Abstract

We address the problem on how newly invented concepts are evaluated with respect to a background ontology of conceptual knowledge so as to decide which of them are to be accepted into a system of familiar concepts, and how this, in turn, may a ect the previously accepted conceptualisation. As technique to tackle this problem we explore the applicability of Paul Thagard's computational theory of coherence. In particular, we propose a formalisation of Thagard's notion of conceptual coherence for concepts represented in the AL description logic and explore by means of an illustrative example the role coherence may play in the process of conceptual blending.

Keywords

• conceptual blending
• coherence
• description logics

Introduction

Combinational creativity |when novel ideas (concepts, theories, solutions, works of art) are produced through unfamiliar combinations of familiar ideas| is, of the three forms of creativity put forward by Boden, the most di cult to capture computationally [ 2 ]. Putting concepts together to generate new concepts is, in principle, not a di cult task; but doing this in a computationally tractable way, and being able to recognise the value of newly invented concepts for better understanding a certain domain, is not as straightforward.

An important development that has signi cantly in uenced the current understanding of the general cognitive principles operating during concept invention is Fauconnier and Turner's theory of conceptual blending [ 6, 7 ]. Fauconnier and Turner proposed conceptual blending as the fundamental cognitive operation underlying much of everyday thought and language, and modelled it as a process by which humans subconsciously combine particular elements and their relations of originally separate conceptual spaces into a uni ed space, in which new elements and relations emerge, and new inferences can be drawn.

The theory has been primarily applied as an analytic tool for describing already existing blends of ideas and concepts in a varied number of elds, such as linguistics, music theory, poetics, mathematics, theory of art, political science, discourse analysis, philosophy, anthropology, and the study of gesture and of material culture [ 20 ]. But it has been also widely recognised to be a theory that can serve as a basis for computational models of creativity [ 5, 9, 10, 15, 21 ].

Copyright © 2016 for this paper by its authors. Copying permitted for private and academic purposes.

To guide the concept invention process, in addition to the blending mechanism per se, at least two additional dimensions need to be considered, namely the origin and destination of concept invention, i.e., from where (and how) input concepts are selected and to whom the concept invention is headed. Confalonieri et al. have proposed a process model for concept invention in which these dimensions are taken into account [ 3 ]. Inputs are selected based on a similarity measure that is computed relative to a Rich Background, and blends are evaluated using an argumentation framework based on value preferences of the audience for which concepts are invented.

In this paper, we aim at showing how Thagard's computational theory of coherence [ 19 ] could also serve as an additional mechanism for triggering concept invention and evaluating newly blended concepts. In [ 18 ], Thagard suggested to use coherence as a model for the closely related cognitive process of conceptual combination, where the focus is primarily on language compositionality such as noun-noun or adjective-noun combinations [ 17 ]. Kunda and Thagard, for instance, show how conceptual coherence can be used for describing how we reason with social stereotypes [ 12 ].

Building upon Thagard's intuitions and principles for modelling coherence, we propose a formalisation of Thagard's notion of conceptual coherence for concepts represented in a description logic |we take the basic description logic AL as a start| and further explore its applicability to conceptual blending. But instead of interpreting coherence or incoherence based on statistical correlations or causal relations (i.e., on frequencies of positive or negative association), we determine coherence and incoherence as dependent on how concept descriptions are stated. Failure to nd conceptual blends that cohere with some given background knowledge leads to a search for alternative conceptual blends that eventually increase the overall coherence of the blend with the background knowledge.

The paper is organised as follows: In Section 2 we give a brief overview of Thagard's computational theory of coherence, in Section 3 we introduce some core de nitions regarding coherence and coherence graphs, and in Section 4 we provide a formalisation of conceptual coherence for the description logic AL.

