THE COMPUTATIONAL REPRESENTATION OF CONCEPTS

IN FORMAL ONTOLOGIES

Some General Considerations

Marcello Frixione and Antonio Lieto

Dipartimento di Scienze della Comunicazione, Università di Salerno

via Ponte Don Melillo, I-84084 Fisciano (Salerno), Italy

Keywords: Concept Representation, Description Logics, Non Monotonic Reasoning, “Dual Process” Theories.

Abstract: Within cognitive science, the “concept of concept” results to be highly disputed and problematic. In our

opinion, this is due to the fact that the notion itself of concept is in some sense heterogeneous, and

encompasses different cognitive phenomena. This results in a strain between conflicting requirements, such

as, for example, compositionality on the one side and the need of representing prototypical information on

the other. This has several consequences also for the practice of knowledge engineering and for the

technology of formal ontologies. In this paper we propose an analysis of this state of affairs. As a possible

way out, in the conclusions we suggest a framework for the representation of concepts, which is inspired by

the so called dual process theories of reasoning and rationality.

1 INTRODUCTION

Computational representation of concepts is a

central problem for the development of ontologies

and for knowledge engineering. Concept

representation is a multidisciplinary topic of

research that involves such different disciplines as

Artificial Intelligence (AI), Philosophy, Cognitive

Psychology and, more in general, Cognitive Science.

However, the notion of concept itself results to be

highly disputed and problematic. In our opinion, one

of the causes of this state of affairs is that the notion

itself of concept is in some sense heterogeneous, and

encompasses different cognitive phenomena. This

results in a strain between conflicting requirements,

such as, for example, compositionality on the one

side and the need of representing prototypical

information on the other. This has several

consequences for the practice of knowledge

engineering and for the technology of formal

ontologies

In this paper we propose an analysis of this

situation. The paper is organised as follows. In sect.

2 we point out some differences between the way

concepts are conceived in philosophy and in

psychology. In sect. 3 we argue that AI research in

some way shows traces of the contradictions

individuated in sect. 2. In particular, the requirement

of compositional, logical style semantics conflicts

with the need of representing concepts in the terms

of typical traits that allow for exceptions. In sect. 4

we review some attempts to resolve this conflict in

the field of knowledge representation, with

particular attention to description logics. In the

conclusions (sect. 5) we sketch a possible way out,

which is inspired by the so called dual process

theories of human reasoning and rationality,

according to which the existence of different types

of reasoning systems is assumed. Indeed, it is our

opinion that a mature methodology to approach

knowledge representation and knowledge

engineering should take advantage also from the

empirical results of cognitive psychology that

concern human abilities.

2 CONCEPTS IN PHILOSOPHY

AND IN PSYCHOLOGY

Within the field of cognitive science, the notion of

concept is highly disputed and problematic.

Artificial intelligence (from now on AI) and, more in

general, the computational approach to cognition

reflect this state of affairs. Conceptual representation

396

Frixione M. and Lieto A..

THE COMPUTATIONAL REPRESENTATION OF CONCEPTS IN FORMAL ONTOLOGIES - Some General Considerations.

DOI: 10.5220/0003095903960403

In Proceedings of the International Conference on Knowledge Engineering and Ontology Development (KEOD-2010), pages 396-403

ISBN: 978-989-8425-29-4

 2010 SCITEPRESS (Science and Technology Publications, Lda.)

seems to be constrained by conflicting requirements,

such as, for example, compositionality on the one

side and the need of representing prototypical

information on the other.

A first problem (or, better, a first symptom that

some problem exists) consists in the fact that the use

of the term “concept” in the philosophical tradition

is not homogeneous with the use of the same term in

empirical psychology (see e.g. Dell’Anna and

Frixione 2010). Briefly

, we could say that in

cognitive psychology a concept is essentially

intended as the mental representations of a category,

and the emphasis is on such processes as

categorisation, induction and learning. According to

philosophers, concepts are above all the components

of thoughts. Even if we leave aside the problem of

specifying what thoughts exactly are, this requires a

more demanding notion of concept. In other words,

some phenomena that are classified as “conceptual”

by psychologists turn out to be “nonconceptual” for

philosophers. There are, thus, mental representations

of categories that philosophers would not consider

genuine concepts. For example, according to many

philosophers, concept possession involves the ability

to make explicit, high level inferences, and

sometimes also the ability to justify them (Peacocke

1992; Brandom 1994). This clearly exceeds the

possession of the mere mental representation of

categories. Moreover, according to some

philosophers, concepts can be attributed only to

agents who can use natural language (i.e., only adult

human beings). On the other hand, a position that

can be considered in some sense representative of an

“extremist” version of the psychological attitude

towards concepts is expressed by Lawrence

Barsalou in an article symptomatically entitled

“Continuity of the conceptual system across species”

