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Deriving Valid Expressions fromOntology Definitions
Yannis Tzitzikas1 2, Nicolas Spyratos3, Panos Constantopoulos1
2
1 Department of Computer Science, University of Crete, Greece2
Institute of Computer Science, ICS-FORTH
3 Laboratoire de Recherche en Informatique, Universite de
Paris-Sud, France
Email :f
tzitzik, panosg
@ics.forth.gr, [email protected]
Abstract. In this paper, we consider an ontology as a set of
terms together
with three binary relations on terms, called synonymy,
subsumption and cross-
reference. We present a model-theoretic interpretation of
ontologies and we
show that this approach although appropriate for deciding the
soundness of
an ontology, is not sufficient for providing a sound and
complete inference
procedure for checking the validity of expressions in an
ontology. Therefore
a different "proof-theoretic" approach which allows checking the
validity ofexpressions is also presented.
1 Introduction
Research on ontologies is becoming increasingly widespread in
the computer science commu-nity and its importance is being
recognized in many diverse research fields and applicationareas,
such as conceptual analysis, conceptual modeling, information
retrieval, informationintegration, agent communication, semantic
annotation (see [11] for a review). There arenumerous definitions
of what an ontology is, revolving around the basic idea that "an
ontologyis a consensual and formal specification of a vocabulary
used to describe a specific domain"
(see [10] for a review).In this paper, we consider an ontology
as a set of terms together with three binary relations
on terms, called synonymy, subsumption and cross-reference.
These ontologies resemble to thestructure of those linguistic
ontologies (such as thesauri) which can accept a clear
semanticinterpretation. They also resemble to the organizational
structures employed by web siteproviders, in order to organize
their contents and to provide browsing and retrieval services,e.g.
the subject hierarchy of Yahoo!. We present a model-theoretic
interpretation of ontologiesinspired by the approach of [25], [26]
which allows checking whether a linguistic ontologyis appropriate
for tasks requiring a semantic interpretation of ontologies. We
show that themodel-theoretic interpretation although appropriate
for deciding the soundness of an ontology,is not sufficient for
providing a sound and complete inference procedure for checking
thevalidity of expressions in an ontology. Therefore a different
"proof-theoretic" approach is also
presented.In section 2 we define our ontologies and in section 3
we discuss their implementation.
In section 4 we discuss the soundness of an ontology and in
section 5 we present the queryexpressions that we consider. In
section 6 we investigate the model theoretic interpretation as
aninference procedure, while in section 7 we present an alternative
inference procedure. In section8 we compare our approach with other
knowledge representation and reasoning approaches,and finally, in
section 9 we review related work and conclude the paper.
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2 Ontologies
Intuitively, an ontologyconsists of a set of words, or terms,
and a set of relationships betweenthe terms. Each term describes
some aspect of a set of objects of interest and the
relationshipsbetween terms reflect corresponding relationships
between the objects. The assignment ofmeaning to terms and the
recording of relationships are the outcome of a formal process
andenjoy the consensus of some community. We conceptualize the
world as a set of objects, thatis, we assume an arbitrary, but
fixed, domain of discourse and a corresponding set of objectsO b j
. The only constraint that we impose on the set O b j is that it
must be a denumerable set.
Def 2.1 A terminology is a finite set of words pertaining to the
objects in a specific domain ofdiscourse. The elements of a
terminology are called terms.
For example, the following set is a terminology for describing
the content of universitycourses:
T = f m a t h e m a t i c s ; h i s t o r y ; c o m p u t e r s
c i e n c e ; h u m a n i t i e s ; : : : g
Here the underlying set O b j is the set of course identifiers,
such as M315, H213, CS265 andso on. The set of objects described by
a term is the interpretation of that term.
Def 2.2 Given a terminology T , we call interpretation of T over
O b j any function I : T ! 2 O b j .
Thus each termt
denotes certain objects inO b j
and its interpretationI ( t )
is the set ofobjects to which the term t is correctly applied.
In our discussion the set O b j will be usuallyunderstood from the
context. So we shall often say simply "an interpretation" instead
of "aninterpretation over O b j ". In the previous example, an
interpretation ofT by a given agent mightassign to the term m a t h
e m a t i c s all course identifiers beginning with M , to h i s t
o r y all courseidentifiers beginning with H , to c o m p u t e r s
c i e n c e all course identifiers beginning with C S , andso on.
However, note that different agents may attach different
interpretations to the same term.Strictly speaking, interpretation,
as defined above assigns to a term denotational or
extensionalmeaning ([30]).
