Many aspects of how humans form and combine concepts are notoriously di cult to capture formally. In this paper, we focus on the representation of three particular such aspects, namely overextension, underextension, and dominance. Inspired in part by the work of Hampton, we consider concepts as given through a prototype view, and by considering the interdependencies between the attributes that de ne a concept. To approach this formally, we employ a recently introduced family of operators that enrich Description Logic languages. These operators aim to characterise complex concepts by collecting those instances that apply, in a nely controlled way, to `enough' of the concept's de ning attributes. Here, the meaning of `enough' is technically realised by accumulating weights of satis ed attributes and comparing with a given threshold that needs to be met.
A green chair is a chair that is also green, it is an untroubled combination of attributes that gives us a conjunctive concept GuC. It is untroubled because the speci c colour of a chair is an attribute that does not a ect the `chairness' of the chair: changing or losing the colour green will not make the chair a non-chair.
However, many combinations of concepts are not easily understood as simple conjunctions, because some de ning features of the combined concepts can enter into a subtle conversation: this paper discusses and formally analyses some of those phenomena studied within psychology and cognitive science, where the combination of two (or more) concepts has a more subtle semantics than intersection of sets.
In the classical view of concepts, the extensions of concepts are precisely de ned: for each object it would be possible to specify whether it falls within the de nition of the concept or not. If the object ts all the necessary and su cient conditions in the de nition of the concept, it will be categorised as an instance of the concept, otherwise it will not. Moreover the combination of concepts is de ned in a compositional way. This means that the meaning of a complex concept functionally depends on the meaning of the primitive concepts
(or predicates) that compose it. In other words, the de nition of a complex
concept is exclusively determined by its basic components, and this is often seen
to explain in part the ease and proli c ability of humans in creating meaningful
phrases. This perspective, that can be traced back to Fre
The crisp view of concepts in the traditional logic-based paradigm excludes indeterminacy, graded membership and typicality e ects from the process of categorisation, while the merely extensional compositionality of concepts does not allow for the representation of a number of cognitive phenomena linked to concept combination.
Overextension in con
We begin with some examples to set the scene. A toothless tiger is a tiger that has no teeth.1 Analysing the literal, rather than the metaphorical, meaning of this conjunctive concept, we include all those tigers which are, although tigers, lacking teeth, thereby losing an important, although not essential property. Similar to the green chair , this can be understood as a simple conjunctive concept when the concept of `tiger' is given in such a way that it allows for grades of typicality (e.g., wrt. a prototype). However, a de nition of tiger in terms of classical logic will exclude toothless tigers as non-tigers.
Now consider a pink panther as compared to the green chair . Whereas a
green chair might be quite like a red chair , just disagreeing in the colour, a pink
panther is quite unlike a black panther . Whilst it is extremely rare for a panther
to be black, it is still an optional colour quality attached to a panther.2 A pink
panther , on the other hand, assumes a colour which, by de nition of the natural
1 The phrase comes from the Chinese cultural sphere, and a `toothless tiger' in English
is essentially synonymous to a `paper tiger' (a tiger made from paper, Chinese phrase
`zhilaohu'): both are metaphorical concepts refering to \something or someone that
claims or appears to be powerful and/or threatening, but is actually ine ectual
and unable to withstand challenge" (see https://en.wikipedia.org/wiki/Pape
This is an example for the rst of our three classi cation problems, overextension, where we include more instances that belong to (the intersection of) the de ning concepts. To see the e ect in a more straightforward example, take the concepts of domestic worker and home appliance: clearly, these do not intersect? Now consider a robotic vacuum cleaner : It is an autonomous domestic robot with a oor cleaning system. It is not a domestic worker in the usual sense of worker , since it is not a person, but often it would be categorised by people as a domestic worker which is also a home appliance .
If we consider the category of the Tools that are also Weapons we could consider di erent examples: hammers, chainsaws, razors, scissors, screwdrivers, pliers and so on. But maybe we are interested in considering only the most typical ones (say: hammer, razor and scissor).
