CONTEXTUALIZING ONTOLOGIES FOR AGENTS

Lito Perez Cruz and John Newsome Crossley

∗

Faculty of Information Technology, Monash University, Melbourne, Australia

Keywords:

Agents, Logics for agent systems, Ontologies and agent systems, Description logic.

Abstract:

It is well accepted that the usefulness of agents is enhanced by ontologies, but a common problem encountered

by agents is the difﬁculty of accessing heterogenous ontologies. This problem is addressed by contextualizing

ontologies, but how? We show how agents can contextualize ontologies that are represented using description

logics. Several attempts have been made in addressing this contextualization problem, but we use the tech-

nique of the Tiered Logic Method (TLM) to build a system that is much simpler, more elegant, and easier to

implement than existing technologies. Moreover, since TLM is a methodology it also has applications in other

types of system. We sketch proofs of soundness, completeness and decidability for such a system, subject only

to simple ﬁniteness constraints, which would be satisﬁed in any practical case. Finally this method solves the

problem of transitive subsumption propagation,which is still unaddressed by other well known proposals.

1 INTRODUCTION

Ontologies play a vital role in applications that in-

volve the cooperation of multiple agents (Obitko and

Maˇr´ık, 2003). How can agents deal with heteroge-

nous ontologies? The problem is solved if the agent

has a map of its ontology to the one that it is about

to read. Like databases, ontologies are described us-

ing a computer language, so how do we contextualize

ontology languages (Bouquet et al., 2004).

Ontology languages such as OWL2 are founded

on Description Logic (DL) (Baader et al., 2003), so

the problem becomes an issue of how DLs may be

contextualized. Most researchers start with the Local

Model Semantics of (Giunchiglia and Ghidini, 2000)

and DDL (Borgida and Seraﬁni, 2002). Here, con-

cepts in one ontology are linked to concepts found

in another ontology using bridge rules. This was

extended in E -Connections (Kutz et al., 2004), us-

ing a relation E, which relates individuals in differ-

ent ontologies to capture the meaning of bridge rules.

Further investigation has led to Packaged-DL (P-DL)

(Bao et al., 2006), akin to the encapsulation found

in Object Oriented programming, allowing one on-

tology to import or reuse concepts from another on-

tology. The drawback is that P-DL syntax requires a

revamp of existing DLs and existing ontology reason-

∗

Thanks to a referee for pointing out IDDL and other

helpful remarks. A full account of technicalities is expected

to be submitted to a journal.

ers need to be re-engineered, and at present there is no

reasoner that implements P-DL. The system IDDL of

(Zimmermann and Duc, 2008) uses local and global

systems related to ours, but has to introduce special

reasoning rules rather than using a standard (proposi-

tional) logic as we do. Except for P-DL and IDDL, all

of the above suffer from one crucial weakness: they

do not support transitive subsumption propagation,

that is to say, the rule: i : A

⊑

−→ j : B and j : B

⊑

−→ k :C,

imply i : A

⊑

−→ k : C.

In this paper we contextualize DLs preserving

transitive subsumption by using the Tiered Logic

Method (TLM) of (Cruz and Crossley, 2009), and we

present a solution which, we believe, is simpler than

others. Moreover, the ideas have been tested in ex-

isting reasoning engines without re-engineering be-

ing required. We apply TLM to contextualize a typical

DL called A L C , which is a a subset of S H I Q (D )

−

the present foundation for ontology languages such

as OWL. However, the principles found here are also

applicable to contextualizing S H I Q (D )

−

DLs - A Review. DLs use the fundamental notion

of concepts which are sets of individuals.

A L C syn-

tax is grouped into statements that are terminological

For simplicity of exposition we do not use roles, at-

tributes or features although the extension required to in-

clude them is routine. For full details see (Cruz, 2010).

147

Perez Cruz L. and Newsome Crossley J..

CONTEXTUALIZING ONTOLOGIES FOR AGENTS.