As we explained earlier, an ontology comprises not only a
terminology, i.e. a set of terms,but also relationships between
those terms. For the purposes of this paper, we shall considerthree
kinds of relationships, as stated in the following definition:
Def 2.3 An ontology is a quadruple A = ( T ; ; ; ) where T is a
terminology and ; ; arebinary relations over T such that:
" " is reflexive, symmetric and transitive (i.e. an equivalence
relation). Conceptually, " "means "synonym of", e.g. m a t h m a t
h e m a t i c s , c o m p u t e r s c i e n c e i n f o r m a t i c
s .
" " is irreflexive, transitive, and asymmetric. Conceptually, "
" means "subsumed by" or"isA" or "covered by", e.g. m a t h s c i e
n c e s , c a n a r i e s b i r d s .
" " is reflexive and symmetric. Conceptually, " " means
"cross-reference" or "related to"in the sense of non-disjoint
interpretations e.g. m a t h c o m p u t e r s c i e n c e , c o m
p u t e r s e l e c t r o n i c s , p e t s p a r r o t s .
Figure 1.(A) shows graphically an ontology for describing the
content of university courses,where terms are represented as nodes
and term relationships as labeled edges. Observe thatedges labeled
by " " are oriented (because " " is asymmetric), while edges
labeled by " ",or " " are not oriented (because the relations " "
and " " are symmetric). Many web catalogs,such as Yahoo!, employ an
ontology in order to organize their contents (for more see [21])
andFigure 1.(B) shows graphically an ontology for describing the
contents of web pages advertisingelectronic products.
Now, let A = ( T ; ; ; ) be an ontology and let I be an
interpretation ofT . Clearly, in orderfor I to make sense in the
ontology, it must also reflect all relationships that exist
betweenterms. This is precisely what is stated in the following
definition of a model.
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Cameras
Still cameras
PhotoCameras
Moving picture c.
Mobile Phones
Twin lens reflex c.
Electronics
Television c.
TV c.
Cine c.
Underwater c.
Stereo cam.
Photography
Reflex c.
Single lens reflex c.
SLR c.
Instant picture c.Miniature c.
35 mm c.Artificial Intelligence6
Databases5
Math
Sciences
Mathematics Computer Science
AI History9
Philosophy
Humanities
10
8
32 4
1
7
A B
~
~
~
Figure 1: Graphical representation of two ontology
definitions
Def 2.4 Let A = ( T ; ; ; ) be an ontology. An interpretation I
of T is called a model of A ifthe following hold, for any terms t ;
t 0 in T :(i) I ( t ) 6= ; , i.e. every term is associated with at
least one object (i.e. with a "witness")(ii) ift t 0 then I ( t ) =
I ( t 0 ) , i.e. synonymy is interpreted as set equality(iii) ift t
0 then I ( t ) I ( t 0 ) , i.e. subsumption is interpreted as
strict set inclusion(iv) ift t 0 then I ( t ) \ I ( t 0 ) 6= ; ,
i.e. term cross-reference is interpreted as nonempty
intersection
Let us now discuss these constraints. Recall that an
interpretation, as defined in Def. 2.2assigns to a term extensional
meaning 1, and an ontology definition actually specifies a
number
of constraints that must hold between these meanings. However we
must clarify that theinterpretation of a term does not refer to its
"extension" in a particular database, but it refers toits extension
in the whole domain O b j , and this is the reason for requiring I
( t ) 6= ; for each termt and model I . On the contrary, I ( t ) =
; would mean that term t cannot by applied to any of theobjects in
O b j , but in this case, t would be useless and should not be
included in the ontologydefinition. Another remark concerns the
subsumption relation " ". We interpret subsumptionby strict set
inclusion ( ) and not by set inclusion ( ) as it is commonly done
(see extensionalsubsumption [17]). Roughly, if subsumption was
interpreted by set inclusion ( ) then a "cycle"(cycles will be
defined formally in section 4) might induce that a term t is
synonym to a termt
0 , although t may have been declared that subsumes t 0 .
However this phenomenon does notfit well to the "axiomatic nature"
of the recorded subsumption relationships of an
ontology.Nevertheless, strict set inclusion ( ) introduces problems
when we want to extend a stored
interpretationI
to a modelI
0
(which is needed in query answering in a materialized
ontology).This issue is discussed in a subsequent article on
materialized ontologies.
Def 2.5 An ontology A is called soundif there is a model of A ,
otherwise it is called unsound.
If A is a sound ontology we will write A = , while if A is an
unsound ontology we will writeA 6 = . We will return to the issue
of soundness in section 4.
3 Implementing an Ontology
The definition of an ontology can be implemented using any of a
number of data models. Forexample, using the relational model [7],
we can implement the definition of an ontology as adatabase schema
consisting of four tables, one for storing the terminology and the
others forstoring the three relationships of the ontology:
TERMINOLOGY(term-id:Int, term-name:Str)
SYNONYM(term1:Int, term2:Int)
ISA(term1:Int, term2:Int)
RELATED(term1:Int, term2:Int)
1In contrast to the intensional meaning of terms (i.e. [14],
[5]).
