KNOWLEDGE REPRESENTATION THROUGH COHERENCE

SPACES

A Theoretical Framework for the Integration of Knowledge Representations

V. Michele Abrusci, Marco Romano

Dipartimento di Filosoﬁa, Universit

a Roma Tre, 234 via Ostiense, Roma, Italy

Christophe Fouquer

LIPN, Universit

e Paris 13 et CNRS, 93430 Villetaneuse, France

Keywords:

Coherence spaces, Folksonomy, Knowledge Representation, Logics, Ontology.

Abstract:

This work is an ongoing research (sponsored with a PhD grant by Epistematica Srl) about the interpretation of

ontologies and their operations on speciﬁc graphs called Ontological Compatibility Spaces (OCS). Such graphs

are particular Coherence Spaces that are used to give a denotational semantics to Linear Logic, hence giving

a solid logical basis to our work. Using a graph framework, we depart from traditional set representation. It

provides us with tools to represent actual relations among resources (and data) over the WorldWideWeb. After

deﬁning such OCSs and their correspondence with ontologies, we show to which extent folksonomies may

also be of use to discover ontologies by means of OCS. Then we brieﬂy discuss what we may obtain thanks to

such an interpretation, conﬁdent that it may beneﬁt the Semantic Web initiative with a tool for dynamic data

and resources exchange.

1 INTRODUCTION

Within the Semantic Web initiative it is assumed that

expressive formal description of data sources will

lead to their interconnection throughout the World-

WideWeb via logical inter-deﬁnition of concepts ap-

pearing in different descriptions. According to this

core idea, the research area of Knowledge Represen-

tation (KR) has been co-opted; the merge of the most

promising KR models (namely ontologies) with stan-

dard markup languages to be used in the Web has

been followed by the adoption of inference engines

to be plugged into knowledge bases (KB) in order to

exploit the expressivity of languages that implement

some Description Logic (DL), namely OWL. Thus,

the interconnection of datasources depends on their

speciﬁcation in a formal way suitable for automatic

reasoning. In the meantime, the social evolution of

the Web (Web2.0) has showed the effects of collective

intelligence and its potential to manage large amounts

of information with user-friendly tools requiring no

speciﬁc (or none at all) skill in KR. We think that

the Knowledge Engineers (KE) community that is de-

veloping the technological layer of the Semantic Web

should strictly cooperate with, and take advantage of,

the Web2.0 communities that spontaneously provide

the Web with collections of resources that are catego-

rized (though roughly) according to some collectively

developed knowledge framework.

Therefore, after a brief discussion of the logical

assumptions claimed or implied with KR involvement

– and of their aptness to account for low-level, un-

trained and spontaneous categorization of resources –

we propose an alternative logical framework, that of

Linear Logic, which, we argue, can trigger the dy-

namic exchange of resources and data through differ-

ent datasources. In particular we expect that it can

provide the tools to describe and realize the passage

from a datasource to any other without the need for

a given-in-advance and usually “hand-made” formal

description of any single datasource involved and of

the mapping between every pair of them. This way,

we could also get Semantic Web closer to Social Web

using directly the knowledge put into Web2.0 envi-

ronments (e.g. tagging spaces) without the step of

ontology extraction that requires the deﬁnition of a

conceptual hierarchy to be validated somehow. In

fact we should build knowledge representations out of

Web2.0 environments using the minimum of the ap-

propriate logic that allows also for the representation

220

Abrusci V., Romano M. and Fouqueré C. (2009).

KNOWLEDGE REPRESENTATION THROUGH COHERENCE SPACES - A Theoretical Framework for the Integration of Knowledge Representations.

In Proceedings of the International Conference on Knowledge Engineering and Ontology Development, pages 220-225

DOI: 10.5220/0002301002200225

 SciTePress

of operations between knowledge representations.