(Barsalou 2005). He refers to knowledge of scream

situations in macaques, which involves different

modality-specific systems (auditory, visual, affective

systems, etc.). Barsalou interprets these data in

favour of the thesis of a continuity of conceptual

representations in different animal species, in

particular between humans and non-human

mammals: “this same basic architecture for

representing knowledge is present in humans. [...]

knowledge about a particular category is distributed

across the modality-specific systems that process its

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Things are made more complex by the fact that also within the

two fields considered separately this notion is used in a

heterogeneous way, as we shall synthetically see in the following.

As a consequence, the following characterisation of the

philosophical and psychological points of view is highly

schematic.

properties” (p. 309). Therefore, according to

Barsalou, a) we can speak of a "conceptual system"

also in the case of non human animals; b) also low-

level forms of categorisation, that depend on some

specific perceptual modality pertain to the

conceptual system. Elizabeth Spelke’s experiments

on infants (see e.g. Spelke 1994; Spelke and Kinzler

2007) are symptomatic of the difference in approach

between psychologists and philosophers. Such

experiments demonstrate that some extremely

general categories are very precocious and

presumably innate. According to the author, they

show that newborn babies already possess certain

concepts (e.g., the concept of physical object). But

some philosophers interpreted these same data as a

paradigmatic example of the existence of

nonconceptual contents in agents (babies) that had

not yet developed a conceptual system.

2.1 Compositionality

The fact that philosophers consider concepts mainly

as the components of thoughts brought a great

emphasis on compositionality, and on related

features, such as productivity and systematicity, that

are often ignored by psychological treatments of

concepts (compositionality, productivity and

systematicity of representations are defined in the

following of this section). On the other hand, it is

well known that compositionality is at odds with

prototypicality effects, which are crucial in most

psychological characterisations of concepts.

Let us consider first the compositionality

requirement. In a compositional system of

representations we can distinguish between a set of

primitive, or atomic symbols, and a set of complex

symbols. Complex symbols are generated starting

from primitive symbols through the application of a

set of suitable recursive syntactic rules (usually,

starting from a finite set of primitive symbols, a

potentially infinite set of complex symbols can be

generated). Natural languages are the paradigmatic

example of compositional systems: primitive

symbols correspond to the elements of the lexicon

(or, better, to morphemes), and complex symbols

include the (potentially infinite) set of all sentences.

In compositional systems the meaning of a

complex symbol s functionally depends on the

syntactic structure of s and from the meaning of

primitive symbols in it. In other words, the meaning

of complex symbols can be determined by means of

recursive semantic rules that work in parallel with

syntactic composition rules. In this consists the so-

called principle of compositionality of meaning,

THE COMPUTATIONAL REPRESENTATION OF CONCEPTS IN FORMAL ONTOLOGIES - Some General

Considerations

397

which Gottlob Frege identified as one of the main

features of human natural languages.

In classical cognitive science it is often assumed

that mental representations are compositional. One

of the most clear and explicit formulation of this

assumption is due to Jerry Fodor and Zenon

Pylyshyn (1988). They claim that compositionality

of mental representations is mandatory in order to

explain some fundamental cognitive phenomena. In

the first place, human cognition is generative: in

spite of the fact that human mind is presumably

finite, we can conceive and understand an unlimited

number of thoughts that we never encountered

before. Moreover, also systematicity of cognition

seems to depend on compositionality: the ability of

conceiving certain contents is related in a systematic

way to the ability of conceiving other contents. For

example, if somebody can understand the sentence

the cat chases a rat, then she is presumably able to

understand also a rat chases the cat, in virtue of the

fact that the forms of the two sentences are

syntactically related. We can conclude that the

ability of understanding certain propositional

contents systematically depends on the

compositional structure of the contents themselves.