DOI: 10.5220/0003053701470152

In Proceedings of the International Conference on Knowledge Engineering and Ontology Development (KEOD-2010), pages 147-152

ISBN: 978-989-8425-29-4

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

Γ |=

)

Γ |=

(Flat)

Γ |=

(Flat-0)

Figure 1: The transfer rules.

ones which comprise the TBox and those that are as-

sertional ones which comprise the ABox. We follow

the standard terminology of (Baader and Nutt, 2003).

We shall write C = {a, b}, etc. as an abbreviation for

a : C, b :C, etc.

Traditionally DL literature deﬁnes an interpreta-

tion structure for a system such as ours as a pair

I = h∆

, •

i. Here the set, ∆

, is the domain

of the interpretation (also known as the abstract do-

main) and •

is the interpretation function. However

we shall switch to a slightly different notation so that

we can keep the superscript to indicate the ‘context’

to which a statement belongs, in line with (Cruz and

Crossley, 2009):

instead of a superscript for the interpretation func-

tion we shall instead use the Greek iota ι. Thus in-

stead of C

we shall write ι(C). Hence our interpre-

tation I will be written as h∆

, ιi.

2 TIERED DESCRIPTION LOGIC

Our underlying methodology, TLM, follows the gen-

eral pattern of (Cruz and Crossley, 2009), but our

approach here is that of description logic and there-

fore we shall be principally concerned with semantic

equivalence rather than the syntactic (viz. provable)

equivalence of that paper.

We start with a number of ontologies or localities,

which we denote by superscripts i, j, k, . . .. Each of

these localities will have its own description logic.

Such a language constitutes the strictly local lan-

guage at each locality in tier-0: the local level. We

then move up to what we call tier-1 which is the

global or system level and combine them using pos-

itive propositional logic. Finally we take the global

statements back down into tier-0 and form the local

language.

In order to move statements between the two tiers

we use the idea of ﬂattening (Buvaˇc et al., 1995). This

entails that once a statement has been made (and its

semantics determined for its own locality) then the

truth or falsehood of the statement is unaffected by

reporting it in another locality, see Figure 1.

In our earlier work, (Cruz and Crossley, 2009), we

used predicate logic at tier-0 and full classical propo-

sitional calculus at tier-1. We can use full propo-

These languages may differ, see Remark 2.

sitional calculus if we do not wish to have ‘Bridge

rules’ or ‘Bridge declarations’ but these are what

gives the tiered logic method its strength. A bridge

declaration is a statement that says that a concept in

one locality is subsumed by a concept in another. For

example, the concept

HOUSE

in an English-speaking

locality can be subsumed under the concept

CASA

a Spanish-speaking one. So a real estate agent sell-

ing in both Britain and Spain may ‘bridge’ between

British and Spanish contexts by a bridge declaration

HOUSE

Britain

⊑

CASA

Spain

Remark 1. When we wish to combine bridge state-

ments involving different localities there is a prob-

lem about negation. Consider a bridge statement

such as A

⊑ B

. We can certainly consider this as

a global statement as we did in the house/casa exam-

ple. But what would ¬(A

⊑ B

) mean in our system?

Viewed from a traditional logic point of view it cer-

tainly means that some members of A are not mem-

bers of B, but we cannot answer the question: Where

are these elements? Elements in A are in the locality

i, but may or may not be in locality j (cf. Remark 2).

So the problem is how to say, in our language: x is

not in j.

We therefore restrict our propositional calculus to

positive propositional calculus (+PL), that is to say,

we only employ the connectives ∨ and ∧, and do not

use ¬ or →. We claim this is still in the spirit of

description logic since, in a DL, it is not possible to

negate a terminological statement.

Note that (Zimmermann and Duc, 2008) deal with

this in a different way, and the price they have to pay

is that, in their system, ‘reasoning on IDDL systems

is not trivial’.

hDL, +PLi Syntax. By Strictly Local Syntax, we

mean the DL syntax as described in Section 1, for a

given locality k. Note that all DL statements, which

we shall call Strictly Local Statements, are sentences.

Remark 2 (Overlap Requirements.). It is possible to

have overlaps of individuals or atomic concepts in the

languages at the different localities. In such cases we

shall impose the requirement that if two atomic state-

ments, from different localities, are syntactically iden-

tical, then they are also semantically identical, and

vice versa. This carries over to more complicated

statements in a straightforward way. We shall also

assume that, apart from such overlaps, there are no

symbols in common between the DLs in different lo-

calities.

