Prototype theories are an important family of cognitive theories of concepts that model the classification under a concept in terms of the proximity of an object to the prototype of the concept. While logic-based definitions of concepts are standard in Description Logics and OWL, prototype-based definitions are not directly available, although they are very useful whenever we need to model commonsense concepts or data dependent classifications. We propose a strategy to define prototypes in OWL enriched with SWRL (Semantic Web Rule Language). By means of data properties we model the weighted features of prototypes, and, by using SWRL constraints, we implement the computation of the proximity of an instance to a prototype. We also leverage a foundational ontology to provide semantics of the features occurring in prototype descriptions. We exemplify our treatment in Protégé.
members of a category form a continuum. Since then, prototype theory is mostly understood as providing a Various cognitive theories have been proposed in the summary representation [1], representing the central tenliterature to define and understand the representation dency of a category as a whole. Although there exist of concepts [1]. One prominent theory among these is several variants of prototype theory, their central core is prototype theory, which has received significant formal that it is possible to represent concepts as a weighted list and empirical development [2, 3, 4]. As a result, it has of the attributes most commonly found among concept served as a foundational inspiration for numerous formal instances. The more attributes an individual satisfies, the systems that aim to provide improved models of human more similar it will be to the prototype, and the more concepts in fields such as Logic and Knowledge Repre- typical it will be for the category. sentation [5, 6, 7, 8]. To formally capture this idea, Masolo and Porello [5]
Prototype theory traces its origins back to Wittgen- propose a logical rendering of prototypes by introducing stein’s concept of family resemblance [9], but its empir- weights into the language of first-order logic. The logicical foundation is due to the findings of Eleanor Rosch based rendering of prototypes paves the way towards [2]. Her experiments showed that many concepts exhibit introducing semantic constraints to make the meaning typicality efects (namely, some class members are more of the features or the attributes explicit. Porello et al. [6] representative than others), and that members and non- translate this approach into Description Logic (DL) by introducing novel operators into the syntax of standard 9StehptWemorbkesrh2o6p, 2o0n2F3o,rBmerallina,nGdeCr moganniytive Reasoning, DL languages. These operators take lists of concept de$ daniele.porello@unige.it (D. Porello); scriptions, associate a weight to each of them, and return Guendalina.Righetti@stud-inf.unibz.it (G. Righetti); complex concepts which apply to the instances of the nicolas.troquard@unibz.it (N. Troquard); domain that satisfy a suficient number of the concept roberto.confalonieri@unipd.it (R. Confalonieri); descriptions. A threshold is set to decide what is deemed olivhetrt.pksu:/t/zd@anuineilbezp.oitr(eOll.o.Knuettz()D. Porello); to be suficient, and a new value function is introduced to https://www.unibz.it/it/faculties/engineering/academic-staf/ define the semantics. This function sums up the weights person/40334-guendalina-righetti (G. Righetti); of the features an instance satisfies and returns a number https://www.inf.unibz.it/~ntroquard/ (N. Troquard); which allows expressing instances’ typicality. When the https://www.math.unipd.it/~confa/ (R. Confalonieri); number reaches the threshold, the instance is classified http://www.inf.unibz.it/~okutz/ (O. Kutz) as a member of the concept.
0000-0003-3655-0218 (D. Porello); 0000-0002-4027-5434 (G. Righetti); 0000-0002-5763-6080 (N. Troquard); Threshold concepts for representing prototypes have 0000-0003-0936-2123 (R. Confalonieri); 0000-0003-1517-7354 also been used by Baader and Ecke [7], who propose (O. Kutz) however a diferent interpretation. The semantics makes © 2023 Copyright for this paper by its authors. Use permitted under Creative Commons License use of a prototype distance function defined over weighted CPWrEooUrckReshdoinpgs IhStpN:/c1e6u1r3-w-0s.o7r3g ACttEribUutRion W4.0oInrtekrnsahtioonpal (PCCroBYce4.0e).dings (CEUR-WS.org) alternating parity tree automata, which assign to each egy for representing prototypes in OWL 2 plus SWRL element of an interpretation a distance value expressing and for computing similarity by means of SWRL conits similarity to the prototype. straints. Section 5 combines the SWRL rendering with
Relatedly, typicality operators [10, 11, 8] have also the background ontology based on dolce. Section 6
exbeen proposed in the DL literature to reason non- emplifies our approach. Section 7 discusses how to learn
monotonically about the typicality of instances. In these the weights and the diagnosticity values from a data set.
cases, it is possible to identify the most typical members Finally, Section 8 concludes and indicates future work.