This is an example for the second of our three classi cation problems, underextension, where we include less instances than belong to the (intersection of the) de ning concepts. One way of making `less' precise is to collect only the (globally) most typical members of the intersection (see below).
Finally, consider a houseboat . A houseboat is both a house and a boat ,
however, it is a rather special house, being able to swim and lacking a solid
fundament, and it is a rather special boat, being able to be inhabited. This is an
example of, on the one hand, overconstraining the conjunction (as in
underextension), as well as prioritising certain aspects of a concept over others. In this case,
we obtain an example of conceptual blendin
Such complex aspects of how humans form and combine concepts, which are
notoriously di cult to capture in Cognitive Science, are largely absent in
logicbased approaches for the representation of concepts. We base the approach of this
paper on the work of Hampton [Hampton, 1987, 1988a,b, 2017] and we develop
an abstract setting to represent the three particular concept formation aspects
just discussed: overextension, when a concept classi es more than what is
constrained by its parts, underextension, when it classi es less, and dominance,
when some parts of a concept dominate the classi cation question. The
motivation for this work is to bridge the traditional techniques for doing knowledge
representation with cognitively grounded views of human classi cation.
of Wakanda, ruled by the Black Panther (see https://www.nationalgeog
This paper contributes to the project of extending Description Logics (DLs)
to model cognitively relevant features of classi cation [Po
The approach to weighted logics that we follow here takes inspiration from
the use of sets of weighted proposition for representing utility functions in
Uckelm
We introduce the weighted logic in Section 2, introduce in more detail the three classi cation problems in Section 3, discuss the formalisation of the problems in Section 4, and conclude in Section 5. 2
Weighted concept combination: The rr-operator logic We introduce a class of m-ary operators, denoted by the symbol rr (spoken `tooth'), for combining concepts. Each operator works as follows: i) it takes a list of concept descriptions, ii) it associates a vector of weights to them, and iii) it returns a complex concept that applies to those instances that satisfy a certain combination of concepts, i.e., those instances for which, by summing up the weights of the satis ed concepts, a certain threshold is met.
An initial technical study of ALCrr was unde
Basic Syntax and Semantics of the rr-Logic. The new logic is denoted by ALCrrR , where weights and thresholds range over real numbers r 2 R. In the following we will refer to the languages for brevity just as ALCrr. To dene the extended language of ALCrr, we add combination operators as follows, which behave syntactically just like m-ary modalities. We assume a vector of m weights w 2 Rm and a threshold value t 2 R. Each pair w, t speci es an operator: if C1; : : : ; Cm are concepts of ALC, then rrtw(C1; : : : ; Cm) is a concept of ALCrr. Note that in this basic de nition, the possible nesting of the operator is excluded.4
For Ci0 2 ALC, the set of ALCrr concepts is then described by the grammar:
C ::= A j :C j C u C j C t C j 8R:C j 9R:C j rrtw(C10; : : : ; Cm0) The semantics of the operator is obtained by extending the de nition of the semantics of ALC as follows. Let I = ( I ; I ) be an interpretation of ALC. We de ne the value of a d 2 I under a rr-concept C = rrtw(C1; : : : ; Cm) by setting: vCI(d) =
X i2f1;:::;mg fwi j d 2 CiI g The interpretation (i.e., the extension) of a rr-concept in I = ( I ; I ) is then: (rrtw(C1; : : : ; Cm))I = fd 2
I j vCI(d) tg (1) (2) To better visualise the weights an operator associates to the concepts, we sometimes use the notation rrt((C1; w1); : : : ; (Cm; wm)) instead of rrtw(C1; : : : ; Cm).
In the following examples, we will consider the value of an object name a (aka individual constant) wrt. a rr-concept for interpretations that satisfy a certain knowledge base K (i.e., a set of formulas).
name of ALC and K an ALC knowledge base. We set vCK(a) :=
X i2f1;:::;mg fwi j K j= Ci(a)g I.e., vCK(a) gives the accumulated weight of those Ci that are entailed by K to satisfy a.