Deﬁnition 1. Bridge Declarations: Syntax. When

we relate a concept term in one ontology such as A

in i to a concept term in another ontology such as B

KEOD 2010 - International Conference on Knowledge Engineering and Ontology Development

148

in j then we call such a declaration a Bridge Declara-

tion and we express this as a terminological statement

⊑ B

, or A

≡ B

as the case may be.

Deﬁnition 2. Basic Global Formulae. If ϕ is a

strictly local statement in locality k, then ϕ

is a basic

global statement, with the intended meaning that ϕ is

true in k. If A

⊑ B

is a bridge declaration, then it is

also a basic global statement.

Global Statements. 1. Basic global statements are

global statements (henceforth designated by bold let-

ters: X, Y, Z etc.). 2. If X and Y are global state-

ments, then (X∨ Y) is a global statement. 3. If X

and Y are global statements, then (X∧Y) is a global

statement. 4. If X is a global statement and k is any

locality, then X

is a global statement.

We now extend strictly local syntax and seman-

tics, referring to the ordinary A L C , with the admix-

ture of global statements, to give our local syntax.

Deﬁnition 3. Local Syntax. For locality k, this is the

inductive closure of: 1. All strictly local statements

are local statements of locality k, 2. If X is a global

statement then it is a local statement in k, 3. If A, B

are local statements, then local statements formed us-

ing positive propositional connectives are also local

statements of k, e.g. (A∧B), (A∨ B). 4. If A is a local

statement and k is a locality, then A

is also a local

statement of k.

hDL, +PLi Semantics. Strictly local semantics has

been coveredin Section 1. A local model for a locality

k is therefore of the form I

= h∆

, ∆

, ι

Deﬁnition 4. Global Models. If K is the set of lo-

calities, we deﬁne a global model to be a structure

M = {I

: k ∈ K} such that I

is a local model for

the locality k.

We write I



ϕ, where ϕ is either a terminolog-

ical or assertion statement in locality k, to mean that

is a model for locality k.

Deﬁnition 5. Semantics of Basic Global State-

ments. If M = {I

: k is a locality}, then we shall

write M |=

[a] to mean that x : A

is true in M

iff A is a concept term in locality j such that, when

x ∈ O

(the set of individuals in locality k) is assigned

the value a i.e. ι

(x) ∈ ∆

, then I

a : A.

Bridge Declaration Semantics. The basic step in

deﬁning global satisfaction is as in (Cruz and Cross-

ley, 2009): a basic global statement of the form ϕ

is true globally if, and only if, the local interpre-

tation I

makes ϕ true. Then use the usual rules

of propositional calculus for combinations of global

statements. What is our corresponding semantic def-

inition for bridge declarations? Intuitively A

⊑ B

translates as ∀x(x : A

→ x : B

), which in turn may

be rendered as ∀x((x ∈ i ∧ x ∈ A) → (x ∈ j ∧ x ∈ B)).

We take the natural semantics for this, which may be

found in clause 2 in the next deﬁnition.

Remark 3. Note that our natural interpretation of

⊑ B

means that if c ∈ O

and c ∈ A, then c must be

in O

. An important consequence of this is that when

we have a bridge declaration A

⊑ B

, which is true

for a given global model, then A

⊑ ⊤

will also be

true, where ⊤

is the universal concept for locality j.

Deﬁnition 6. Global Semantics. If M is a global

model, we deﬁne global satisfaction inductively on the

complexity of a global statement X.

1. If X is a basic global statement ϕ

, where ϕ a

strictly local statement in k, then X = ϕ

is glob-

ally satisﬁed in M, written M |=

, iff I

ϕ.

In this case we also say that ϕ is locally satisﬁed

at k, and write M |=

2. if X= (A

⊑ B

), then M |=

X iff it is the case that

for all objects c ∈ O

, I

A[c] implies c ∈ O

and I

B[c]. The analogous deﬁnition applies

for the case of the ≡ connective.

3. If X = (Y∨Z), then M |=

X iff M |=

Y or M |=

Z, and analogously for X = (Y∧ Z).