Conceptual blending in AL is described in Section 5, and coherence is applied to blending in Section 6. We conclude in Section 7. 2

Thagard's Computational Theory of Coherence Thagard addresses the problem of determining which pieces of information, such as hypotheses, beliefs, propositions or concepts, to accept and which to reject based on how they cohere and incohere among them, given that, when two elements cohere, they tend to be accepted together or rejected together; and when two elements incohere, one tends to be accepted while the other tends to be rejected [ 19 ].

This can be reformulated as a constraint satisfaction problem as follows.

Pairs of elements that cohere between them form positive constraints, and pairs of elements that incohere between them form negative constraints. If we partition the set of pieces of information we are dealing with into a set of accepted elements and a set of rejected elements, then a positive constraint is satis ed if both elements of the constraint are either among the accepted elements or among the rejected ones; and a negative constraint is satis ed if one element of the constraint is among the accepted ones and the other is among the rejected ones.

The coherence problem is to nd the partition that maximises the number of satis ed constraints.

Note that in general we may not be able to partition a set of elements as to satisfy all constraints, thus ending up accepting elements that incohere between them or rejecting an element that coheres with an accepted one. The objective is to minimise these undesired cases. The coherence problem is known to be NP-complete, though there exist algorithms that nd good enough solutions of the coherence problem while remaining fairly e cient.

Depending on the kind of pieces of information we start from, and on the way the coherence and incoherence between these pieces of information is determined, we will be dealing with di erent kinds of coherence problems. So, in explanatory coherence we seek to determine the acceptance or rejection of hypotheses based on how they cohere and incohere with given evidence or with competing hypotheses; in deductive coherence we seek to determine the acceptance of rejection of beliefs based on how they cohere and incohere due to deductive entailment or contradiction; in analogical coherence we seek to determine the acceptance or rejection of mapping hypotheses based on how they cohere or incohere in terms of structure; and in conceptual coherence we seek to determine the acceptance or rejection of concepts based on how they cohere or incohere as the result of the positive or negative associations that can be established between them. Thagard discusses these and other kinds of coherence.

Although Thagard provides a clear technical description of the coherence problem as a constraint satisfaction problem, and he enumerates concrete principles that characterise di erent kinds of coherences, such as those discussed later in Section 4 for conceptual coherence, he does not clarify the actual nature of the coherence and incoherence relations that arise between pieces of information, nor does he suggest a precise formalisation of the principles he discusses.

Joseph et al. have proposed a concrete formalisation and realisation of deductive coherence [ 11 ], which they applied to tackle the problem of norm adoption in normative multi-agent systems. In this paper, we shall focus on the problem of conceptual coherence and its applicability to conceptual blending. 3

Preliminaries: Coherence Graphs

In this section we give precise de nitions of the concepts intuitively introduced in the previous section.

De nition 1. A coherence graph is an edge-weighted, undirected graph G = hV; E; wi, where:

1. V is a nite set of nodes representing pieces of information.

2. E V (2) (where V (2) = ffu; vg j u; v 2 V g) is a nite set of edges representing the coherence or incoherence between pieces of information.

3. w : E ! [ 1; 1] n f0g is an edge-weighted function that assigns a value to the

coherence between pieces of information.

Edges of coherence graphs are also called constraints.

When we partition the set V of vertices of a coherence graph (i.e., the set of pieces of information) into a set A of accepted elements and a set R = V n A of rejected elements, then we can say when a constraint |an edge between vertices| is satis ed or not by the partition.

De nition 2. Given a coherence graph G = hV; E; wi, and a partition (A; R) of V , the set of satis ed constraints C(A;R) E is given by:

C(A;R) = nfu; vg 2 E u 2 A i v 2 A; whenever w(fu; vg) > 0 o u 2 A i v 2 R; whenever w(fu; vg) < 0

All other constraints (i.e., those in E n C(A;R)) are said to be unsatis ed.

The coherence problem is to nd the partition of vertices that satis es as much constraints as possible, i.e., to nd the partition that maximises the coherence value as de ned as follows, which makes coherence to be independent of the size of the coherence graph.

