Charting the Potential of Description Logic for the Generation of Referring Expressions

The generation of referring expressions (GRE), an important subtask of Natural Language Generation (NLG) is to generate phrases that uniquely identify domain entities. Until recently, many GRE algorithms were developed using only simple formalisms, which were taylor made for the task. Following the fast development of ontology-based systems, reinterpretations of GRE in terms of description logic (DL) have recently started to be studied. However, the expressive power of these DL-based algorithms is still limited, not exceeding that of older GRE approaches. In this paper, we propose a DL-based approach to GRE that exploits the full power of OWL2. Unlike existing approaches, the potential of reasoning in GRE is explored. 1 GRE and KR: the story so far Generation of Referring Expressions (GRE) is the subtask of Natural Language Generation (NLG) that focuses on identifying objects in natural language. For example, Fig.1 depicts the relations between several women, dogs and cats. In such a scenario, a GRE algorithm might identify d1 as “the dog that loves a cat”, singling out d1 from the five other objects in the domain. Reference has long been a key issue in theoretical linguistics and psycholinguistics, and GRE is a crucial component of almost every practical NLG system. In the years following seminal publications such as (Dale and Reiter, 1995), GRE has become one of the most intensively studied areas of NLG, with links to many other areas of Cognitive Science. After plan-based contributions (e.g., (Appelt, 1985)), recent work increasingly stresses the human-likeness of the expressions generated in simple situations, culminating in two evaluation campaigns in which dozens of GRE algorithms were compared to human-generated expressions (Belz and Gatt, 2008; Gatt et al., 2009). Figure 1: An example in which edges from women to dogs denote feed relations, from dogs to cats denote love relations. Traditional GRE algorithms are usually based on very elementary, custom-made, forms of Knowledge Representation (KR), which allow little else than atomic facts (with negation of atomic facts left implicit), often using a simple 〈Attribute : V alue〉 format, e.g 〈Type : Dog〉. This is justifiable as long as the properties expressed by these algorithms are simple one-place predicates (e.g., being a dog), but when logically more complex descriptions are involved, the potential advantages of “serious” KR become overwhelming. (This point will become clearer in later sections.) This realisation is now motivating a modest new line of research which stresses logical and computational issues, asking what properties a KR framework needs to make it suitable to generate all the referring expressions that people can produce (and to generate them in reasonable time). In this new line of work, which is proceeding in tandem with the more empirically oriented work mentioned above, issues of human-likeness are temporarily put on the backburner. These and other empirical issues will be brought to bear once it is better understood what types of KR system are best suitable for GRE, and what is the best way to pursue GRE in them. A few proposals have started to combine GRE with KR. Following on from work based on labelled directed graphs (cf. (Krahmer et al., 2003)) – a well-understood mathematical formalism that offers no reasoning support – (Croitoru and van Deemter, 2007) analysed GRE as a projection problem in Conceptual Graphs. More recently, (Areces et al., 2008) analysed GRE as a problem in Description Logic (DL), a formalism which, like Conceptual Graphs, is specifically designed for representing and reasoning with potentially complex information. The idea is to produce a formula such as Dog u ∃love.Cat (the set of dogs intersected with the set of objects that love at least one cat); this is, of course, a successful reference if there exists exactly one dog who loves at least one cat. This approach forms the starting point for the present paper, which aims to show that when a principled, logic based approach is chosen, it becomes possible to refer to objects which no existing approach to GRE (including that of Areces et al.) has been able to refer to. To do this, we deviate substantially from these earlier approaches. For example, while Areces et al. use one finite interpretation for model checking, we consider arbitrary (possibly infinite) interpretations, hence reasoning support becomes necessary. We shall follow many researchers in focussing on the semantic core of the GRE problem: we shall generate descriptions of semantic content, leaving the decision of what words to use for expressing this content (e.g., ‘the ancient dog’, or ‘the dog which is old’) to later stages in the NLG pipeline. Furthermore, we assume that all domain objects are equally salient (Krahmer and Theune, 2002). As explained above, we do not consider here the important matter of the naturalness or efficacy of the descriptions generated. We shall be content producing uniquely referring expressions whenever such expressions are possible, leaving the choice of the optimal referring expression in each given situation for later. In what follows, we start by explaining how DL has been applied in GRE before (Sec. 2) , pointing out the limitations of existing work. In Sec.3 we discuss which kinds of additional expressivity are required and how they can be achieved through modern DL. In Sec.4 we present a generic algorithm to compute these expressive REs. Sec.5 concludes the paper by comparing its aims and achievements with current practise in GRE. 2 DL for GRE 2.1 Description Logics Description Logic (DLs) come in different flavours, based on decidable fragments of firstorder logic. A DL-based KB represents the domain with descriptions of concepts, relations, and their instances. DLs underpin the Web Ontology Language (OWL), whose latest version, OWL2 (Motik et al., 2008), is based on DL SROIQ (Horrocks et al., 2006). An SROIQ ontology Σ usually consists of a TBox T and an ABox A. T contains a set of concept inclusion axioms of the formC v D, relation inclusion axioms such as R v S (the relation R is contained in the relation S), R1 ◦ . . . ◦ Rn v S, and possibly more complex information, such as the fact that a particular relation is functional, or symmetric; A contains axioms about individuals, e.g. a : C (a is an instance of C), (a, b) : R (a has an R relation with b). Given a set of atomic concepts, the entire set of concepts expressible by SROIQ is defined recursively. First, all atomic concepts are concepts. Furthermore, if C and D are concepts, then so are > | ⊥ | ¬C | C uD | C tD | ∃R.C | ∀R.C | ≤ nR.C | ≥ nR.C | ∃R.Self | {a1, . . . , an}, where > is the top concept, ⊥ the bottom concept, n a non-negative integer number, ∃R.Self the self-restriction ((i.e., the set of those x such that (x, x) : R holds)), ai individual names and R a relation which can either be an atomic relation or the inverse of another relation (R−). We call a set of individual names {a1, . . . , an} a nominal, and use CN , RN and IN to denote the set of atomic concept names, relation names and individual names, respectively. An interpretation I is a pair 〈∆I , I〉 where ∆I is a non-empty set and I is a function that maps atomic concept A to AI ⊆ ∆I , atomic role r to rI ⊆ ∆I × ∆I and individual a to aI ∈ ∆I . The interpretation of complex concepts and axioms can be defined inductively based on their semantics, e.g. (C uD)I = CI ∩DI , etc. I is a model of Σ, written I |= Σ, iff all the axioms in Σ are satisfied in I. It should be noted that one Σ can have multiple models. For example when T = ∅,A = {a : A t B}, there can be a model I1 s.t. ∆I1 = {a}, aI1 = a,AI1 = {a}, BI1 = ∅, and another model I2 s.t. ∆I2 = {a}, aI2 = a,BI2 = {a}, AI2 = ∅. In other words, the world is open. For details, see (Horrocks et al., 2006). The possibly multiple models indicate that an ontology is describing an open world. In GRE, researchers usually impose a closed world. From the DL point of view, people can (partially) close the ontology with a DBox D (Seylan et al., 2009), which is syntactically similar to the ABox, except that D contains only atomic formulas. Furthermore, every concept or relation appearing in D is closed. Its extension is exactly defined by the contents of D, i.e. if D 6|= a : A then a : ¬A, thus is the same in all the models. The concepts and relations not appearing in D can still remain open. DL reasoning can be exploited to infer implicit information from ontologies. For example, given T = {Dog v ∃feed−.Woman} (every dog is fed by some woman) and A = {d1 : Dog,w1 : Woman}, we know that there must be some Woman who feeds d1. When the domain is closed as D = A we can further infer that this Woman is w1 although there is no explicit relation between w1 and d1. Note that the domain ∆I in an interpretation ofD is not fixed, but it includes all the DBox individuals. However, closing ontologies by means of the DBox can restrict the usage of implicit knowledge (from T ). More precisely, the interpretations of the concepts and relations appearing inD are fixed therefore no implicit knowledge can be inferred. To address this issue, we introduce the notion of NBox to support Negation as Failure (NAF): Under NAF, an ontology is a triple O = (T ,A,N ), where T is a TBox, A an ABox and N is a subset of CNorRN . We callN an NBox. NAF requires that O satisfy the following conditions: 1. Let x ∈ IN and A ∈ N uCN . Then (T ,A) 6|= x : A implies O |= x : ¬A. 2. Let x, y ∈ IN and r ∈ N u RN . Then (T ,A) 6|= (x, y) : r implies O |= (x, y) : ¬r. Like the DBox approach, the NBox N defines conditions in which “unknown” should be treated as “failure”. But, instead of hard-coding this, it specifies a vocabulary on which such treatment should be applied. Different from the DBox approach, inferences on this NAF vocabulary is still possible. An example of inferring implicit knowledge with NAF will be shown in later sections. 