4. if X = Y

, M |=

X iff M |=

Local Semantics.

1. If ϕ is a strictly local statement in k, then M |=

iff I

ϕ.

2. If a basic global statement Φ = ϕ

(which is by

deﬁnition a local statement too), then M |=

Φ iff

ϕ.

3. If Φ is a bridge declaration, A

⊑ B

, then we

deﬁne M |=

Φ iff M |=

Φ as in Deﬁnition 6,

clause 2. Similarly for A

≡ B

4. If Φ = Ψ∨ Θ, then M |=

Φ iff M |=

Ψ or M |=

Θ, and analogously for Φ = Ψ ∧ Θ.

5. If Φ = Ψ

, then M |=

Φ and M |=

Φ iff M |=

Ψ.

Note here the change from γ to i.

If I

A we say M locally satisﬁes A in locality

For the simpliﬁcation of statements we have se-

mantic equivalents, as opposed to the syntactic equiv-

alents of (Cruz and Crossley, 2009). We use the ordi-

nary rules of (positive) propositional calculus for tier-

1 and the usual DL rules for tier-0. Between the tiers

we use the semantic version of the ﬂattening rules of

(Buvaˇc et al., 1995). hDL, +PLi is a system with local

ontologies which we treat as localities. These locali-

ties are assumed to be consistent with each other.

CONTEXTUALIZING ONTOLOGIES FOR AGENTS

149

For soundness (and consistency) of a set Γ of

statements we may assume Γ contains only global

statements initially since any statement A local to k

is semantically equivalent to the global statement A

Deﬁnition 7. A global set of statements, Γ, is said

to be (a) globally consistent, (b) locally consistent or

strict locally consistent to mean that there is, respec-

tively, (a) a global, (b) a local model for Γ, respec-

tively.

Layered Tabelaux. In the DL world, logic is

treated not as a system of axioms and rules of de-

duction but in terms of tableaux in the same style as

modal logics are often treated. Our aim is to show

that, given a consistent set of global statements, we

can extend these to a complete tableau.

We now introduce thetableau construction method

for hDL, +PLi which, inter alia builds an ordinary DL

tableau for each locality using the Tableau Comple-

tion rules, TCR, as found in (Haarslev et al., 2000)

and earlier in (Buchheit et al., 1993). From now on,

an ABox A will be superscripted with its locality: if

O(k) is the ontology in locality k, then its ABox is

; likewise for TBoxes. The necessary additional

rules for our tabelaux are in Section 2. We call the

tableaux of hDL, +PLi layered tableaux because the

tableaux for hDL, +PLi are trees of forests of trees.

The main tree deals with global statements but

the global statements are evaluated to eventually wind

down to the local tableaux they inﬂuence. So our

tableau expansion rules are of two kinds: a) those that

govern global statements and b) those that govern lo-

cal statements.

We recall the relevant results from (Horrocks

et al., 2000) and (Cruz and Crossley, 2009) for local

tableaux.

Deﬁnition 8. An ABox A , is consistent iff there is an

interpretation I of its TBox such that it is an inter-

pretation of all the assertions in A .

Theorem 1. A concept C is satisﬁable in A iff A ∪

{a :C} is consistent with A .

Deﬁnition 9. A tableau is clash free iff none of its

nodes contains a clash. It is complete if no tableau

completion rules can be applied to it.

Theorem 2. Satisﬁability and subsumption of con-

cepts is reducible to testing consistency of ABoxes (cf.

(Horrocks et al., 2000), Theorem 1).

Theorem 3. An ABox, A , is consistent iff it has a

complete tableau. This also implies that because it

has a complete tableau, A has a model, cf. (Haarslev

et al., 2000).

Theorem 4. Every global statement is semantically

equivalent to a propositional combination (using only

∨ and ∧) of basic global statements (including bridge

declarations).

The proof is easily established as a semantic ver-

sion of Theorem 1 of (Cruz and Crossley, 2009).

Because we are starting off with a consistent set of

formulae, conﬂicts that may occur due to bridge rule

declarations will be detected. Furthermore, the issue

of blocking is handled by the local tableau completion

rules, following (Haarslev et al., 2000).