This can be easily accounted for if we assume that

mental representations have a structure similar to a

compositional language.

2.2 Against "Classical" Concepts

Compositionality is less important for many

psychologists. In the field of psychology, most

research on concepts moves from the critiques to the

so-called classical theory of concepts, i.e. the

traditional point of view according to which

concepts can be defined in terms of necessary and

sufficient conditions. Rather, empirical evidence

favours those approaches to concepts that accounts

for prototypical effects. The central claim of the

classical theory of concepts (i.e.) is that every

concept c is defined in terms of a set of features (or

conditions) f

, ..., f

that are individually necessary

and jointly sufficient for the application of c. In

other words, everything that satisfies features

, ..., f

is a c, and if anything is a c, then it must

satisfy f

, ..., f

. For example, the features that

define the concept bachelor could be human, male,

adult and not married; the conditions defining

square could be regular polygon and quadrilateral.

This point of view was unanimously and tacitly

accepted by psychologists, philosophers and

linguists until the middle of the 20th century.

The first critique to the classical theory is due to

a philosopher: in a well known section from the

Philosophical Investigations, Ludwig Wittgenstein

observes that it is impossible to individuate a set of

necessary and sufficient conditions to define a

concept such as GAME (Wittgenstein, 1953, § 66).

Therefore, concepts exist, which cannot be defined

according to classical theory, i.e. in terms of

necessary and sufficient conditions. Rather, concepts

like GAME rest on a complex network of family

resemblances. Wittgenstein introduces this notion in

another passage in the Investigations: «I can think of

no better expression to characterise these similarities

than “family resemblances”; for the various

resemblances between members of a family: build,

features, colour of eyes, gait, temperament, etc. etc.»

(ibid., § 67).

Wittgenstein's considerations were corroborated

by empirical psychological research: starting from

the seminal work by Eleanor Rosch, psychological

experiments showed that common-sense concepts do

not obey to the requirement of the classical theory

usually common-sense concepts cannot be defined

in terms of necessary and sufficient conditions (and

even if for some concept such a definition is

available, subjects do not use it in many cognitive

tasks). Rather, concepts exhibit prototypical effects:

some members of a category are considered better

instances than others. For example, a robin is

considered a better example of the category of birds

than, say, a penguin or an ostrich. More central

instances share certain typical features (e.g., the

ability of flying for birds, having fur for mammals)

that, in general, are not necessary neither sufficient

conditions.

Prototypical effects are a well established

empirical phenomenon. However, the

characterisation of concepts in prototypical terms is

difficult to reconcile with the requirement of

compositionality. According to a well known

argument by Jerry Fodor (1981), prototypes are not

compositional (and, since concepts in Fodor's

opinion must be compositional, concepts cannot be

prototypes). In synthesis, Fodor's argument runs as

follows: consider a concept like PET FISH. It results

from the composition of the concept PET and of the

concept FISH. But the prototype of PET FISH

cannot result from the composition of the prototypes

of PET and of FISH. For example, a typical PET is

furry and warm, a typical FISH is greyish, but a

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On the empirical inadequacy of the classical theory and on the

psychological theories of concepts see (Murphy 2002).

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typical PET FISH is not furry and warm neither

greyish.

Moreover, things are made more complex by the

fact that, also within the two fields of philosophy

and psychology considered separately, the situation

is not very encouraging. In neither of the two

disciplines does a clear, unambiguous and coherent

notion of concept seem to emerge. Consider for

example psychology. Different positions and

theories on the nature of concepts are available

(prototype view

, exemplar view, theory theory),

that can hardly be integrated. From this point of

view the conclusions of Murphy (2002) are of great

significance, since in many respects this book

reflects the current status of empirical research on

concepts. Murphy contrasts the approaches

mentioned above in relation to different classes of

problems, including learning, induction, lexical

concepts and children’s concepts. His conclusions

are rather discouraging: the result of comparing the

various approaches is that “there is no clear,

dominant winner” (ibid., p. 488) and that “[i]n short,

concepts are a mess” (p. 492). This situation

persuaded some scholars to doubt whether concepts

constitute a homogeneous phenomenon from the

point of view of a science of the mind (see e.g.