2 KR AND WEB RESOURCES

About ten years after the adoption of KR to support

the building of the Semantic Web it is apparent that

there are some major problems that make it a long and

hard way to walk. Indeed, to have knowledge repre-

sentations suitable for use with current technologies

and according to the theoretical background of KE,

we need both given in advance formal descriptions of

the datasources we would like to put together over the

Web, and the provision of rules for the mapping be-

tween every couple of datasources to have them really

exchanging data and resources. As regards the need

for formal descriptions, it requires both specialists

to deﬁne the domain ontologies (Gruber, 1995) and

some kind of professionals to give them the formal

dressing. As regards the mapping issue, it prevents to

compare any two datasources (i.e. to use them in the

sense of Semantic Web) unless someone has provided

the mapping instructions. Put in this way, the Seman-

tic Web looks like too ambitious a research initiative,

since it expects enormous efforts to properly deﬁne

suitable ontologies, while as a commercial initiative

(as it is at least partially) it follows a completely top-

down and anti-dynamic approach since ontology def-

inition invariably freezes the knowledge about some-

thing according to its understanding on the part of a

restricted group of experts, thus behaving in an oppo-

site way to the dynamic and unpredictable character

of Web communities.

By contrary, within the Web2.0 movement have

emerged some systems for quick and easy classiﬁca-

tion of resources by both the authors and the users

that we can generally indicate with the term tagging,

which is studied as the form of emerging categoriza-

tion called folksonomy (Vander-Wal, 2007). Com-

pared to ontologies, such bottom-up and popular sys-

tems are expressively very poor and usually gener-

ate no taxonomy but ﬂat spaces where, even though

there may be some hierarchy among the concepts that

are identiﬁed by tag-terms, it is neither showed nor

recorded when tagging. We will not hold that tagging

spaces are ﬁrst class knowledge representations since

they completely lack any interest for the global con-

sistency and awareness of the complete resulting ter-

minology or concept-scheme. Nevertheless tagging

is the form of “naive categorization”, made by stick-

ing labels to resources in the Web, where folksonomy

comes from. Folksonomies are the aggregations of

tag-terms, i.e. what can be used to ﬁnd out the re-

sources that match some tag and, with it, also the

corresponding concept expressed by the term, pre-

cisely as it should be for Semantic Web ontologies.

Secondly, ﬂat tag spaces are classifying – yet poorly,

but they are really doing it day by day – greater and

greater amounts of web resources. So they seems able

to do for Semantic Web that part of the work that on-

tologies cannot manage, i.e. large scale categoriza-

tion. If we then consider that KR has entered the Web

in order to help in making machine-readable the in-

formation in it, and not primarily to develop the ﬁnest

descriptions of particular worlds, we should exploit

any contribution able to cooperate in achieving Se-

mantic Web.

2.1 A Logical Model to Rely On

Being a major result of KE, ontologies and especially

DL ontologies have a precise logical meaning, i.e. a

deﬁnite semantic interpretation (Baader et al., 2003).

Let’s observe what is their semantic interpretation.

It relies on Set theory and associates to the ideas ex-

pressed by concept names (and their logical deﬁni-

tions) the corresponding set of “all the objects that

. . . ” within the speciﬁc domain of interest that the

ontology is designed to describe.

More formally, let I be the interpretation function

from an ontology to a non-empty set ∆, which is the

domain of interpretation. An atomic concept A is in-

terpreted as A

⊆ ∆ while a role R as R

⊆ ∆ × ∆.

Let C, D be concepts; R a role between C and D; a, b

individuals; the semantics of an ontology results as:

I (>) = ∆; I (⊥) =

0; I (C) = C

(⊆ ∆); I (D v C) =

⊆ C

; I (C(a)) = a ∈ C

; I (R) = R

⊆ {

x, y

|x ∈

∧ y ∈ D

}; I (R(a, b)) =

a, b

∈ R

. The infer-

ences that a reasoner can draw from such KBs are

generated by the axioms offered as concept deﬁni-

tions and possibly also presented as A-box assertions

about individuals. Legal inferences are those which

are valid for all the possible models based on the par-

ticular domain ∆ and function I chosen. By the way

we remark that generally the interpretation domain ∆

is never given any concreteness, so that operations be-

tween ontologies affect primarily the intensional level

and then the semantic interpretation is produced sim-

ply as a by-product of formalization, since it is al-

ways possible to declare which set should be the re-

sult of some operation. This would not be a problem

if ontologies were not to assist interchange of data in

a Web of concrete resources. But Semantic Web on-

tologies are precisely to describe what is within dif-

ferent datasources and to enable the jump from each

other so that what exactly is ∆ is not a minor prob-

lem. In order to achieve a working Semantic Web,

we think we should talk about applied ontologies, i.e.