De nition 3. Given a coherence graph G = hV; E; wi, the coherence of a partition (A; R) of V is given by X

jw(fu; vg)j (G; (A; R)) = fu;vg2C(A;R) jEj

Notice that there may not exist a unique partition with a maximum coherence value. Actually, at least two partitions have the same coherence value, since (G; (A; R)) = (G; (R; A)) for any partition (A; R) of V . 4

Conceptual Coherence in Description Logics Thagard characterises conceptual coherence with these principles [ 19 ]: Symmetry: Conceptual coherence is a symmetric relation between pairs of concepts.

Association: A concept coheres with another concept if they are positively associated, i.e., if there are objects to which they both apply.

Given Concepts: The applicability of a concept to an object may be given perceptually or by some other reliable source.

Negative Association: A concept incoheres with another concept if they are negatively associated, i.e., if an object falling under one concept tends not to fall under the other concept.

Acceptance: The applicability of a concept to an object depends on the applicability of other concepts.

To provide a precise account of these principles we shall formalise Association and Negative Association between concepts expressed in a description logic, since these are the principles de ning coherence and incoherence. We shall assume coherence between two concept descriptions when we have explicitly stated that one subsumes the other (\there are objects to which both apply"); and we shall assume incoherence when we have explicitly stated that they are disjoint (\an object falling under one concept tends not to fall under the other concept").

De nition 4. Given a Tbox T in description logic AL and a pair of concept

descriptions C; D 62 f>; ?g, we will say that: { C coheres with D, if C v D 2 T , and that { C incoheres with D, if C v :D 2 T or C u D v ? 2 T .

In addition, coherence and incoherence between concept descriptions depend on

the concept constructors used, and we will say that, for all atomic concepts A, atomic roles R, and concept descriptions C; D 62 f>; ?g: { :A incoheres with A; { C u D coheres both with C and with D; { 8R:C coheres (or incoheres) with 8R:D, if C coheres (or incoheres) with D.1

Symmetry follows from the de nition above, and Acceptance is captured by the aim of maximising coherence in a coherence graph. For this we need to de ne how a TBox determines a coherence graph, and, in order to keep the graph nite, we express coherence and incoherence only between non-trivial concept descriptions (i.e., excluding > and ?) that are explicitly stated in the TBox.

De nition 5. Let T be a TBox in AL. The set of non-trivial subconcepts of

T is given as

sub(T ) = [

sub(C) [ sub(D)

CvD2T where sub is de ned over the structure of concept descriptions as follows: sub(A) = fAg sub(?) = ; sub(>) = ; sub(:A) = f:A; Ag sub(C u D) = fC u Dg [ sub(C) [ sub(D) sub(8R:C) = f8R:Cg [ sub(C) sub(9R:>) = f9R:>g 1 Note that since AL allows only for limited existential quanti cation we cannot provide a general rule for coherence between concept descriptions of the form 9R:>.

De nition 6. The coherence graph of a TBox T is the edge-weighted, undi

rected graph G = hV; E; wi whose vertices are non-trivial subconcepts of T (i.e., V = sub(T )), whose edges link subconcepts that either cohere or incohere according to De nition 4, and whose edge-weight function w is given as follows: w(fC; Dg) = ( 1 if C and D cohere

1 if C and D incohere

5

Conceptual Blending in AL We follow the modelling principles and techniques of [ 4 ], where the process of conceptual blending is characterised by the notion of amalgams [ 1, 14 ]. According to this approach, the process of conceptual blending can be described as follows: 1. We take a taxonomy of concepts described in a background ontology expressed as a Tbox T . 2. A mental space of an atomic concept A is modelled, for the purpose of conceptual blending, by means of a subsumption A v C specifying the necessary conditions we are focusing on. 3. The new concept to be invented is represented by the concept description that conjoins the atomic concepts to be blended. 4. With amalgams we generalise the input spaces based on the taxonomy in our TBox until a satisfactory blend is generated.