The Completion Procedure. At the top we have

a propositional tableau, but we then extract each lo-

cal component and throw it to the appropriate local

DL tableau for its local handling. Conversely, with-

out simpliﬁcation techniques, some of these compo-

nents will affect other localities and need to be put

back into the collection of global statements. This

process may be circumvented by ﬁrst ﬂattening the

global statements, and then expressing them as propo-

sitional combinations of basic global statements.

We now give a formal description.

We write T for the tree whose root is Γ and whose

nodes are sets of statements producedby the construc-

tion operation on global statements. After a ﬁnite

number of steps the global statements will all have

been ‘reduced’ to propositional combinations of basic

global statements. The basic global statements with

superscript k are then treated locally, in their relevant

localities, using the TCR in the usual way. For each

k this will create a standard DL tableau. The process

terminates when no more ‘reductions’ can be made.

The ﬁrst two steps, which are always to be

rechecked, ensure that we do not generate inconsis-

tent tableaux.

1. If at any stage we would be adding a contradictory

atomic statement, e.g. a : ¬A to a branch in a local

tableau containing a :A, then we abort that branch.

2. If all the branches in a local tableau are aborted,

then remove the branch above that. Note that

this will remove all the tableaux coming from that

branch. This is equivalent to a local contradiction

translating into a global one.

3. If we split at a node then we replicate the tableau

that we have at that node before adding the new

items to the two emanating branches.

4. Reduce the number of superscripts in a global

statement to a minimum by ﬂattening.

5. By Theorem 4 we can reduce every global state-

ment to a propositional combination of basic

global statements and bridge declarations.

KEOD 2010 - International Conference on Knowledge Engineering and Ontology Development

150

Figure 2: Our tableau rules for layered tableau T.

Rule Name and Operation

Formula in Γ

Flat rule (X

)

Add X

to T.

Flat-0 rule for

a basic global

statement ϕ

Send ϕ to the tableau for locality

j (see Step 8).

α rule (X

∧ Y)

Add X

, Y

to T.

β rule (X

∨ Y)

Add X

as a left node to T, Y

as a right node to T. Note. This

will form a split in our layered

tableau.

Bridge rule

for A

⊑ B

For each c in locality i, if c ∈ O

and c : A is in the tableau for i

then add c : B to the tableau for

j. Conversely, if c : ¬B is in the

tableau for j and c ∈ O

, then

add c : ¬A to the tableau for i.

Only do these steps if consistent.

6. Apply the rules in Figure 2 to develop the tableau.

(a) Peculiar to our case are Flat and Flat-0. These

rules allow us to ‘enter’ a locality. Further-

more, what is of critical importance in these

rules are the β and Flat-0 rules. β will cause

a branch to split in the layered tableau. Flat-

0 will ‘send’ a statement into a local (sub-

)tableau (see Step 8).

(b) The next two are the usual propositionaltableau

rules, see e.g. (Smullyan, 1968).

to its appropriate locality after stripping the su-

perscript.

If we have c : A for c in O

in the (local) tableau

for i, then we add c : B to the tableau for j. Note

that if c was not already in O

then we add it to

that locality as a new name. On the other hand

if c : ¬B is in that tableau for j, and c ∈ O

, then

we add c : ¬A to the tableau for i. Of course ei-

ther of these procedures might produce a clash.

In this case we abort this branch (see Step 1

above).

7. Next we put each concept term into negation nor-

mal form, see (Haarslev et al., 2000), i.e. where

all negation signs are only applied to atomic con-

cepts.

8. At this stage we use the TCR in each locality, to

develop the individual tableaux. However, each

time we add a new constant to a tableau we must

then use any applicable bridge declaration (see

Figure 2 and Step 6c) again because such an ad-

dition may add new statements to a different local

tableau.The TCR will add new atomic assertions

to a local tableau.

9. Finally we repeat all of the above steps fairly un-

til only atomic assertions and bridge declarations

remain.

Remark 4. We cannot eliminate the bridge declara-

tions since, whenever another rule introduces a state-

ment such as c : A

, we must check whether any bridge

declaration for A

⊑ B

is applicable (see Figure 2

and Step 6c).