Machery 2005 and 2009; Frixione 2007).

3 CONCEPT REPRESENTATION

IN AI

The situation sketched in the section above is in

some sense reflected by the state of the art in AI and,

more in general, in the field of computational

modelling of cognition. This research area seems

often to hesitate between different (and hardly

compatible) points of view. In AI the representation

of concepts is faced mainly within the field of

knowledge representation (KR). Symbolic KR

systems (KRs) are formalisms whose structure is, in

a wide sense, language-like. This usually involves

that KRs are assumed to be compositional.

In a first phase of their development (historically

corresponding to the end of the 60s and to the 70s)

many KRs oriented to conceptual representations

tried to keep into account suggestions coming from

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Note that the so-called prototype view does not coincide with

the acknowledgement of prototypical effects: as said before,

prototypical effects are a well established phenomenon that all

psychological theories of concepts are bound to explain; the

prototype view is a particular attempt to explain empirical facts

concerning concepts (including prototypical effects). On these

aspects see again Murphy 2002.

psychological research. Examples are early semantic

networks and frame systems. Frame and semantic

networks were originally proposed as alternatives to

the use of logic in KR. The notion of frame was

developed by Marvin Minsky (1975) as a solution to

the problem of representing structured knowledge in

AI systems

. Both frames and most semantic

networks allowed the possibility to characterise

concepts in terms of prototypical information.

However, such early KRs where usually

characterised in a rather rough and imprecise way.

They lacked a clear formal definition, and the study

of their meta-theoretical properties was almost

impossible. When AI practitioners tried to provide a

stronger formal foundation to concept oriented KRs,

it turned out to be difficult to reconcile

compositionality and prototypical representations.

As a consequence, they often choose to sacrifice the

latter.

In particular, this is the solution adopted in a

class of concept-oriented KRs which had (and still

have) wide diffusion within AI, namely the class of

formalisms that stem from the so-called structured

inheritance networks and from the KL-ONE system

(Brachman and Schmolze 1985). Such systems were

subsequently called terminological logics, and today

are usually known as description logics (DLs)

(Baader et al. 2002).

A standard inference mechanism for this kind of

networks is inheritance. Representation of

prototypical information in semantic networks

usually takes the form of allowing exceptions to

inheritance. Networks in this tradition do not admit

exceptions to inheritance, and therefore do not allow

the representation of prototypical information.

Indeed, representations of exceptions can be hardly

accommodated with other types of inference defined

on these formalisms, concept classification in the

first place (Brachman 1985). Since the

representation of prototypical information is not

allowed, inferential mechanisms defined on these

networks (e.g. inheritance) can be traced back to

classical logical inferences.

In more recent years, representation systems in

this tradition have been directly formulated as

logical formalisms (the above mentioned description

logics, Baader et al., 2002), in which Tarskian,

compositional semantics is straightly associated to

the syntax of the language. Logical formalisms are

paradigmatic examples of compositional

representation systems. As a consequence, this kind

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Many of the original articles describing these early KRs can be

found in (Brachman & Levesque 1985), a collection of classical

papers of the field.

THE COMPUTATIONAL REPRESENTATION OF CONCEPTS IN FORMAL ONTOLOGIES - Some General

Considerations

399

of systems fully satisfy the requirement of

compositionality. This has been achieved at the cost

of not allowing exceptions to inheritance. By doing

this we gave up the possibility of representing

concepts in prototypical terms. From this point of

view, such formalisms can be seen as a revival of the

classical theory of concepts, in spite of its empirical

inadequacy in dealing with most common-sense

concepts.

Nowadays, DLs are widely adopted within many

application fields, in particular within the field of the

representation of ontologies. For example, the OWL

(Web Ontology Language) system is a formalism in

this tradition that has been endorsed by the World

Wide Web Consortium for the development of the

semantic web

4 NON-CLASSICAL CONCEPTS

IN COMPUTATIONAL

ONTOLOGIES

Of course, within symbolic, logic oriented KR,

rigorous approaches exist, that allow to represent

exceptions, and that therefore would be, at least in

principle, suitable for representing “non-classical”

concepts. Examples are fuzzy logics and non

monotonic formalisms. Therefore, the adoption of

logic oriented semantics is not necessarily

incompatible with prototypical effects. But such

approaches pose various theoretical and practical

difficulties, and many unsolved problems remain.