Note that for positive weights, a given name a and a xed interpretation I such that I j= K, we always have that vCK(a) vCI(aI ).
Example 1. Consider the set of concepts C = fRed; Round; Colouredg and the concept C de ned by means of the rr operator
C = rrt((Red t Round; w1); (9above:Coloured; w2)) 4 In a more ne-grained de nition ALCirrK , i 0, is the logic with i levels of allowed nesting and where weights and thresholds range over K; we will comment on this further below. The de nition of C means that the relevant information to establish the categorisation under C of an object is whether (i) it is red or round, and (ii) it is above a coloured thing.
Consider the following knowledge base K = fRed(a); 9above:Blue(a); Blue v Colouredg, i.e., an agent knows that the object a is red and it is above a blue thing and that blue things are coloured, Blue v Coloured.
The value of a returned by vCK is computed as follows. Firstly, if a satis es Red, then a satis es Red t Round, so the weight w1 can be obtained. Moreover, since Blue v Coloured 2 K and a satis es 9above:Blue, then a satis es 9above:Coloured, so also the weight w2 can be obtained. Thus, vCK(a) is w1 + w2. If w1 + w2 t, then a is classi ed under C.
Having de ned the basic rr operator, a number of derived operations are easily obtained. First, it is possible to de ne the set of instances that at most reach a given threshold: rr t((C1; w1); : : : (Cm; wm)) rr t((C1; w1); : : : ; (Cm; wm)) (3) Given this, we can de ne the concept of instances that exactly score a certain threshold value t: rr=t((C1; w1); : : : (Cm; wm))
rrt((C1; w1); : : : ; (Cm; wm)) u rr t((C1; w1); : : : (Cm; wm)) (4)
Finally, it is interesting to consider the set of entities that maximally satisfy
a combination of concepts C1; : : : Cm of ALC. That is, we may de ne an operator
with the following semantics:
rrmax((C1; w1); : : : (Cm; wm)) I
= fd 2
j vCI(d)
vCI(d0) for all d0 2
g
(5)
De ning rrmax in terms of rrt would require to use a universal role, which
signi cantly increases the expressiv
What happens to ordinary concepts when they are subjected to the logical
operation of conjunction? In the 80s Hampton started a series of experiments to
evaluate how people interpret conjunctive phrases such as sports witch are also
games or games which are also sports
Hampton's proposal is to model the combination of such concepts in terms
of intensional composition which takes into account the features that de ne the
combined concepts. When a complex concept arises as a conjunction, the features
that de ne it
will be derived in a compositional fashion as a disjunction of the features
of the two con
Conjunctive phrases such as sports which are also games should then return a
composite prototype resulting from aggregating all the features commonly
associated with either of the two conjuncts (p. 98). The new set of features would then
be modi ed according with Hampton's model for attribute inheritance
[Hampton, 1987]. This model of the construction of a conjunctive concept would also
predict overextension in conjunctions. Roughly, Hamptons argument runs as
follows: let us imagine that, in order to be categorised under a concept, an object
must show at least two out of three features of that concept. So, let us suppose
that to be categorised in a conjunction, it should show four out of six features of
the component concepts. If an object presents three features of the rst object
but just one feature of the second object it would be categorised as an instance
of the conjunction, even if it is not an instance of both components. This would
re ect overextension in con
Interestingly, Hampton's experiments also showed that this kind of relative clause form is non-commutative: the concept games which are also sports is not identical to the concept sports which are also games. The tooth operator can capture some of these insights and distinctions, as we will show in more detail in Section 4. 3.2
Overextension's specular, or dual, phenomenon is that of underextension, that
is, when considered in the context of conjunction, the phenomenon in which
objects categorised as instances of two categories are not classi ed as belonging to
the (underextending) conjunction of the categories. For instance, an item could
be categorised as sport and as game, but could be not considered an instance
of sports which ar
Hampton's experiments showed another interesting phenomenon, namely the dominance e ect. Roughly speaking, in the simplest case of the composition of just two concepts, one of the two concepts `weighs more' than the other in the de nition of the conjunction. More precisely, the dominance e ect was analysed with respect to typicality. Hampton's aim was in fact to predict the typicality of an item in a conjunction of two classes on the basis of the typicality of the item in each of the conjuncts. In his analyses, he discovered that the weights of the two constituent concepts in the determination of an item's typicality were usually not equal. On the contrary, dominance of a concept with respect to a second was frequently observed. For instance, the weight of the concept sports was almost twice the weight of games in determining the typicality of items in both of the concepts sports which are also games and games which are also sports [Hampton, 1987]. Following these insights, one can distinguish between a concept Head (sports in the example), and a concept Modi er (e.g. games).