of a concept by assuming a preference order over the
elements of the domain of interpretation: depending on
the system, either the individuals further down or the 2. Prototype theories
ones higher up in the order are deemed as the more
typical instances of a concept. Moreover, prototypes and Prototype theories of concepts are models proposed to
typicality in ontologies have also been approached in explain human conceptualisations and classification of
[12, 13]. objects under concepts [
In this paper, we propose to model prototypes in Accordingly, a concept is defined by a prototype, a sumOWL 2 using SWRL rules [14].1 The Semantic Web Rule mary, abstract representation that captures the central Language (SWRL) is an extension of OWL (Web Ontology tendency of a category, namely the most typical instances Language), the most widespread language for authoring or objects belonging to the class. Classification is not alontologies. SWRL allows for the specification of Horn- ways deemed to be crisp: the degree of classification of like rules within an OWL ontology. an object under a concept is instead rendered by means
SWRL is particularly suitable for our purposes because of proximity, or similarity, of the object to the prototype.
it combines OWL class and property expressions with Notwithstanding this common core, there exist several
built-in relations, such as equality, inequality, and arith- variants of prototype theories.
metic operations. These will be instrumental for the According to Rosch and Mervis [2], a prototype is an
representation of prototypes within the ontology and to (unstructured) list of the features or attributes usually
compute the similarity measure that enables classifica- found across the instances of the concept we are
considtions, cf. [
After discussing three variants of prototype theories importance in describing the concept. These weights are
with their measures of similarity, we present our pro- determined based on the frequency distribution among
posal to model prototypes by means of data properties exemplars of the concept, where features more frequently
and SWRL constraints. To ascribe meanings to features observed receive higher weights.
or attributes occurring in prototype descriptions, and to Suppose that the features or attributes that describe the
enable reasoning with them, we introduce a background prototype are given by Q = {1, . . . , }. The prototype
foundational ontology (in particular, we use dolce, cf. for a concept is then represented by the following set of
[
The main novelty of our approach, in comparison to The similarity of an exemplar to a concept is meathe previously mentioned related work, lies in the com- sured through its degree of family resemblance, which can bination of three key ingredients: ) the use of SWRL simply be computed by using the following sum, where to implement computations of similarities, ) the use of the 1 , . . . , are the weights in {1, . . . , } assoa foundational background ontology to provide seman- ciated to prototypical features that are also exhibited by tics of features, and ) the use of threshold concepts object . to interface prototypes with ontologically crisp information. The threshold mechanism allows for importing = ∑︁ (1) prototype-based classifications without altering the se- ∈{1,...,} mantics or the reasoning services of OWL. Hampton’s version of prototype theory is similar to
The remainder of this paper is organised as follows. the one originally proposed by Rosch and Mervis, see In Section 2, we overview prototype theories. In Sec- [4]. However, Hampton suggests a slightly more elabotion 3, after a brief overview of the ontology dolce, we rate way to compute the similarity between objects and discuss how to represent the attributes and the attribute- prototypes, which also takes into account the degree to values of a prototype in terms of qualities and quality which an object exhibits a feature. structures of dolce. Section 4 presents a general strat- Accordingly, the similarity between objects and prototypes is now computed by the following formula:
= ∑=︁1( × (,)) (2) rS{eh(dRaepdea,) pispCol“eltohre),ins(uRmoruebnperrde,osfeSnvhotaeptede)s}",ibnywfahvetohrueer ofCdobelseocrinrigp(rtReioespdn.
According to Hampton, and (,) range as follows: (resp. Round), e.g. the number of instances that are 0 ≤ ≤ 1 and indicates the importance of the -th deemed to be red, which in principle could be diferent feature of the prototype for the classification, and − 1 ≤ from the salience established by the prototype (cf. [3]) . (,) ≤ 1 represents the degree to which the instance To compute Tversky’s contrast rule, we define, for each has , see [4]. attribute , three sets of values (in N), cf. [3], p. 491.
A more structured representation of prototypes is pro- For each attribute-value , lists the number of posed by Smith and colleagues. In [3], a prototype is defined by three ingredients, namely: vthoetensuambobuetrofvtohtaetsatrheactoamrempornoptoertoand an;d ¯not tloists, ) an attribute-value structure, that is, a set of attributes while ¯ lists the number of votes proper to and not of objects associated with values (e.g. colour-red, weight- to . heavy, shape-round), ) an indication of the salience of the attribute values, = { | ( , ′) ∈ , ( , ′′) ∈ and = ) an indication of the diagnosticity of the attributes for (′, ′′)} the classification under the concept.