Formally, the notion of amalgams can be de ned in any representation language L for which a subsumption relation between formulas (or descriptions) of L can be de ned, and therefore also in the set of all AL concept descriptions with the subsumption relation vT .

To formally specify an amalgam we rst need to introduce some notions. Let NC be a set of concept names, NR be a set of role names, and L(T ) be the nite set of all AL concept descriptions that can be formed with the concept and role names occurring in an AL TBox T . Then:

De nition 7. Given two descriptions C1; C2 2 L(T ):

{ A most general specialisation (MGS) is a description Cmgs such that Cmgs vT

C1 and Cmgs vT C2 and for any other description D such that D vT C1

and D vT C2, then D vT Cmgs. { A least general generalisation (LGG) is a description Clgg such that C1 vT

Clgg and C2 vT Clgg and for any other description D such that C1 vT D and C2 vT D, then Clgg vT D.

Intuitively, an MGS is a description that has some of the information from both original descriptions C1 and C2, while an LGG contains what is common to them.

An amalgam or blend of two descriptions is a new description that contains parts from these original descriptions and it can be formally de ned as follows.

De nition 8 (Amalgam). Let T be an AL TBox. A description Cam 2 L(T )

is an amalgam of two descriptions C1 and C2 (with LGG Clgg) if there exist two descriptions C1 and C2 such that: C1 vT C1 vT Clgg, C2 vT C2 vT Clgg, and Cam is an MGS of C1 and C2.

The number of blends that satis es the above de nition can be very large and selection criteria for ltering and ordering them are therefore needed. Fauconnier and Turner discussed optimality principles [ 7 ], however, these principles are di cult to capture in a computational way, and other selection strategies need to be explored. Since we use a logical theory such as AL, one way to evaluate a blend is consistency checking. Another alternative, that we will investigate in this paper, is to evaluate blends in terms of conceptual coherence.

The LGG and the generalised descriptions, needed to compute the amalgam as de ned above, are obtained by means of a generalisation re nement operator that allows us to nd generalisations of AL concept descriptions. Roughly speaking, a generalisation operator takes a concept C as input and returns a set of descriptions that are more general than C by taking a Tbox T into account.

In order to de ne a generalisation re nement operator for AL, we de ne the upward cover set of atomic concepts. In the following de nition, sub(T ) (De nition 5) guarantees the following upward cover set to be nite.

De nition 9. Let T be an AL TBox with concept names from NC . The upward cover set of an atomic concept A 2 NC [ f>; ?g with respect to T is given as:

UpCov(A) := fC 2 sub(T ) [ f>; ?g j A vT C (1) and there is no C0 2 sub(T ) [ f>; ?g We can now de ne our generalisation re nement operator for AL as follows. De nition 10. Let T be an AL TBox. We de ne the generalisation re nement operator inductively over the structure of concept descriptions as follows: (A) = UpCov(A) (>) = UpCov(>) = ; (?) = UpCov(?) (8r:C) = (9r:>) = ; (C u D) = fC0 u D j C0 2 (C)g [ fC u D0 j D0 2 (D)g [ fC; Dg f>g f8r:C0 j C0 2 (C)g whenever (C) 6= ;

otherwise.

House v Object Resident v Person Boat v Object Passenger v Person Land v Medium Person u Medium v ? Water v Medium Object u Medium v ?

Water u Land v ? Object u Person v ?

We should notice at this point that can return concept descriptions that are equivalent to the concept being generalised. One possible way to avoid this situation is to discard these generalisations [ 4 ]. Given a generalisation re nement operator , AL concepts are related by re nement paths as described next.