It is clear from the above rules that we obtain ever

shorter statements when we start from a propositional

combination of global statements, provided we are

dealing with ﬁnite statements and a ﬁnite number of

localities.

Lemma 1. (a) Invariance of A

, cf. (Haarslev et al.,

2000)). Assume that Γ together with each ontology

O(k) is system-wide consistent, then our sending op-

eration preserves system-wide consistency. I.e. for all

k, , if A

′

is derived from A

by application of our

rules for layered tableaux, then A

′

is also consistent

whenever A

is.

(b) Model Existence If our hDL, +PLi yields a

clash free and complete tableau, then our hDL, +PLi

has a model.

Proof. We shall rely on what happens in the local

tableaux and we shall use the results of (Haarslev

et al., 2000), especially Deﬁnition 27, in construct-

ing a canonical interpretationI

for each locality and

then follow Theorem 28 of the same work.

Let T be a layered tableau for our hDL, +PLi

which is complete and clash free. Then for each of

the ﬁnal ABoxes A

(for any local ontology i), the

local tableau is clash free and complete. Hence by

Theorem 3, we can construct a canonical interpre-

tation structure I

as in (Haarslev et al., 2000) for

each of these A

and know that I

|= A

. Now let

M = {I

|i ∈ K}: this is our global model. We then

prove that M satisﬁes every global statement X ∈ Γ

by induction on the structure of X.

We only consider here M |=

X where X = (A

⊑

) as this is the only non-trivial case. Since we have

a layered tableau that is clash free and complete, then

if c : A was in the tableau for i, c : B was added suc-

cessfully to j, and conversely, if c : ¬B was in the

tableau for j, then a : ¬A was added to the tableau

for i, and moreover, these were only added provided

consistency was maintained. So if I

|= a : A in

the canonical interpretation for the local tableau of i,

then I

|= c : B. Conversely, by De Morgan’s laws if

M |=

a : ¬B then a ∈ O

and M |=

a : A.

CONTEXTUALIZING ONTOLOGIES FOR AGENTS

151

Completeness. Since a tableau is complete if it has

no clashes and no more tableau completion rules can

be applied to it., we deﬁne a layered tableau to be

complete if no transformation rule (tableau comple-

tion rule) can be applied to it and it is clash free.

Theorem 5. Let A

be an augmented ABox in the

tableau of locality k. If A

is consistent then there

exists at least one completion A

′

of A

computed

by applying our completion rules.

Use Lemma 1 (a) for each local tableau, then these

local tableaux determine the truth values of the atomic

propositions (which are basic global statements) in

tier-1. If the set of global formulae were inconsis-

tent, then it would contain (a : C)

and (a : ¬C)

for

some concept term C in k. But then A

would be

inconsistent, which is a contradiction.

Theorem 6. (a) Our hDL, +PLi is system-wide con-

sistent iff our hDL, +PLi has a layered tableau which

is clash free and complete.

(b) (Decidability) Checking for the consistency of

our hDL, +PLi system is a decidable problem pro-

vided Γ and the number of localities is ﬁnite.

Theorem 7 (Transitive Subsumption Propagation).

Every complete model of a set Γ of global formulae

containing the Bridge Rules, A

⊑ B

and B

⊑ C

, is

a model of A

⊑ C

The proof is a straightforward exercise in seman-

tics.

3 CONCLUSIONS

This paper serves to support the contention that the

tiered logic method TLM transfers the nice properties

of local logics into the tiered scheme. hDL, +PLi,

using the TLM, exhibits soundness, completeness and

decidability in a similar way to DL in tier-0. TLM pro-

vides a contextualized DL system without the over-

head of heavy duty theoretical machinery in con-

trast with DDL and E -Connections. hDL, +PLi has

been simulated using the distributed reasoning tool

RACER (Haarslev and M¨oller, 2001) (now called

RacerPro) with consistent results.

On a more recherch´e note, the problem of the

negation of Bridge Declarations (see Remark 1),

seems an interesting one; one which has been treated

in a different way in (Zimmermann and Duc, 2008).

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