We can naturally model such phenomenon assigning a higher weight to the `dominant concept' (the Head), or speci c features thereof, or by considering only the most prototypical instances of the dominant concept, as discussed in more detail below. 4
rr-Modelling of Over/Underextension and Dominance We start by proposing how to represent prototype-based classi cations by means of the rr operator. We then proceed to, rstly, show how we can capture simple intersections of concepts de ned in terms of prototypes. Secondly, we present a simple modelling of all three phenomena that we have previously discussed (overextension, underextension, and dominance e ect).
The general picture is informally illustrated in Fig. 1, the formal modelling of which we undertake in the this section.
The main motivation is the simple elegance of the approach, which allows to compactly de ne speci c instances of the three problems. Although not strictly more expressive, replacing extremely large disjunctions with short tooth-based expressions may be arguably seen as a more cognitively adequate modelling. Another, related point, is that the relative weights of the participating features are being made explicit.
The rr operators can represent the cognitive approach to concepts based on
p
C = rrt((P1 ; w1); : : : ; (Pm ; wm)) (6) The classi cation under C applies to the objects that have \enough" features in common with the prototype, so to exceed the threshold t. This is a very simple way to model the idea of being close enough with respect to the prototype. We can also individuate the prototypical instances of C as the objects (if they exist) that satisfy all the Pj in C. A di erent and more liberal approach is to collect all those instances that reach the maximal possible value con rming a number of the Pj : we call this set the core of C and compute it as rrmax(C).
In the following paragraphs, we assume that each concept is de ned by means of a prototype, cf. Eq. (6). We focus on two concepts, which are thus de ned by: C = rrs((P1 ; w1); : : : ; (Pm ; wm)) and D = rrt((Q1 ; v1); : : : ; (Qn ; vn)). 4.1
The logical conjunction of two concepts, which provides the intersection of the extension of the concepts, can be simply de ned by means of the u operator of ALC: Example 2. Suppose now we de ne the two concepts Sport and Game as follows: Sport = rrs((DoneForFun; w1); (IsCompetitive; w2);
(DoneToGetFit; w3); (NeedsAthleticAbilities; w4)) (8) Game = rrt((DoneForFun; w1); (IsCompetitive; w2);
(NotSeriousGoal; w5); (InvolvesConcentration; w6)) (9) Let us further assume the following knowledge base:
K1 = fDoneForFun(f); IsCompetitive(f); NeedsAthleticAbilities(f); NotSeriousGoal(f)g It allows one to categorise `football' (f) as an instance of both Sport and Game (i.e., w1 + w2 + w4 s and w1 + w2 + w5 t). In this case, football would get both the weight for being a Sport and the weight for being a Game, therefore belonging to the simple intersection of the two concepts as de ned above. On the other hand, consider a di erent knowledge base, about beerpong (bp):
K2 = fDoneForFun(bp); IsCompetitive(bp); NotSeriousGoal(bp)g In this case, we could imagine that w1 + w2 + w5 t but w1 + w2 would then not enter in the simple intersection of Sport and Game. s: beerpong 4.2
A combination of concepts that returns a set larger than the intersection of the two concepts can be approached in a number of ways. We discuss two strategies. Firstly, we can lower the threshold for the classi cation under C or under D. That is, in the combination of C and D the importance of the list of features de ning C and D is relaxed. So, more instances shall fall under either concepts, returning a larger intersection. For example, when combining boat and house, into a house boat we are not requiring that the instance of the combined concept satisfy all the attributes of the composing concepts, only a su cient number.