In [3], the salience value of an attribute-value (e.g. ¯ = { | ( , ′) ∈ , ( , ′′) ∈ and = colour-red) captures “the subjective frequency with ′ − ′′, if ∈ N, 0 otherwise} which the value occurs in instances of the concepts, and the perceptibility of the value” (p. 489). Salience values ¯ = { | ( , ′) ∈ , ( , ′′) ∈ and = are introduced to account for attribute-values that are ′′ − ′ if ∈ N, 0 otherwise} closely related to the concept. E.g., people are faster at establishing that apples are red than apples are round, The similarity between an instance and a prototype suggesting that red is more salient than round in the is then computed by the Tversky rule: prototype for apple, cf. [3].
By contrast, diagnosticity values measure how useful = ∑︁ { × (∑︁ − ∑︁ ¯ − ∑︁ ¯ )} (5) the attribute is in separating instances of the concept ∈A from instances of contrasting concepts.
In this setting, a representation of a prototype can be Intuitively, Tversky’s measure evaluates similarity by given by means of a set of weighted attribute-values. Sup- taking into account commonalities and diferences bepose that the attributes of interest are A = {1, . . . , }. tween the description of the prototype and the descripMoreover, for each attribute , assume that the possible tion of the instance . values of attribute are Q = {1 , . . . , }. We in- We sketch an example discussed in [3]. We restrict dicate by the diagnosticity of some attributes . By ourselves to two attributes, Color and Shape, with val , we denote the salience of the -th value of attribute ues of color in {Red, Brown} and values for shapes in . A prototype can then be specified by the following {Round, Squared}. A prototype for the concept Apple set of triples, for some ∈ A and 1 , . . . , includes then the following triples: ∈ Q (cf. [3], Figure 1).
= {(1 , , 1 ), . . . , ( , , )}
(3) described as follows:
In this setting, the similarity between an object and the prototype is computed employing Tversky’s contrast rule (see [3]). To employ this rule, the authors also represent objects, or instances, as descriptions in terms of sets of weighted attribute-values. Object descriptions are sets of pairs of attribute-values and salience values (while diagnosticity specifically operates for prototypes). For some ∈ A and 1 , . . . ,
∈ Q , an instance is = {(1 , ), . . . , ( , )} (4) (Red, Color, Color), (Brown, Color, BCroolowrn)
Red (Round, Shape, SRhoaupned), (Squared, Shape, SSqhuaapered)
Color for classifying apples is higher than the diagnosticity of the attribute Shapes, while the salience of the value Red for apples is higher than the salience of Brown.
The similarity measures that we have presented define a value for the proximity of an instance to the prototype.
As such, classification under a concept is graded. To obtain the set of entities that falls under a concept (i.e. the extension of the concept) by means of the prototype mechanism of classification, a threshold is set, cf. [4]. Definition 1. Given a threshold , an object belongs to a quality space, which is modelled as an abstract region, the extension of the concept if and only if i.e. a subclass of Abstract. Qualities are also divided into ≥ . (6) tqyupaelist,yssupcahceassacsosolocria,twedeitgoheta, cshhqapuae,liteytct.ypqeuaalrietiienss.pTirhede
We shall use the threshold mechanism to introduce a by Gärdenfors quality dimensions, cf. [
We conclude this section by noticing that a prototype Figure 1. may include complex features that are not reduced to attribute value pairs. For instance, complex properties such as “has legs” or “you can drive it” are used by [2] in the summary description. We shall use the word features when we wish to emphasize general properties that may occur in the description of a prototype.
and the concept defined by the prototype within an ontological setting. Then, in the next section, we will approach how to render prototypes in OWL 2 plus SWRL.
By embedding prototypes within a rich ontological framework, we can provide a formal representation of the semantics of the features involved in the description of the prototype and this, in turn, enables reasoning about the content of those features. Figure 1: Fragment of the dolce taxonomy and relations for
We place our discussion within the foundational ontol- quality and qualia.
ogy dolce (Descriptive Ontology for Linguistic and
Cognitive Engineering), see [
Qualities are properties of objects that can be perceived individual qualities of the object, and the qualia of the
or measured. Qualities are dependent entities, in that quality, i.e. the corresponding quality regions.