De nition 11. A nite sequence C1; : : : ; Cn of AL concepts is a concept re ne

ment path C1 ! Cn from C1 to Cn of the generalisation re nement operator i Ci+1 2 (Ci) for all i : 1 i < n. (C) denotes the set of all concepts that can be reached from C by means of in a nite number of steps. The repetitive application of the generalisation re nement operator allows us to nd a description that represents the properties that two or more AL concepts have in common. This description is a common generalisation of AL concepts, the so-called generic space that is used in conceptual blending.

De nition 12. An AL concept description G is a generic space of the AL concept descriptions C1; : : : ; Cn if and only if G 2 0 (Ci) for all i = 1; : : : ; n.

5.2

An Example: The House-Boat Blend The process of conceptual blending in terms of amalgams can be illustrated by means of a typical blend example: the house-boat [ 7, 8 ]. The precise formalisation is not critical at this point, di erent ones exist [ 9, 15 ], but all provide similar distinctions.

The AL theories for House and Boat introduce the axioms modelling the mental spaces for house and boat.

House v 8usedBy:Resident u 8on:Land

Boat v 8usedBy:Passenger u 8on:Water

The House and Boat theories cannot be directly blended since they generate an inconsistency. This is due to the background ontology stating that the medium on which an object is situated cannot be land and water at the same time (Figure 1). Therefore, some parts of the House and Boat descriptions need to be generalised in a controlled manner before these concepts can be blended. The generic space between a house and a boat|an object that is on a medium and used-by a person|is a lower bound in the space of generalisations that need to be explored in order to generalise these concepts and to blend them into a house-boat. The generic space is obtained according to De nition 12 by applying the re nement operator . House

Boat House

Example 1. Let us consider the House and Boat concepts. Their generic space is: 8usedBy:Personu 8on:Medium and is obtained as follows. In the House concept, the subconcepts 8usedBy:Resident and 8on:Land are generalised to 8usedBy:Person and 8on:Medium respectively. In the Boat concept, the subconcepts 8usedBy: Passenger and 8on:Water are generalised in a similar way.

From a conceptual blending point of view, the house-boat blend can be created when the medium on which a house is situated (land) becomes the medium on which boat is situated (water), and the resident of the house becomes the passenger of the boat. This blend can be obtained when the input concepts house and boat are generalised as follows:

House v 8usedBy:Resident u 8on:Medium

Boat v 8usedBy:Person u 8on:Water The house-boat blend is obtained by conjoining the generalised mental spaces House and Boat (Figure 2). It is easy to see that House u Boat is an amalgam according to De nition 8. 6

Evaluating the Coherence of Conceptual Blends This section describes how coherence is used to evaluate blends. That is, how coherence graphs are built, and how the di erent coherence values are to be interpreted. The overall idea is to compute the coherence graph and maximsing partitions for each blend, and use the maximal coherence degree of the coherence graphs to rank the blends.

Let T be the TBox of the background ontology, let A v C and B v D be the axioms representing our mental spaces, and let A u B be the new concept we would like to invent. The process of evaluating blends according to conceptual coherence can be described as follows: 1. Given the mental spaces, we generate a candidate blend according to De nition 8. 2. We form the coherence graph for T [fA v C; B v Dg, including node AuB, according to De nition 6. 3. We compute the coherence maximising partitions according to De nition 3 and we associate it to the blend. 4. We repeat this procedure for all the blends that can be generated from the mental spaces.

Once the maximising partitions are computed, the coherence of the blend could be measured in terms of the coherence value of the coherence-maximising partitions. The degree of the coherence graph directly measures how much a blend coheres with the background ontology.

De nition 13. Let G = hV; E; wi the coherence graph of a blend B and let P the set of partitions of G. The maximal coherence value of B of G is deg(B) = maxf (G; P )g.

P 2P This maximal coherence value can be used to rank blends as follows.

De nition 14. Let T be a TBox of a background ontology, let A v C and

B v D be the axioms representing mental spaces, let B be the set of blends that

can be generated from them. For each b1; b2 2 B, we say that b1 is preferred to b2 (b1 b2) if and only if deg(b1) deg(b2).