2.2 Background Assumptions When applying DL to GRE, people usually impose the following assumptions. • Individual names are not used in REs. For example, “the Woman who feeds d1” would be invalid, because d1 is a name. Names are typically outlawed in GRE because, in many applications, many objects do not have names that readers/hearers would be familiar with. • Closed World Assumption (CWA): GRE researchers usually assume a closed world, without defining what this means. As explained above, DL allows different interpretations of the CWA. Our solution does not depend on a specific definition of CWA. In what follows, however, we use NAF to illustrate our idea. Furthermore, the domain is usually considered to be finite and consists of only individuals appearing in A. • Unique Name Assumption (UNA): Different names denote different individuals. If, for example, w1 and w2 may potentially be the same woman, then we can not distinguish one from the other. We follow these assumptions when discussing existing works and presenting our approach. In addition, we consider the entire KB, including A, T and N . It is also worth mentioning that, in the syntax of SROIQ, negation of relations are not allowed in concept expressions, e.g. you cannot compose a concept ∃¬feed.Dog. However, if feed ∈ N , then we can interpret (¬feed)I = ∆I ×∆I \ feedI . In the rest of the paper, we use this as syntactic sugar. 2.3 Motivation: DL Reasoning and GRE Every DL concept can be interpreted as a set. If the KB allows one to prove that this set is a singleton then the concept is a referring expression. It is this idea (Gardent and Striegnitz, 2007) that (Areces et al., 2008) explored. In doing so, they say little about the TBox, appearing to consider only the ABox, which contains only axioms about instances of atomic concepts and relations. For example, the domain in Fig.1 can be described as KB1: T1 = ∅, A1 = {w1 : Woman, w2 : Woman, d1 : Dog, d2 : Dog, c1 : Cat, c2 : Cat, (w1, d1) : feed, (w2, d1) : feed, (w2, d2) : feed, (d1, c1) : love} Assuming that this represents a Closed World, Areces et al. propose an algorithm that is able to generate descriptions by partitioning the domain.1 More precisely, the algorithm first finds out which objects are describable through increasingly large conjunctions of (possibly negated) atomic concepts, then tries to extend these conjunctions with complex concepts of the form (¬)∃R1.Concept, then with concepts of the form (¬)∃R2.(Concept u (¬)∃R1.Concept), and so on. At each stage, only those concepts that have been acceptable through earlier stages are used. Consider, for instance, KB1 above. Regardless of what the intended referent is, the algorithm starts partitioning the domain stage by stage as follows. Each stage makes use of all previous stages. During stage (3), e.g., the object w2 could only be identified because d2 was identified in stage (2): 1. Dog = {d1, d2}, ¬Dog uWoman = {w1, w2}, ¬Dog u ¬Woman = {c1, c2}. 2. Dog u ∃love.(¬Dog u ¬Woman) = {d1}, Dogu¬∃love.(¬Dogu¬Woman) = {d2}. 3. (¬Dog u Woman) u ∃feed.(Dog u ¬∃love.(¬Dog u ¬Woman)) = {w2}, (¬Dog u Woman) u ¬∃feed.(Dog u ¬∃love.(¬Dog u ¬Woman)) = {w1}. As before, we disregard the important question of the quality of the descriptions generated, other than whether they do or do not identify a given referent uniquely. Other aspects of quality depend in part on details, such as the question in which order atomic concepts are combined during phase (1), and analogously during later phases. However this approach does not extend the expressive power of GRE. This is not because of some specific lapse on the part of the authors: it seems to have escaped the GRE community as a whole that relations can enter REs in a variety of alternative ways. Furthermore, the above algorithm considers only the ABox, therefore background information Areces et al. (Areces et al., 2008) consider several DL fragments (e.g., ALC and EL). Which referring expressions are expressible, in their framework, depends on which DL fragment is chosen. Existential quantification, however, is the only quantifier that was used, and inverse relations are not