they are strictly attached to their objects, their bearers The ascription of an attribute value to an entity is then
(they inhere in their object). For this reason, dolce terms rendered via the ql and qt relations. For an axiomatisation
them individual qualities. The color of this rose is never of these relations in dolce, we refer to [
To use the representation of quality and quality struc- the strategy in Protégé on a concrete case in Section 6. tures of dolce in OWL, we shall simplify it a little.4 We We assume that the information provided by a protoview the ontology in OWL as providing a snapshot at type is an input of our representation, that is, we assume given time of the information about a domain of objects. to have: the set of attributes or features, F = {1, . . . , So, we omit the time argument in ql, which remains im- }, their values, Q = { , . . . , }, the diagnos1 plicit. The reason behind this choice is to avoid dealing ticity of each features , the salience of each with ternary relations in OWL, which makes the render- value, and the values (,), the degree to which each ing more complex. Thus, ql(, ) is now defined as a bi- entity has feature value .5 nary relation, it relates an individual quality to a quale , We sometimes use the labels , for general features, while the time of the association remains implicit. We can not to restrict to attribute values, according to [4], while also introduce a defined relation hasQuale, which is the we use labels to intend attributes as quality structures. composition of ql and qt. For instance, hasQuale(, red) We remind the definition of SWRL atoms and rules, cf. is the composition of the relations qt(, ) and ql(, red). [14]. The syntax of SWRL extends the usual syntax of We shall use these relations to simplify the rendering in OWL by adding (individual) -variables (as we shall see, OWL. they range over the individuals of the domain) and (data)
As usual in ontological modelling, dolce provides bi- -variables (which range over data values). nary classifications of objects under properties and rela- An -object is an individual name or an individual tions; as such, an entity has or has not a certain quale. By variable, a -object is a data literal or a data variable. contrast, prototype theories use graded attribute values, Atoms are then: ) applications of concept description to those measured by the function (,), e.g. in Equation 2. -objects, ) applications of an object property to a pair Thus, to enable the ontology to acquire the bits of infor- of -objects, ) the application of a data property to a mation about the quale of an entity, we need a threshold pair consisting of an -object and a -object. Variables to state that if (,) is “large enough” (e.g. (,) = 1), are denoted by ?x. SWRL rules consist of an implicathen the entity has the quale indicated by the attribute tion between an antecedent (the body) and a consequent value . We shall see how to render this aspect in the (the head). The antecedent and the consequent contain next section. sets of atoms, interpreted conjunctively. The antecedent
As we mentioned, prototypes may include general provides the conditions that need to be satisfied for the
properties, besides attribute-values. The ontological ren- rule to be applicable. The consequent specifies the
logdering of such properties is a matter for a dedicated anal- ical consequences that must hold when the antecedent
ysis. E.g. “has legs” may be represented by using the “part is satisfied. For an exhaustive overview of SWRL syntax
of” relation, where the semantics of such a relation is and semantics, we refer to [14]. A SWRL rule has the
captured by a mereological theory, which is included in following form, where a1, ..., an and b1, ..., bn are atoms,
dolce. We refer to [
The contribution of a foundational ontology such as a1 ^ ... ^ an -> b1 ^ ... ^ bn dolce to our approach is intended to provide the semantics to reason about qualities and qualia – and more gen- To import the information required by the prototype erally about features – and to interface the classifications theory, we shall introduce number of data properties. To provided by means of prototypes with background infor- make the domain of such properties semantically meanmation about the domain. For instance, it is by means of ingful, we shall leverage a background ontology, however the information provided by the ontology that we can we postpone this discussion to Section 5. For now, we infer that an entity cannot be wholly red and brown at implicitly assume that they associate numbers to objects the same time, or that it cannot have both a round and a of the ontology. squared shape.
We illustrate the general mechanism for representing in SWRL the computations required by the prototype theories. In the next section, then, we discuss the integration with a background ontology and we exemplify
php/dolce/
We start by representing the Tversky formula, cf. Equation 5, which is the most involving case. To do that, we assume that the prototype is reified , i.e. that there is an individual name to which we can ascribe certain attribute-values. In this setting, as we discussed in
ence in prototype theories. We discuss how to retrieve them by means of learning algorithms in Section 7. Section 2, a prototype is described by salience numbers swrl:greaterThanOrEqual, swrl:equal implefor attribute-values and by diagnosticity numbers of at- ment subtraction, addition, ≥ and = (respectively). tributes. This information can be rendered by means of Given the values of hasPI-AJ(a,?s), the following instantiated data properties (intended as hasP’I-AJ(a,?s), and hasPI’-AJ(a,?s), we facts of the ontology), for each attribute , attribute- can compute the full value of salience of an attribute by value , salience , of the prototype . hasAJSal, as follows.