To exemplify how the coherence degree can be used to evaluate blends, we consider the house-boat example. According to the amalgams process of conceptual blending described in the previous section, several blends can be generated by blending the mental space of House and Boat. In particular, the concept House u Boat is a valid blend.

The coherence graph blending the House and Boat directly is shown in Figure 3. As expected the concepts House and Boat positively coheres with the axioms representing the mental spaces and with the concept House u Boat, which is representing the blend. The incoherence relation between 8on:Land and 8on:Water is due to the fact that the concepts Water and Land incohere, since the background ontology contains the disjointness axiom Water u Land v ?. The coherence graph of House and Boat has a maximal coherence value of 0:84.

For the sake of our example, we generate new blends by generalising the axioms modelling our mental spaces. For instance, by applying the generalisations seen in the previous section that lead to the creation of the house-boat blend, we obtain the coherence graph in Figure 4.2 The coherence graph of blending House and Boat has a maximal coherence value of 0:9. This graph yields a higher coherence degree since generalising 8on:Land to 8on:Medium prevents the appearance of the incoherence relation between 8on:Land and 8on:Water.

It is easy to see that the blend House u Boat is preferred to House u Boat since it has a maximal coherence degree that is higher. 2 Concepts belonging to the background ontology are omitted. + + 8usedBy:Residentu 8on:Land

Passenger + Resident + +

Person

Medium Object + + + + House

Boat + + +

8on:Water Land Water 8usedBy:Passengeru 8on:Water + + 8usedBy:Resident 8usedBy:Residentu 8on:Medium

+

House

Boat + + This paper should be seen as a rst attempt to (a) provide a formal account of conceptual coherence for a particular concept representation language, and (b) to explore its applicability for guiding the process of conceptual blending.

With respect to (a), we proposed a formalisation of conceptual coherence between concept descriptions expressed in the basic AL description logic. This is only a starting point, and obviously this formalisation exercise should be carried out for more expressive concept representation languages. Usually, coherence and incoherence are not treated only in binary terms, but it is also natural to take certain degrees of coherence or incoherence into account. This, for instance, has also been the approach of Joseph et al. when formalising deductive coherence [ 11 ]. Although there is not an obvious way to do so with the formalisation of conceptual coherence of AL proposed in this paper, we do not discard that this could be done for more expressive concept representation languages. One could imagine that description logics with number restrictions or nominals, such as SROIQ for instance, would allow for expressing degrees of concept overlap that could be interpreted as degrees of coherence or incoherence.

With respect to (b), we have so far only focused on how the coherence values of a graph of concept descriptions were evolving dependent on how these descriptions were changing in our amalgam-based conceptual blending process. However, we have not discussed yet another important aspect of coherence theory, namely how to interpret the two parts of a coherence-maximising partition: the set of accepted and of rejected concepts. The information that a particular concept description falls in the set of accepted concepts or in the set of rejected concepts could also be taken into account to decide the acceptance or rejection of newly invented concepts; or even of already existing concepts in the background knowledge, in the light of newly invented concepts. With the formalisation in AL given in this paper we could not see yet a clear way to provide such an interpretation of acceptance and rejection, but we think this aspect might become clearer as a wider range of concept representation languages is explored.

In this paper we attempted to see how coherence could be used as another tool for guiding the process of conceptual blending and for evaluating conceptual blends in the task of concept invention; an additional technique to those already proposed, such as optimality principles [ 16 ], logical consistency [ 13 ], and values of audiences [ 3 ]. We believe it is worth to further study the proper combination of these techniques and to carry out a comprehensive evaluation.

An implementation of conceptual coherence presented in this paper using the OWL API and Answer Set Programming is available at: https://rconfalonieri@ bitbucket.org/rconfalonieri/coinvent-coherence.git.

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