considered. will not be used. It follows that the domain always has a fixed single interpretation/model. Consequently the algorithm essentially uses modelchecking, rather than full reasoning. We will show that when background information is involved, reasoning has to be taken into account. For example, suppose we extend Fig.1 with background (i.e., TBox) knowledge saying that one should always feed any animal loved by an animal whom one is feeding, while also adding a love edge (Fig.2) between d2 and c2: Figure 2: An extended example of Fig.1. Edges from women to cats denote feed relations. Dashed edges denote implicit relations. If we close the domain with NAF, the ontology can be described as follows: KB2: T2 = {feed ◦ love v feed}, A2 = A1 ∪ {(d2, c2) : love}, N2 = {Dog,Woman, feed, love} The TBox axiom enables the inference of implicit facts: the facts (w1, c2) : feed, (w2, c1) : feed, and (w2, c2) : feed can be inferred using DL reasoning under the above NBox N2. Axioms of this kind allow a much more natural, insightful and concise representation of information than would otherwise be possible. Continuing to focus on the materialised KB2, we note another limitation of existing works: if only existential quantifiers are used then some objects are unidentifiable (i.e., it is not possible to distinguish them uniquely). These objects would become identifiable if other quantifiers and inverse relations were allowed. For example, • The cat which is fed by at least 2 women = Catu ≥ 2feed−.Woman = {c1}, • The woman feeding only those fed by at least 2 women = Woman u ∀feed. ≥ 2.feed−.Woman = {w1}, • The woman who feeds all the dogs = {w2}. It thus raises the question: which quantifiers would it be natural to use in GRE, and how might DL realise them? 3 Beyond Existential Descriptions In this section, we show how more expressive DLs can make objects referable that were previously unreferable. This will amount to a substantial reformulation which allows the approach based on DL reasoning to move well beyond other GRE algorithms in its expressive power. 3.1 Expressing Quantifiers in OWL2 Because the proposal in (Areces et al., 2008) uses only existential quantification, it fails to identify any individual in Fig.2. Before filling this gap, we pause to ask what level of expressivity ought to be achieved. In doing so, we make use of a conceptual apparatus developed in an area of formal semantics and mathematical logic known as the theory of Generalized Quantifiers (GQ), where quantifiers other than all and some are studied (Mostowski, 1957). The most general format for REs that involves a relation R is, informally, the N1 who R Q N2’s, where N1 and N2 denote sets, R denotes a relation, and Q a generalized quantifier. (Thus for example the women who feed SOME dogs.) An expression of this form is a unique identifying expression if it corresponds to exactly one domain element. Using a set-theoretic notation, this means that the following set has a cardinality of 1: {y ∈ N1 : Qx ∈ N2 | Ryx} where Q is a generalized quantifier. For example, if Q is the existential quantifier, while N1 denotes the set of women, N2 the set of dogs, and R the relation of feeding, then this says that the number of women who feed SOME dog is one. If Q is the quantifier at least two, then it says that the number of women who feed at least two dogs is one. It will be convenient to write the formula above in the standard GQ format where quantifiers are cast as relations between sets of domain objects A,B. Using the universal quantifier as an example, instead of writing ∀x ∈ A | x ∈ B, we write ∀(AB). Thus, the formula above is written {y ∈ N1 : Q(N2{z : Ryz)}}. Instantiating this as before, we get {y ∈Woman : ∃(Dog{z : Feed yz)}}, or “women who feed a dog”, where Q is ∃, A = Dog and B = {z : Feed yz} for some y. Mathematically characterising the class of all quantifiers that can be expressed in referring expressions is a complex research programme to which we do not intend to contribute here, partly because this class includes quantifiers that are computationally problematic; for example, a quantifiers such as most (in the sense of more than 50%), which is not first-order expressible, as is well known. To make transparent which quantifiers are expressible in the logic that we are using, let us think of quantifiers in terms of simple quantitative constraints on the sizes of the sets A∩B, A−B, and B−A, as is often done in GQ theory, asking what types of constraints can be expressed in referring expressions based on SROIQ. The findings are summarised in Tab.1. OWL2 can express any of the following types of descriptions, plus disjunctions and conjunctions of anything it can express. Table 1: Expressing GQ in DL