KNOWLEDGE REPRESENTATION THROUGH COHERENCE SPACES - A Theoretical Framework for the Integration

of Knowledge Representations

221

ontologies together with the collection of data or re-

sources that are accounted for within the ontologies,

so that also the operations between ontologies should

be considered primarily as affecting the resources.

Now let’s observe the semantic interpretation of

folksonomies. To be honest, they have no deﬁnite se-

mantic interpretation, but it seems quite easy to adopt

once again Set theory and consider them as very poor

ontologies without concept deﬁnitions. It would be

straightforward to take some domain ∆ and interpret

all the resources that share a speciﬁc tag as a particular

subset of ∆ identiﬁed with the concept lying behind

the tag-term. As regards the calculus they support, no

inference is allowed, but mere resource retrieval by

using tag-terms as search keys. Operations between

folksonomies have not even been conceived.

2.2 One Step Further with More

Structure

We propose a theoretical framework for KR spe-

cially conceived for use within the Semantic Web sce-

nario. Behind this proposal there is a doubt about

the adequacy of the current approach based on Set

theory (for KR) and linguistic analysis of KBs’ in-

tensional level (for the discovery of compatible re-

sources) with respect to the full achievement of the

Semantic Web. Thus, instead of focusing on concep-

tual schemes of ontologies, we propose to focus on

the extensional level of KBs, i.e. on “real” objects,

by adopting a logical framework capable to geomet-

rically represent relations among resources, possibly

discovering the concepts from resource aggregations

that are actually in the Web. In particular we sug-

gest to consider another kind of semantic interpreta-

tion that relies on structures richer than sets, the co-

herence spaces, where the interpretation of a concept

(or tag-term) produces graph theoretical objects along

with the determination of the extensional counterpart

within the collection of resources that is the domain

of interpretation.

Coherence spaces (Girard et al., 1989) are webs

whose points may or not be linked to each other ac-

cording to a binary reﬂexive relation called coher-

ence. They allow for the deﬁnition of a denotational

semantic, so that we can get one also for data ex-

change within the WorldWideWeb and for a deﬁni-

tion of operations between ontologies that is primarily

focused on resources. Coherence spaces come from

Linear Logic (LL) and have been the ﬁrst semantic

interpretation of that logical system. What is more is

that it is not truth-valued semantics, useful only for

talking about formulas – as it always happens with

Classical Logic (LK) – but precisely denotational se-

mantics, useful for talking also about proofs. Indeed,

they offer the domain of interpretation of the objects

manipulated by the logical calculus, i.e. proofs.

LL (Girard, 1987) is a logical system developed

by imposing some restrictions on the use of struc-

tural rules for the construction of deductions within

Gentzen’s proof calculus for ﬁrst order logic (FOL)

known as sequent calculus. The affected rules are

Contraction and Weakening which deal with the num-

ber of times formulas may be used within the same

proof. In LL they are re-deﬁned in form of logical

rules (instead of structural) so that their usage has to

be marked with speciﬁc connectives, called exponen-

tials. This way everything that holds in LK also holds

in LL, although LL is able to better describe what is

happening in a proof: the ability to mark for which

formulas it is licit to have weakening and contraction

means that one can control the times resources are

used. We note, by the way, that this one looks like

an interesting property to have at hand while work-

ing for Semantic Web. As a consequence of the con-

trol on contraction and weakening, LL deconstructs

the connectives ∧ and ∨ doubling them in the multi-

plicative and additive variants, since their behaviour is

different according to the possible uses of the context

(i.e. the other formulas) where the formulas in which

they occur are interacting with. To put it in a nutshell,

the multiplicative connectives operate on the coher-

ence space resulting as the product of the coherence

spaces corresponding to the proofs of the connected

formulas, while the additives on their disjoint union.

Moreover, LL has developed a geometrical represen-

tation of proofs by means of graphs called proof-nets.