{hasAJQISal(p,sij), hasAJDia(p,dj)} hasAJDia associates to objects the number (the diagnosticity of the attribute ), while hasAJQISal associates the salience value (the salience of the -th To simplify the presentation, we are using -?si to value of attribute ). indicate the inverse of the value of ?si, instead of adding
In this setting, an instance is also represented by a set a further rule. of data properties, which associate the salience values. Then, we multiply the value of hasAJSal by the diagnosticity of , to obtain hasAJVal. hasPI-AJ(a,?s1) ^ hasP’I-AJ(a,?s2) ^ hasPI’-AJ(a,?s2) ^ swrl:add(?s,?s1,-?s2,-?s3) -> hasAJSal(a, ?s)
To represent Equation (5) in SWRL, we need to com- hasAJDia(a,?d) ^ hasAJSal(a,?s) pute the values of ∑︀ , ∑︀ ¯ , and ∑︀ ¯ . ^ swrl:multiply(?v,?d,?s) -> hasAJVal(a,?v)
To do that, we introduce the following new data properties.
Finally, the Tversky similarity, represented by the data property hasTVal, is computed by the following rule: hasPI-AJQI associates the salience for common attribute values in hasPI-AJ associates the value of ∑︀ . hasPI’-AJQI associates the values for in ¯ . hasPI’-AJ associates the value of ∑︀ ¯ . hasP’I-AJQI associates the values for in ¯ . hasP’I-AJ associates the value of ∑︀ ¯ . hasAJQISal(a,?s’) ^ hasAJQISal(p,?s’’) ^ minimum(?s,?s’,?s’’) -> hasPI-AJQI(a,?s)
sure, cf. Equation (2). We introduce the data properties hasQJWei to introduce the weights of each feature
To compute the values of those data properties, we value and hasQJDeg to set, for each instance , the value introduce the following SWRL constraints. (,). The data property hasQJVal serves to compute the products of weights and degrees, by means of the following rule. hasA1Val(a,?v1) ^ ... ^ hasAMVal(a,?vm) ^ swrl:add(?t,?v1,...?vm) -> hasTVal(a,?t) 4.2. Hampton’s similarity hasPI-AJQ1(a,?s1) ^...^ hasPI-AJQM(a,?sm) ^ swrl:add(?s,?s1,...,?sm) -> hasPI-AJ(a,?s) hasAJQISal(a,?s’) ^ hasAJQISal(p,?s’’) ^ swrl:subtract(?s,?s’,?s’’) ^ swrl:greaterThanOrEqual(?s,0) -> hasP’I-AJQI(a,?s) hasAJQISal(a,?s’) ^ hasAJQISal(p,?s’’) ^ swrl:subtract(?s,?s’,?s’’) ^ swrl:lessThan(?s,0) -> hasP’I-AJQI(a,0) hasQJDeg(?x,?d) ^ hasQJWei(?x,?w) ^ swrlb:multiply(?v,?d,?w) -> hasQJVal(?x,?v)
Then, we compute its value by means of the following SWRL constraint: hasQ1Val(?x,?v1) ^ ... ^ hasQMVal(?x,?vm) ^ swrlb:add(?v,?v1,...,?vm) -> hasHVal(?x,?v) hasP’I-AJQ1(a,?s1) ^...^ hasP’I-AJQM(a,?sm) ^ swrl:add(?s,?s1,...,?sm) -> hasP’I-AJ(a,?s) Notice that the case of (,) ∈ {0, 1} corresponds
Analogous rules are set for hasPI’-AJ. The con- quite closely to the threshold operators defined in [
5/57/SWRL-IQ_manual.pdf
axioms of the ontology to reason about attribute-values ascriptions. For instance, the reasoner shall exclude that objects might have two qualia of the same type.
In the previous section, we introduced a number of data In Smith and Osherson [3], statements properties to associate the values required by the compu- hasAJQuale(?x,qij) are bivalent, therefore the tations of similarities. Those data properties are simply inputs of the prototype mechanism can be retrieved devices to implement by SWRL the prototype-based clas- by means of qualia ascriptions. Once hasAJQuale is sifications. That is, at that level, they are not attached to instantiated for certain in POB and its quale , the any ontological, or semantic, information. following rules populate the data properties by ascribing
To link the features and attributes occurring in proto- to the pertinent data about salience and diagnosticity. type descriptions to their ontological understanding, we proceed as follows. We present the approach for features hasAJQuale(?x,qij) -> hasQIJSal(?x,sij) that are construed as attribute-values, which can be rep- hasAJQuale(?x,qij) -> hasAJDiag(?x,dj) resented ontologically in terms of qualities and qualia.