Given C = rrs((P1 ; w1); : : : ; (Pm ; wm)) and D = rrt((Q1 ; v1); : : : ; (Qn ; vn)), a family of overextending compositions of C and D can be introduced: C uO D = rrx((P1 ; w1); : : : ; (Pm ; wm)) u rry((Q1 ; v1); : : : ; (Qn ; vn)) (10) where s x or t y or both. In this case, as for the standard intersection, we assume that the (marginal) contribution of the two concepts C and D is equal (they both have to be satis ed). This re ects that the overextending composition is in fact a composition of the two concepts.
The e ect of lowering the threshold for classifying an entity under the composing concepts has the consequence of raising the importance of a (number of) features for the composed concept (whether it is their salience or their diagnosticity). Globally lowering the threshold amounts to assume no information about which speci c feature is important in the combination, whereas directly raising the weight of a number of features entails knowing that those are the important bits, when combining the concepts. Thus, a second strategy for combining concepts that overextends the intersection is by raising the weights of some of the features in C and D.
Example 3. Suppose we de ne again the two concepts Sport and Game as in Equation 8 and Equation 9. Let us also assume the following knowledge base:
K3 = fDoneForFun(w); IsCompetitive(w); NeedsAthleticAbilities(w)g which allows one to categorise `wrestling' (individual w) as an instance of Sport (i.e., w1 + w2 + w4 s) but does not allow to categorise it as a Game (i.e., w1 + w2 < t). The simple intersection would not allow to represent `wrestling' as an instance of the intersection of Sport and Game. To overextend the intersection of the two concepts in our setting, one should represent the intersection as follow: Sport uO Game = rrx (DoneForFun; w1); (IsCompetitive; w2); (DoneToGetFit; w3); (NeedsAthleticAbilities; w4) u rry (DoneForFun; w1); (IsCompetitive; w2); (NonSeriousGoals; w3); (InvolvesConcentration; w4) (11) where the threshold y is set in a way that allows to classify `wrestling' as a Game. 4.3
In a similar manner, we can treat the case of underextensions, that is, by raising the threshold required for classifying the composing concepts. One interpretation of this case is that when combining two concepts, only instances that are more typical of the two concepts can enter the combination of the two concepts. By means of the rrmax, we can de ne underextension operators as follows. Given C = rrs((P1 ; w1); : : : ; (Pm ; wm)) and D = rrt((Q1 ; v1); : : : ; (Qn ; vn)), we set: C uU D = rrmax((P1 ; w1); : : : ; (Pm ; wm)) u rrmax((Q1 ; v1); : : : ; (Qn ; vn)) (12) Example 4. Suppose we de ne the two concepts Sport and Game as before in Equation 8 and Equation 9. Let us also assume that both `basketball' (individual bb) and `ping pong' (individual pp) are in the intersection of Sport and Game, according to the following knowledge base
KB = fDoneForFun(bb); IsCompetitive(bb); NeedsAthleticAbilities(bb);
DoneToGetFit(bb); NotSeriousGoal(bb); InvolvesConcentration(bb);
DoneForFun(pp); IsCompetitive(pp); NeedsAthleticAbilities(pp);
NotSeriousGoal(pp); InvolvesConcentration(pp)g Suppose that we de ne underextending composition of Sport and Game as in Equation (12). Then, basketball would be selected by our operator, as it maximises both Sport and Game, while ping pong will not, as it does not maximise Sport (by failing to satisfy DoneToGetFit).