It exploits graph structures to compose partial proofs

and provides graph properties to determine when a

proof structure is correct. Such graph structures have

a model in coherence spaces, where the denotation

of the proof of a formula is a set of pairwise coher-

ent points, called clique. The operations between co-

herence spaces interpret the composition of formu-

las and their proofs according to LL connectives. In

order to account for the interpretation of ontologies

through coherence spaces we need a particular class

of them, with a typical support set (see below). In ad-

dition, because of the context of Semantic Web and its

need for datasource integration, we propose to call the

structuring relation of such coherence spaces compat-

ibility rather than coherence, and to call this special

class of coherence spaces “Ontological Compatibility

Spaces” (OCS).

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3 ONTOLOGICAL

COMPATIBILITY SPACES

Let A be a semantic web ontology in a language like

OWL. Let L (A) be the set of all the symbols for in-

dividuals, concepts and roles of A (i.e. individual

symbols, unary predicate symbols, binary predicate

symbols), M a set of data, φ a valuation through suit-

able mapping from L (A) to M, i.e. if c is an indi-

vidual symbol of L (A) then φ(c) is an element of

M, if P is a unary predicate symbol of L (A) then

φ(P) ⊆ M, if P is a binary predicate symbol of L(A)

then φ(P) ⊆ M × M.

This deﬁnes an applied ontology A

and we can

represent every concept, role and individual of A

by means of a coherence space [A, M, φ] with sup-

port |[A, M, φ]| = M ∪ (M × M) provided with a co-

herence relation noted ¨

[A,M,φ]

between its points:

x ¨

[A,M,φ]

y iff ∃P ∈ L (A) such that

{

x, y

}

⊆ φ(P). For

sake of clarity, we may note simply ¨ when the co-

herence space is obvious. This coherence relation for-

malizes the notion of compatibility emerging when-

ever an ontology is applied to a set of data. This way

we have deﬁned an OCS.

Indeed, from an abstract point of view, the re-

trieved values for any predicate symbol P of A form

some subset of M ∪ (M × M) whose elements share

with each other something more than all the other el-

ements of M ∪ (M × M). Such a property, instead

of being named according to any speciﬁc symbol P

occurring in the ontology, may be rewarded as the

compatibility between all the points of that subset

of |[A, M, φ]|. Such a group a of pairwise compati-

ble points of [A, M, φ] is called a clique and is noted

a @ [A, M, φ].

Deﬁned as a coherence relation, the compatibil-

ity relation is reﬂexive, symmetric and not transitive.

Reﬂexivity (∀x ∈ |[A, M, φ]|, x ¨ x) assures that every

point of the coherence space is compatible with itself.

Since we deal with a coherence space whose web is

provided by the set M ∪ (M × M) we have also pairs

y, z

as points of |[A, M, φ]|. Thanks to reﬂexivity, ev-

ery point is a clique and may be considered as a min-

imal class of compatibility. As an example, single-

tons that are retrieved from M through φ (for individ-

uals of A) are interpreted in such cliques. Symmetry

(∀x, y ∈ |[A, M, φ]|, x ¨ y ⇐⇒ y ¨ x) expresses the

core of the idea of compatibility, since for compati-

bility we mean the possibility to put two “objects” to-

gether based on their sharing of some common prop-

erty – not necessarily an expressed one – and such

a commonality has to be inevitably a reciprocal fact.

Non-transitivity prevents the overwhelming distribu-

tion of compatibility that would mix up different Ps

of A whenever they have some common points. How-

ever transitivity may appear under certain conditions,

e.g. within a clique, where the points of a clique are

all pairwise compatible.

A coherence space can be represented by a graph

whose set of nodes is given with the web and the

edges reﬂect the relations of coherence of every point

with the others. Also an OCS can be represented

through a graph. The graph G(V, E) of our OCS

[A, M, φ] may be deﬁned by the set of nodes V =

|[A, M, φ]| and that of edges E ⊆ |[A, M, φ]|

with the

constraint that for every two points x, y of |[A, M, φ]|

we have

{

x, y

}

∈ E ⇐⇒ x ¨ y that is to say

{

x, y

}

∈

E ⇐⇒ ∃P ∈ L(A) s.t.

{

x, y

}

⊆ φ(P). Based on the

deﬁnition of coherence, when looking at the graph of

an OCS, a clique a @ [A, M, φ] turns out to be a com-

pletely connected portion of the graph (a subgraph).