Each type of feature, e.g. parthood, would in principle In Hampton’s view, we may not have discrete
inrequire a dedicated ontological analysis, see [
We define the background ontology to be an excerpt hasQJDeg(?x,?y) as inputs of the mechanism. of dolce, see Section 3. Accordingly, we partition the By means of SWRL constraints, as we have seen in class PhysicalRegion into the types of attributes that Section 4, we can compute the proximity to the prototype we are considering (e.g. ColorRegion, ShapeRegion, (according to Tversky or Hampton), using the relevant etc.). The elements of those classes are ontologically data that are stored by the instantiated data properties. qualia, corresponding to the attribute-values listed in To integrate SWRL computations within the OWL reathe prototype description. Thus, given attribute types soner – and in doing so, enable reasoning by using the A = {1, . . . , }, PhysicalRegion is partitioned information provided by the background ontology – we into disjoint classes of attribute-values. add to the ontology the class C, corresponding to the
In usual ontological modelling, classifications are bi- set of entities that are classified under C by means of its nary, an entity has or has not a certain attribute-value prototype. To obtain the extension of the concept, we (an entity is red or it is not). To embed soft information have to set a threshold . We may do that by adding the into the discrete world of ontologies, we can again ex- following SWRL constraints: ploit a thresholding mechanism. We introduce the binary relations (i.e. the object properties) hasAJQuale, one for hasHVal(?x,?y) each attribute type, which relate entities to qualia of the ^swrlb:greaterThanOrEqual(?y,t) -> C(?x) (H) specific type, see Section 3. 7 hasTVal(?x,?y)
More specifically, the domain of hasAJQuale is POB ^swrlb:greaterThanOrEqual(?y,t) -> C(?x) (T) (physical objects) and the range is the -th element of the partition of PhysicalRegion . The hasAJQuale By means of those constraints, the class C of the ontolrelations are set as functional, an object has only one ogy can be populated with all the instances that meet the value for each attribute type (at a time). threshold, according to the similarity equations, either
When we allow for degrees of ascriptions of attribute- (2) or (5), and to the equivalence (6). The class C is in values, we have to decide a threshold for inferring that general placed as a subclass of POB in dolce taxonomy. an entity has a quale , whenever the entity has it up The reason is that C contains entities that have physito a certain degree. A very cautious way of setting this cal qualities, which must be ascribed to physical objects threshold is expressed by the following SWRL constraint: according to dolce. However, if we know that we are classifying more specific types of entities, a more refined hasQJDeg(?x,1) -> hasAJQuale(?x,qj) placement within dolce hierarchy can be decided.
E.g., if we know that we are classifying animals, C can
By means of this constraint, we can populate the object be placed as a subclass of AgentivePhysicalObject,
property hasAJQuale of all those pairs (of entities and which refines POB and enables more inferences, cf. [
Once the class C is populated by means of the SWRL rules, then we can treat it as a usual class of the ontology, and use any reasoner to obtain the properties of C-entities that are entailed wrt. the background ontology.
Note that, to execute any SWRL rules, each variable occurring in the antecedent has to be instantiated. Since hasQJDeg range in [− 1, 1], according to Hampton, instances that are deemed not have the -th attribute value, might be set to -1. In the case of binary (yes/no) ascription of attribute values, hasQJDeg ranges in {0, 1}. Stating that an entity does not have feature amounts to setting the value of hasQJDeg to 0, which entails not weighing the in the computation of proximity.
We have proposed a theory based on OWL and SWRL to enable reasoning about the attributes and attributevalues occurring in prototype theories. As usual in applied ontology, ontological theories are used to specify meanings of concepts, mainly by excluding unintended models, see [23]. For this reasons, it is interesting to have a look at the models of the theory.
An (abstract) interpretation of OWL, cf. [14], is given by:
ℐ = (, , , , , ) where is a non-empty set of resources (the domain of ℐ), ⊆ is a set of typed literals (for datatypes), is a mapping from classes and datatypes to and (respectively), is a mapping from (object and data) properties to relations on , is a mapping from typed literals to elements of , and is a mapping from individual names to (owl: Thing) ⊆ .
We refer to [24] and [25] for the usual semantics of OWL. An interpretation satisfies an ontology if it satisifes all axioms and facts in it. We rephrase how to extend the interpretation to SWRL constraints, cf. [14].
To interpret SWRL atoms, the interpretation ℐ is extended to ℐ′, assigning an interpretation also to -variables and -variables, as follows: ) () ∈ (owl: Thing), for any -variable ; ) () ∈ , for any -variable . The conditions of satisfaction of SWRL atoms, given ℐ′, are the following ones, for concept description , object property , and data property : ℐ′ satisfies an antecedent a1, ..., an of a rule if it is empty ( = 0) or ℐ′ satisfies every atom ai. ℐ′ satisfies a consequent b1, ..., bn if it is empty or ℐ′ satisfies every atom bi.
A rule is satisfied by an interpretation ℐ if, for every extension ℐ′ of ℐ, if ℐ′ satisfies the antecedent, then ℐ′ satisfies the consequent.