Another way of introducing underextension is to constrain the intersection of C and D by imposing that the instances must maximise the combination of the features of C and D: CuD urrmax (P1 ; w1); : : : ; (Pm ; wm); (Q1 ; w1); : : : ; (Qn ; vn) .
This de nition is non-equivalent to (12) only in the case of mutually incompatible attributes de ning C and D. We leave this aspect for future work. In the case of dominance, as we have seen, the head of the combination outweighs the other composing concepts. Given C = rrs((P1 ; w1); : : : ; (Pm ; wm)) and D = rrt((Q1 ; v1); : : : ; (Qn ; vn)), we present three strategies to model dominance operators. A rst strategy is to de ne dominance operators as follows: C u1D D = rrs0 (P1 ; w1); : : : ; (Pm ; wm) u rrt0 (Q1 ; v1); : : : ; (Qn ; vn) (13) where s0 s and t0 t.5 In this case, globally raising the threshold of the dominant concept amounts to include the typical instances of C into the composition, whereas globally lowering the threshold of the non-dominant concept includes possibly non-D instances. This means that a sport that is also a game must be primarily a sport .
A second strategy is to unpack (or atten) the de nitions of C and D and raise the weights of the most salient features ? 6= S(C) f1; : : : ; mg of the dominant concept C for the composition, whilst including some required elements R(D) f1; : : : ; ng of D to compute a new minimal threshold. We may set: C u2D D = rr (C;D) (P1 ; w1 ); : : : ; (Pm ; wm); (Q1 ; v1 ); : : : ; (Qn ; vn) (14) where is a function that re-assigns weights such that (wi) = wi > wi for all i 2 S(C) (and (x) = x otherwise) and (C; D) = i2S (wi) + j2R (vj ) is a new threshold that accumulates the in ated weights for the salient features of C and the required weights from D.
A third strategy to de ne a dominant concept would be to collect those instances that maximise the sum of weights of the dominant concept (using e.g. the rrmax operator).
Notice that these types of compositions are generally non-commutative, as the rst combined concept (the head) is more important than the second. Example 5. Suppose we de ne again the two concepts Sport and Game as in Equation 8 and Equation 9, and consider again the knowledge base in Example 5 Here the choice of making the rst concept dominant is merely formal, we are not making any assumption on the linguistic phrasing of this complex concept. 3. If we consider the concept Sport as the dominant one, we could apply Equation 13 to include wrestling in the intersection of the concepts Sport and Game, simply lowering the threshold for being included in Game. This is again a case of overextension, but the idea behind this choice here is that having the most important features of the concept Sport `count more' to enter in the conjunctive concept.
We have a similar result if we apply Equation 14:
Sport u2D Game = rr (Sport;Game) (DoneForFun; w1 ); (IsCompetitive; w2 ); (DoneToGetFit; w5 ); (NeedsAthleticAbilities; w6 )(DoneForFun; v1 ); (IsCompetitive; v2 ); (NotSeriousGoal; v3 ); (InvolvesConcentration; v4 ) (15) where we set (Sport; Game) = (w1)+ (w2)+ (w6)+ (v1)+ (v2). It is worth noticing here that just the weights associated to the concept Sport are raised by the function , as required in Eq. 14: since Sport is the dominant concept, its salient features are considered the most important in this combination. 5
We studied three cognitively and psychologically interesting and important
phenomena that address classi cation problems with a focus on conjunctive
operations on concepts, namely overextension, underextension, and dominance. To
address these phenomena from a formal modelling point of view, we employed
the logic ALCrr int
Future work will be dedicated to further investigate the logical properties of
the operators and their natural extensions, and to employ them in a systematic
fashion to the modelling of salient cognitive features of concepts, concept
combinations in general, and conceptual blending in particular [Eppe et al., 2018;
Confalonie
Another line of research will deepen the comparison with the formal studies
on typicality
Further, we hope to pursue further a deeper exploration of the
correspondence theory between tooth logic, statistical learning and classi cation models,
and concept learning in DL,