In this section we overview some recent proposal

of extending concept-oriented KRs, and in particular

DLs, in order to represent non-classical concepts.

Recently different methods and techniques have

been adopted to represent non-classical concepts

within computational ontologies. They are based on

extensions of DLs and of standard ontology

languages such as OWL. The different proposals

that have been advanced can be grouped in three

main classes: a) fuzzy approaches, b) probabilistic

and Bayesan approaches, c) approaches based on

non monotonic formalisms.

a) As far as the integration of fuzzy logics in DLs

and in ontology oriented formalisms is concerned,

see for example Gao and Liu 2005, and Calegari and

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The problem of representing of non-classical concepts in DLs is

related with, but independent from, another theoretical and

practical knowledge representation problem, namely the problem

of representing instances in taxonomies. The notion of "instance"

itself in taxonomic formalisms is problematic (for some aspects

see e.g. Frixione et al. 1989). However, we do not face these

problems here.

Ciucci 2007. Stoilos et al. (2005) propose a fuzzy

extension of OWL,

f-OWL, able to capture imprecise and vague

knowledge, and a fuzzy reasoning engine that lets f-

OWL reason about such knowledge. Bobillo and

Staccia (2009) propose a fuzzy extension of

OWL 2 for representating vague information in

semantic web languages. However, it is well known

(Osherson and Smith 1981) that approaches to

prototypical effects based on fuzzy logic encounter

some difficulty with compositionality.

b) The literature offers also several probabilistic

generalizations of web ontology languages. Many of

these approaches, as pointed out in Lukasiewicz and

Straccia (2008), focus on combining the OWL

language with probabilistic formalisms based on

Bayesian networks. In particular, Da Costa and

Laskey (2006) suggest a probabilistic generalization

of OWL, called PR-OWL, whose probabilistic

semantics is based on multi-entity Bayesian

networks (MEBNs); Ding et al. (2006) propose a

probabilistic generalization of OWL, called Bayes-

OWL, which is based on standard Bayesian

networks. Bayes-OWL provides a set of rules and

procedures for the direct translation of an OWL

ontology into a Bayesian network. A general

problem of these approaches could consist in

avoiding arbitrariness in assigning weights in the

translation from traditional to probabilistic

formalisms.

c) The role of non monotonic reasoning in the

context of formalisms for the ontologies is actually

a debated problem. According to many KR

researches, non monotonic logics are expected to

play an important role for the improvement of the

reasoning capabilities of ontologies and of the

Semantic Web applications. In the field of non

monotonic extensions of DLs, Baader and Hollunder

(1995) propose an extension of ALCF system based

on Reiter’s default logic

. The same authors,

however, point out both the semantic and

computational difficulties of this integration and, for

this reason, propose a restricted semantics for open

default theories, in which default rules are only

applied to individuals explicitly represented in the

knowledge base. Since Reiter’s default logic does

not provide a direct way of modelling inheritance

with exceptions in DLs, Straccia (1993) proposes an

extension of H-logics (Hybrid KL-ONE style logics)

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The authors pointed out that “Reiter's default rule approach

seems to fit well into the philosophy of terminological systems

because most of them already provide their users with a form of

‘monotonic’ rules. These rules can be considered as special

default rules where the justifications - which make the behavior of

default rules nonmonotonic – are absent”.

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able to perform default inheritance reasoning (a

kind of default reasoning specifically oriented to

reasoning on taxonomies). This proposal is based on

the definition of a priority order between default

rules. Donini et al. (1998, 2002), propose an

extension of DL with two non monotonic epistemic

operators. This extension allows one to encode

Reiter’s default logic as well as to express epistemic

concepts and procedural rules. However, this

extension presents a rather complicated semantics,

so that the integration with the existing systems

requires significant changes to the standard

semantics of DLs. Bonatti et al. (2006) propose an

extension of DLs with circumscription. One of

motivating applications of circumscription is indeed

to express prototypical properties with exceptions,

and this is done by introducing “abnormality”

predicates, whose extension is minimized. Giordano

et al. (2007) propose an approach to defeasible

inheritance based on the introduction in the ALC DL

of a typicality operator T

, which allows to reason

about prototypical properties and inheritance with

exceptions. This approach, given the non monotonic

character of the T operator, encounters some

problems in handling inheritance (an example is

what the authors call the problem of irrelevance).