An interpretation satisfies an ontology containing (SWRL) rules if it satisfies every axioms and facts of the ontology and every rules.
Let be a concept name populated by means of (e.g.) Hampton’s similarity and by means of the thresholding constraint, cf. rule (H) with threshold , where ∈ . Given an interpretation ℐ, the extension of in ℐ can be computed by means of ℐ′ and it shall include the set of entities that satisfy the rule (H), for every extension ℐ′.
ℐ′ () ⊇ { ∈ ℐ′ (owl: Thing) |
there exists ∈ , (, ) ∈ ℐ′ (hasHVal) and (, ) ∈ ℐ′ (swrlb:greaterThanOrEqual)}
tion of the concept is termed knowledge-independent in
[
Our interpretations of interest are those that satisfy the background ontology dolce as well as the instantiated data properties. We denote by the ontology obtained by joining dolce with the required facts about the instantiated data properties. Let ℐ any interpretation that satisfies . The rules that we have introduced are intended to be interpreted wrt. interpretations that satisfy , i.e ℐ′. However, deciding a single particular interpretation is too demanding. For instance, ℐ′ may associate values of ℐ′ (hasHVal) for non-named individuals.
What we wish is to allow for any interpretation that satisfies and that applies the prototype classifications explicitly to those individuals for which we have actually instantiated the data properties.
This view of the interpretation of C is termed
knowledge-dependent in [
interpreted as data literals. There is a more standard
way to present the definition of a knowledge-dependent
interpretation of in terms of interpretations, cf. [
We render the prototype-based concept Elephant in ton’s and Rosch’s similarity measures. The diference OWL plus SWRL, discussing, for reasons of space, only in the two measures is in the range of the data properthe variants proposed by Hampton and Rosch, while we ties establishing degrees: while in the case of Hampton, leave Tversky’s rule for future applications, cf. Section the value ranges in the interval [− 1, +1], in the case of 2.9 Rosch it is in {0,1} (1, if the instance has the feature, 0 if
In this case, a prototype is a list of features associated it doesn’t). with weights, expressing their importance. Simplifying, To link data properties with object properties, we may suppose that the prototype description of Elephant is: add the following rules: hasGreyDeg(?x,1) -> hasColorQuale(?x,grey) hasLargeDeg(?x,1) -> hasSizeQuale(?x,large) hasTrunkDeg(?x,1) -> hasPartSomeTrunk(?x) Elephant = {(hasTrunk, 3), (Large, 2), (Grey, 2)}
interpret as two attributes (Size and Color) with
attribute-values (large and grey). Our background Where hasPartSomeTrunk is a defined class,
characontology includes the taxonomy of dolce and it par- terised by ∃hasPart.Trunk.10
titions the class PhysicalRegion into two subclasses Then, the data property hasElephantVal is
introSize and Color, containing as elements (individual duced to compute the Hampton similarity of instances
names) large and grey (respectively). We also have with the prototype.
a feature hasTrunk, which can be interpreted ontologi- The values associated with this data property are
comcally in terms of parthood, rather than as a quality struc- puted by the SWRL constraint:
ture. The feature hasTrunk is interpreted by the
deifned class ∃hasPart.Trunk, where Trunk is a sub- hasTrunkVal(?x,?v1) ^ hasLargeVal(?x,?v2)
class of POB and hasPart is the mereological relation ^ hasGreyVal(?x,?v3) ^
of dolce. Moreover, we introduce the class Elephant swrlb:add(?v,?v1,?v2,?v3)
and we may assume that it is placed under the class -> hasElephantVal(?x,?v)
AgentivePhysicalObject of dolce, cf. [
To render prototypical classifications of Elephant by ship to the class defined by the prototype. We can then means of SWRL, we introduce the following (functional) populate the class Elephant by means of the following data properties: SWRL constraint: hasTrunkWei, hasLargeWei, and hasGreyWei hasTrunkDeg, hasLargeDeg, and hasGreyDeg hasTrunkVal, hasLargeVal, and hasGreyVal hasTrunkDeg(?x,?d) ^ hasTrunkWei(?x,?w) ^ swrlb:multiply(?v,?d,?w) -> hasTrunkVal(?x,?v) 9The ontology implementing this toy example can be found at https://github.com/diporello/OntologyForPrototypesSWRL/blob/ main/TestPrototypeTheoryOntologySWRL.owx. The documentation for the SWRL plug-in can be found at https://github.com/ protegeproject/swrltab-plugin. hasTrunkValue(dumbo,3), hasLargeValue(dumbo,0), hasGreyValue(dumbo,2) 10This definition is required to run the SWRL plug-in for Protégé. swrlb:greaterThanOrEqual(?y,t) ^ hasValueElephant(?x,?y) -> Elephant(?x)
Regarding Hampton’s similarity measure, we might Based on this encoding, a data set can be defined conhave, e.g.: taining positive and negative examples about the concepts that may appear or not as the concepts required by hasTrunkDeg(dumbo,1), the description of an ‘Elephant’11. hasLargeDeg(dumbo,0.3), Then, the task will be to learn a linear classification hasGreyDeg(dumbo,0.9). model of the sort () = 1, if ∑︀ () + ≥ 0, In this case, we obtain: () = 0, otherwise, that will approximate on the basis of 1, . . . , . Once the weights 1, . . . , and hasTrunkVal(dumbo,3), the constant term are learned, these can be used to dehasLargeVal(dumbo,0.6), ifne the prototype concept as {(1, 1), . . . , (, )} hasGreyVal(dumbo,1.8) where = − is set as the threshold for the membership to the class defined by .