We observe that the interpretation of ontologies as

OCSs successes: i) every individual of A

is repre-

sented within the OCS as a clique of a single point

(the smallest class of compatibility); ii) every concept

or role of A

is represented as a class of compatible

points (a general clique). Following the inverse di-

rection, we observe that every clique of the OCS is

the denotation of: i) some concept or role of A

; ii)

or some subconcept (or subrole) not identiﬁed by any

predicate of L (A) yet recognizable in A

as a subset

of some concept (or role) identiﬁed by some predicate

in L (A); iii) or an individual of A

4 OCS VS ONTOLOGY

We have designed OCSs specially for representing

ontologies. We now discuss the advantages that our

proposal may bring to the development and usage of

folksonomies and suggest to which extent all this may

beneﬁt Semantic Web.

4.1 Tagging and Ontology Extraction

Nowadays when looking at the set of tag-terms

adopted within a community it is expected to recon-

struct a formal ontology out of that, establishing a

neat and formal hierarchy among concepts, useful for

resource retrieval through progressive speciﬁcation.

The major side effect of such a reduction is the loss

of the dynamic aspect of Web2.0.

We may observe that building the Semantic Web

by means of ontologies requires predeﬁned sets of

metadata (the schemes) to be adopted and respect-

fully obeyed. Their usefulness – and the wealth of

Semantic Web itself as the workplace of autonomous

agents (Berners-Lee et al., 2001) – will depend in

KNOWLEDGE REPRESENTATION THROUGH COHERENCE SPACES - A Theoretical Framework for the Integration

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223

fact on the number of resources whose set of meta-

data matches one or more of the predeﬁned schemes

so that programs specially written according to the

same scheme(s) will be able to use those resources.

So the Web of data that W3C indicates turns out to

be something like a huge database where a neat deﬁ-

nition of the logical scheme (even composed of many

different ontologies) can be achieved only thanks to

the standardization of the metadata tags to be used to

describe resources. In the opposite direction goes the

practice of free tagging, so that folksonomies emerge

as everlasting works in progress where concept-terms

institution and resource description and classiﬁcation

always happen in the same time, with no hope for

standardization. In fact, when people tag they freely

choose and establish their own categories in an unend-

ing process of ontology elaboration. Moreover, while

using their very personal categories people also ex-

press their own “world’s understanding” so that tag-

ging spaces are not only useful for classiﬁcation, but

also convenient for collective intelligence to share

knowledge. We remark that tagging spaces publish

enormous amounts of resources with some kind of

classiﬁcation while providing a cognitive framework

that has not the claims of ontology but that is pow-

erful enough to let one recognize and ﬁnd classes of

resources that are compatible, i.e. similar to some ex-

tent. Maybe such a cognitive framework is a lower

quality contribution, in comparison to formal DL on-

tologies, to have the content of the Web surely rec-

ognizable, but it seems to show a more feasible way

to do that. In fact, since it relies on no pre-emptive

requirements, no standardized label to tag resources,

it preserves the dynamical behaviour of Web2.0 and

lets Semantic Web to be a “common people affair”

as it has been for last years for WorldWideWeb, and

for its big boom. However, it still lacks an appropri-

ate theoretical framework to be successfully exploited

to the beneﬁt of Semantic Web. Therefore we pro-

pose one: to consider tagging spaces without any at-

tempt to reduce them to formal ontologies. Instead of

the usual techniques for tags clustering and concept

extraction, we may exploit OCSs in order to recover

from tagging spaces a description of the resources in a

datasource that is formal enough to be useful for data

exchange but that does not need ad hoc speciﬁcation

of a conceptual hierarchy. We look only for compat-

ibility between resources observing the connections

given within an OCS. Since operations between co-

herence spaces have already been largely studied in

LL, we have almost ready-to-use precision tools to

talk about usual operations (union of ontologies, mod-

ule extraction, . . . up to ontology merging) via their in-

terpretation as operations between OCSs using the LL

connectives – and thus opening to a new role to play

in Computer Science for LL (Ehrhard et al., 2004).

Our proposal is then to give more logical dignity to

ﬂat tagging spaces without rising too high in formal

complexity so as to prevent large contribution from

common web-users. We simply aim to describe ﬂat

spaces in such a way that it makes sense to talk about

operations between them.

