):

(5) Two politicians spied on someone from every city




Fig.9 - Simplified dependency tree for

Two politicians spied on someone from every city

Just two semdep arcs can vary: the one exiting two, and the one exiting some (). This reduces the 13 acceptable readings to just 6, whose corresponding graphs are shown in fig.10.




(a) (b) (c)




(d) (e) (f)


Fig. 10 - The six disambiguated readings for

Two politicians spied on someone from every city


In [Joshi et al. 03], underspecification is dealt with by means of a formalism similar to MRS [Copestake et al. 99] that makes use of quantifier sets. It accepts just 4 readings for (5), expressed, according to the nested quantifier scopings, as:

2, 2, 2, 2

while 2 and 2 are considered as unacceptable (see also [Larson 85]). The correspondence with our graphs is as follows:

(a): 2

(b): 2

(c): not available ( in the scope of , but not of 2, with 2 having widest scope; this and the next three readings are called “inverse linking”)

(d): 2

(e): 2

(f): 2

(***) 2 is ruled out by our constraint 3.

It is clear that the predictions are rather different. Let us see the paraphrases:

  1. Susan and Harry spy John, who comes from every city.

  2. Susan spies John and Harry spies Ada; John and Ada come from every city.

  3. Susan and Harry spy John (coming from C1), Ada (from C2), and so on for every city.

  4. Susan and Harry spy John and Ada (coming from C1) respectively, two other politicians spy two other persons (coming from C2), and so on for every city.

  5. John (coming from C1) is spied by Susan and Harry, Ada (coming from C2) is spied by other two politicians, and so on for every city.

  6. Susan spies John (coming from C1), Ada (coming from C2), and so on for every city, while Harry spies Marcus (coming fom C1), Juliet (from C2), and so on for every city

(***) John (coming from every city) is spied by Susan and Harry (wrt C1), by Marcus and Juliet (wrt C2), and so on for every city.

It seems that reading (a) is the most natural. According to [Joshi et al. 03] “it may be banned due to general architectural reasons” (p.3). The logical representation they propose for (a) is [Joshi et al. 03, footnote 2]:

y [person(y)  2 x [politician(x) 

z [city(z)  from(y,z)  (spy(x,y)]]]2

And the comment is that “2 involves having the quantifier y separated from its restrictor from(y,z), a configuration that should probably be banned …”.

On the contrary, our translation is :

(4f1) x1x2 y

[x1x2 

x [(x=x1  x=x2)  politician’(x)] 

person’(y) 

z [city’(z)  from’(y, z)] 

x [(x=x1 x=x2)  spy’(x,y)]]

And here, the separation mentioned above does not arise. However, the interpretation of the numerical quantifier [Joshi et al. 03, footnote 1] is analogous to the one adopted here (apart from the use of a variable ranging over sets, which we replace with two separate individual variables). So, it seems that after a shift of existential quantifiers, the sentence gets an acceptable form, thus matching the intuition.

Reading (d) is more complex, but it seems possible to express that situation via (5), although [Larson 85] sees it as unacceptable. We observe, however, that it would be possible to simulate the stacked quantifier storage adopted by Larson by means of a constraint stating that a node (as two in fig.10d) cannot occur in the middle of a path connecting two nodes one of which dominates the other in another dependency subtree (as  and  in fig. 10d).

On the contrary, the absence of a predicate linking the politicians to the cities makes it hard to imagine how the spying politicians may vary according to the particular city John comes from.3

In general, most approaches to underspecification use flat representations (MRS, but also LUD [Bos et all, 96], UDRSs [Reyle, 1993], …) in which the semantic atoms are isolated and, for disambiguating the sentence, some constraints are added to the representation (for example, in MRS labels are assigned to holes). It seems that there is a strict correspondence between the atoms of the flat representation and the nodes of the dependency tree. For example, we can associate MRS atoms to the dependency tree as shown in fig.11.

Fig.11 - The syntactic/semantic tree of

Every student hates some course


hate'

HATES

STUDENT

student'

h5: some (y, h6, h7)

h1: every (x, h2, h3)

h4: student (x)

h8: course (y)

h2 =q h4

h6 =q h8

h0 =q h9



SOME

EVERY



COURSE

course'

h0 : hates




It is apparent that the dependency tree contains the same information as MRS or other flat under-specified representations, and that the MRS constraints are expressed in DTS via semdep arcs. But the correspondence between licensed semdep arcs and MRS constraints deserves further studies.


7. Conclusions

This paper describes an underspecified semantic representation based on dependency trees. It shows that a base dependency tree may be viewed as an underspecified representation: the different un-ambiguous readings are obtained by simply adding, according to some rules and constraints, new arcs to the tree. We claim that the representations obtained according to this procedure correspond to the actual possible readings for the sentences.

Currently, two algorithms are available, the first of which extends a basic dependency tree by adding all and only the semdep arcs that respect the constraints described in the paper. A second algorithm generates gets as input a completely specified DTS and produces the corresponding logical formula.

An extensive listing of example DTS, and the Java implemention of the algorithms mentioned in the previous paragraph are available from http://www.di.unito.it/lesmo/DTS.html.


8. References

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[Bosco & Lombardo 03] C. Bosco, V. Lombardo: “A relation-based schema for treebank annotation”. In A. Cappelli, F.Turini (eds).: Advances in Artificial Intelligence, Springer Verlag, Berlin, 2003, 462-473.

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[Debusmann et al. 04] R. Debusmann, D. Duuchier, A. Koller, M. Kuhlmann, G. Smolka, S. Thater: “A Relational Syntax-Semantics Interface Based on Dependency Grammars”, 2004.

[Hobbs and Shieber 87] J. R. Hobbs, S.M. Shieber: “An Algorithm for Generating Quantifier Scoping”, Computational Linguistics 13, 1987, 47-63.

[Hudson 90] R. Hudson: English word grammar. Basil Blackwell, Oxford and Cambridge, MA, 1990.

[Joshi and Schabes 97] A.K. Joshi, Y. Schabes: Tree-Adjoining Grammars. In G. Rozenberg, A. Salomaa (eds.): Handbook of Formal Languages. Springer Verlag, Berlin, 1997, 69-123.

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[Joshi et al. 03] A. K. Joshi, L. Kallmeyer, M. Romero: “Flexible Composition in LTAG: Quantifier Scope and Inverse Linking”, In R. Musken and H. Bunt (eds) Computing Meaning, Vol. 3, Kluwer, 2003.

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[Niehren et al. 1997] J. Niehren, M. Pinkal, P. Ruhrberg: “A Uniform Approach to Underspecification and Parallelism“. Proc. ACL 97, Madrid.

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[Reyle 93] U. Reyle: “Dealing with ambiguities by Underspecification: Construction, Representation and Deduction”, Journal of Semantics 10 (2), 1993.

[van Deemter and Peters 96] K. van Deemter, S. Peters: “Semantic Ambiguity and Underspecification”, CSLI, Stanford, Ca., 1996.

1 However, there is an exception to this rule on universals (see §5.2).

2 We exchanged the y and z variables wrt. to [Joshi et al. 03] in order to enforce the consistency with our formulas

3 It is assumed (here as in the original papers) that the PP ‘from every city’ is syntactically part of the NP ‘someone …’ and not a verbal dependent.


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