If the threshold is set to 4 (() = 4), then the SWRL For the case of the Tversky measure, that requires to engine can infer that the threshold is met and that Dumbo diferentiate the contribution of salience and diagnosis thus included (in both cases) as an instance of the class ticity, we could assume that the salience of a feature is Elephant. That is, the class Elephant is populated by ‘learned’ from domain experts, e.g. by asking them what the instance dumbo, which then inherits the ontological attribute-values make the diference in classifying a cerdescription coming from the axioms of the background tain concept, or obtained from the numbers of instances ontology. For crisp ascriptions of degrees, the ontology that are present in the data. The diagnosticity could be will also populate the following object properties: treated, instead, as the weight of a feature. In this way the previous learning scenario could be applied. hasTrunkDeg(dumbo,1) -> hasPartSomeTrunk(dumbo) A diferent setting was illustrated in [ 27] where the hasGreyDeg(dumbo,1) -> hasColorQuale(dumbo,grey) authors proposed to learn Tversky’s similarity (and a weighted variant of it) of two images based on the concept features contained in the images. In their setting, the 7. Discussion: Learning Weights learning data are represented by pairs of sets of features which inform about the semantic similarity or dissimilarIn the above examples, it was assumed that the weights ity between objects. Then, the problem of learning the associated with the prototype concepts (modelled accord- semantic similarity between two objects is modelled as ing to the diferent variants) were given. We depict a an optimization problem using a contrastive loss funcscenario where we can learn a prototype concept from tion. This approach could be generalised to the case in data. If the prototype concept to be learned is treated which we learn a similarity metric from a given set of as a linear classification model, then it is possible to use instances of a prototype. The similarity measure could standard linear classification algorithms (such as the per- then be used as a threshold classifier to decide for the ceptron algorithm, logistic regression, or linear SVM) to membership of an instance to the class defined by a given learn its weights and its threshold, given a set of asser- prototype concept . tions or facts about individuals. A recent related work is dedicated to study the com
This setting was introduced in [
Then, each individual in the knowledge base induces a pair ⟨⃗(), ()⟩ where ⃗() ∈ {0, 1} is such that its -th element ⃗() = 1 if |= (), and it is 0 otherwise; likewise, () = 1 if |= (), and () = 0 otherwise. This setting applies to the case where classifications under features or attribute-values are binary. 11The paper [26] proposes a method for learning the weights in a
way that such requirements can be captured.
Research Project PRIN2020 2020SSKZ7R: BRIO – Bias, Risk and Opacity in AI, https://sites.unimi.it/brio/ We presented an environment based on OWL 2 and SWRL to implement the main aspects of prototype theories. We used data properties to store information about weights, degrees, and salience, and we used rules and the builtin SWRL operators to compute the similarity between instances. As we have demonstrated, the SWRL setting is suficient to implement the main aspects of prototype theories. SWRL constraints allow also for populating the ontology (its classes and object properties) with information that is deemed suficiently crisp. By adopting a background ontology, we enabled a certain level of formal description of the attributes, the attribute-values, and the general features occurring in the prototype descriptions. Also, we introduced the means to integrate prototypical classifications with ontological, feature-independent information.
Future work is planned in three directions. Firstly, we are interested in exploring the learning algorithms for weights, salience, and diagnosticity values. Secondly, we plan to approach threshold and typicality operators, cf. [7, 11, 8, 6], to implement them, or approximate them, by means of SWRL, and to extend our approach to more expressive threshold operators, e.g. including counting capabilities [29]. Finally, we are interested in exploring the SWRL capacity to provide an ontological rendering of the exemplar view of concepts, cf. [30].