Katz and Parsia argue that ALCK, a non monotonic

DL extended with the epistemic operator K

(that

can be applied to concepts or roles) could represent a

model for a similar non monotonic extension of

OWL. In fact, according to the authors, it would be

possible to create “local” closed-world assumption

conditions, in order the reap the benefits of non

monotonicity without giving up OWL’s open-world

semantics in general.

A different approach, investigated by Klinov and

Parsia (2008), is based on the use of the OWL 2

annotation properties (APs) in order to represent

vague or prototypical, information. The limit of this

approach is that APs are not taken into account by

the reasoner, and therefore have no effect on the

inferential behaviour of the system (Bobillo and

Straccia 2009).

5 CONCLUSIONS: A “DUAL

PROCESS” PROPOSAL

Though the presence of a relevant field of research,

there is not, in the scientific community, a common

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For any concept C, T(C) are the instances of C that are

considered as “typical” or “normal”.

The K operator could be encoded in RDF/XML syntax of OWL

as property or as annotation property.

view about the use of non monotonic and, more in

general, non-classical logics in ontologies. For

practical applications, systems that are based on

classical Tarskian semantics and that do not allow

for exceptions (as it is the case of “traditional” DLs),

are usually still preferred. Some researchers, as, for

example, Pat Hayes (2001), argue that the non

monotonic logics (and, therefore, the non monotonic

“machine” reasoning for Semantic Web) can be

maybe adopted for local uses only or for specific

applications because it is “unsafe on the web”.

Anyway, the question about which “logics” must be

used in the Semantic Web (or, at least, until which

degree, and in which cases, certain logics could be

useful) is still open. At the same time, the empirical

results from cognitive psychology show that most

common-sense concepts cannot be characterised in

terms of necessary/sufficient conditions. Classical,

monotonic DLs seem to capture the compositional

aspects of conceptual knowledge, but are inadequate

to represent prototypical knowledge.

As seen before, cognitive research about

concepts seems to suggest that concept

representation does not constitute an unitary

phenomenon from the cognitive point of view. In

this perspective, a possible solution should be

inspired by the experimental results of empirical

psychology, in particular by the so-called dual

process theories of reasoning and rationality

(Stanovich and West 2000, Evan and Frankish

2008). In such theories, the existence of two

different types of cognitive systems is assumed. The

systems of the first type (type 1) are

phylogenetically older, unconscious, automatic,

associative, parallel and fast. The systems of the

type 2 are more recent, conscious, sequential and

slow, and are based on explicit rule following. In our

opinion, there are good prima facie reasons to

believe that, in human subjects, classification, a

monotonic form of reasoning which is defined on

semantic networks, and which is typical of DL

systems, is a task of the type 2 (it is a difficult, slow,

sequential task). On the contrary, exceptions play an

important role in processes such as categorization

and inheritance, which are more likely to be tasks of

the type 1: they are fast, automatic, usually do not

require particular conscious effort, and so on.

Therefore, a reasonable hypothesis is that a

concept representation system should include

different “modules”: a monotonic module of type 2,

involved in classification and in similar “difficult”

tasks, and a non monotonic module involved in the

management of exceptions. This last module should

be a "weak" non monotonic system, able to perform

THE COMPUTATIONAL REPRESENTATION OF CONCEPTS IN FORMAL ONTOLOGIES - Some General

Considerations

401

only some simple forms of non monotonic

inferences (mainly related to categorization and to

exceptions inheritance). This solution goes in the

direction of a “dual” representation of concepts

within the ontologies, and the realization of hybrid

reasoning systems (monotonic and non monotonic)

on semantic network knowledge bases. Some of the

proposal reviewed in the sect. 4 above could be

probably interpreted in this perspective. It could be

objected that proposals based on non monotonic

extensions of classical DLs are not suitable to model

type 1 reasoning systems, since their computational

properties are even worst of those of traditional,

monotonic DLs. In this perspective, an alternative

solution should be combining ontologies and logic

programming rules, endowed with usual semantics

for non monotonic logic programs (see e.g. Eiter et

al. 2008).

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