4.2 Folksonomies out of OCS

In order to have an OCS out of a ﬂat tagging space

we need nothing more than what has been stated for

ontologies, since we will consider it as a very sim-

ple ontology, with just some niceties: the language

L (F) of such a folksonomy F will be the pair (T, I)

where T is the set of tag-terms and I the set of URIs.

The web of the OCS [F, I, φ] will be |[F, I, φ]| = I. φ

is the pointer from a tag-term t to the set φ(t) ⊆ I of

the resources tagged with t – i.e. it is a query. The

compatibility relation is slightly revised as x ¨

[F,I,φ]

y ⇐⇒ ∃t ∈ L (F) s.t.

{

x, y

}

⊆ φ(t). We observe

some characteristics of such an OCS: i) for every tag-

term t: t ∈ T → ∃x ∈ I x ∈ φ(t); ii) if we assume that

in a tagging space there cannot be a resource that

has no tag, we have also the inverse, for every URI

i: i ∈ I → ∃v ∈ T i ∈ φ(v). These mean that we have

no empty concepts and that the graph associated to a

folksonomy may be a fully labelled graph. Moreover,

we remark that thanks to the compatibility relation we

are now able to read and build concepts out of the

tagging space in an original way. In fact we do not

search for interesting concepts looking at recurring

couplings of certain tag-terms stuck to different re-

sources (looking for synonymy or some subsumption

between terms). Quite the contrary, while searching

for cliques we look at recurring couplings of certain

resources under different tags, i.e. among the URIs

retrieved for the tags. We look for compatibility be-

tween resources and try to collect all the classes of

compatibility, what turns out to be a new tool for con-

cept discovery. Based on the deﬁnition of compati-

bility, we may have a class of compatible resources

which are not all together within the retrieved URIs

for one single tag, yet they are all pairwise compati-

ble. It is such a case that of a possible new concept not

explicitly recognized on the part of the tagging com-

munity but implicitly present as an underlying idea of

compatibility. Whether such a concept can be identi-

ﬁed with some linguistic term or not, it is not impor-

tant: we are abstracting from linguistic determination

of compatibility classes and we can grasp some new

concept which one can look for other elements com-

patible with.

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5 CONCLUSIONS

In conclusion, we recall some of the most challeng-

ing aspects that are to be further investigated in our

research along with the results that we can see at

present. Recurring to coherence spaces to describe

operations between ontologies (and folksonomies),

we have to face some deep questions that emerge

from aspects inner to LL and coherence spaces the-

ory. Indeed, coherence spaces and their cliques rep-

resent proofs of (multisets of) formulas (from here on

we use formula while meaning multiset). Each co-

herence space offers the place for the denotation of

one formula and shows a correspondence regarding a

clique, a formula and its proof, while operations be-

tween coherence spaces provide the denotation for the

calculus when composing formulas. When adopting

coherence spaces for ontologies, we have to ﬁnd the

right place to put other elements in this correspon-

dence, i.e. concepts and resources, together with on-

tology. We have to clearly set the circle of correspon-

dences and a ﬁrst attempt is to compare the formulas

appearing in the right-hand side of a sequent in LL

with an ontology seen as the union of its concepts, so

that the proof satisfying one of the concepts satisﬁes

also the whole ontology.

For the time being, we are proposing something

like an alternative descriptive language to represent

data sources in the WorldWideWeb. With such a

language we are given the easy of use of tagging

space, the ontology/folksonomy extraction procedure

described above and a calculus based on LL connec-

tives, that are a little more and a little ﬁner than those

usually adopted working with ontologies. We have

already mentioned the doubling of some connectives

(∧ and ∨, in their multiplicative and additive formu-

lation) in LL, together with the appearing of two new

connectives, the exponentials, that control the use of

contraction and weakening, i.e. the use of resources.

Without doubt it is worth to assess the usefulness of

such “logical controllers” in order to account for op-

erations in the Web that involve resources. If we con-

sider that the multiplicative fragment of LL shows in-

teresting computational properties and that reasoners

for LL are already available, we are further encour-

aged in developing such a language in order to assess

its capability to satisfy Semantic Web needs.

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