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" " over the set T = as follows: for all c ; c 0 in T = , c c 0
iff there is t 2 c and t 0 2 c 0 such thatt t
0 . We shall use the same symbol for both " " and its extension
over T = , as the distinctionwill be clear from the context.
BA
(III)(II)(I)(III)(II)(I)
a b
c
a b a b
cd
a,b
d,ca,c
b
a,b~~
~
~
Figure 4: Three unsound ontologies
Def 4.1 An ontology A = ( T ; ; ; ) is acyclic if " " is acyclic
over T = . Otherwise theontology A is cyclic.
Recall that a binary relation R over a set N is acyclic, if for
every element n 1 of N , thereis no sequence n 1 ; : : : ; n k with
k 1 such that n 1 = n k and n i R n i + 1 for all i = 1 ; : : ; k ?
1. Forexample, the ontologies of Figure 4.(A) are cyclic, because
in each graph of Figure 4.(B) the setof nodes is the set T = and "
" is cyclic over T = .
Proposition 4.1 Every cyclic ontology is unsound.Proof:
Let A = ( T ; ; ; ) be a cyclic ontology, and assume that there
exists a model I of A .
It follows from Def. 2.4(ii) that all terms of an equivalence
classc 2 T =
have the sameinterpretation, and let us denote this
interpretation by I ( c ) . It follows from Def. 2.4(iii) that ifc
c
0 then I ( c ) I ( c 0 ) . Since A is cyclic there exists a
sequence of equivalence classes c 1 ; : : : ; c kwith k 1 such that
c = c 1 c k = c . This implies that I ( c ) = I ( c 1 ) I ( c k ) =
I ( c ) .Obviously, this is impossible, hence A is unsound.
Proposition 4.2 Every acyclic ontology A is sound.
We will prove this proposition by providing below an algorithm
(Algorithm 4.1) which takesas input an acyclic ontology A and
produces a model m of A . In this algorithm, we assume afunction
witness that takes as argument either a term t or a pairt t 0 of
different but related termsand returns a single object from O b j .
We assume that witness is injective everywhere except onsymmetric
pairs, i.e. pairs of the form t t 0 and t 0 t with t 6= t 0 . For
such pairs witness returns thesame object, i.e. witness(t t 0 ) =
witness(t 0 t ). Note that the existence of such a function
requiresthat the set O b j is "adequately" large. Specifically, we
must have: card( O b j ) card(T ) + 1/2 *card( f t t 0 t 6= t 0 g
), where "card" stands for "cardinality". The correctness of the
algorithmthat follows, and therefore the proof of Proposition 4.2,
pre-supposes satisfaction of the aboveconstraint.
Let us apply the algorithm to the ontology shown in Figure 5,
assuming O b j to be the set ofall positive integers, and the
function witness to be defined as follows:
w i t n e s s ( a ) = 1 w i t n e s s ( b ) = 2 w i t n e s s (
c ) = 3 w i t n e s s ( d ) = 4w i t n e s s ( e ) = 5 w i t n e s
s ( f ) = 6 w i t n e s s ( b d ) = 7
Below we describe each step of the algorithm, and Figure 6 shows
the interpretations of theterms as they are being updated in each
step of the algorithm.
Step 1: In this step a distinct integer is assigned to the
interpretation of each term:m 1 ( a ) = f 1 g , m 1 ( b ) = f 2 g ,
m 1 ( c ) = f 3 g , m 1 ( d ) = f 4 g , m 1 ( e ) = f 5 g , m 1 ( f
) = f 6 g
Step 2: In this step, the relationship b d causes the following
assignments:m 2 ( b ) = m 1 ( b ) f 7 g = f 2 g f 7 g = f 2 7 gm 2
( d ) = m 1 ( d ) f 7 g = f 4 g f 7 g = f 4 7 g
Step (3): In this step, the relationship c b causes the
following assignments:m 3 ( c ) = m 2 ( c ) m 2 ( b ) = f 3 g f 2 7
g = f 2 3 7 gm 3 ( b ) = m 3 ( c ) = f 2 3 7 g
Step (4): In this step, the relationships b a , e f and d e
cause the assignments:m 4 ( a ) = m 3 ( a ) m 3 ( b ) = f 1 2 3 7
g
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(i) m ( t ) 6= ; . This holds due to Step 1.
(ii) if t t 0 then m ( t ) = m ( t 0 ) . This holds due to Step
3.
(iii) if t t 0 then m ( t ) m ( t 0 ) . This holds due to Step
4. If t t 0 the algorithmperforms the assignment m ( t 0 ) : = m (
t 0 ) m ( t ) . This assignment guarantees only thatm ( t ) m (
t
0
) . However m ( t ) m ( t 0 ) actually holds because m ( t )
does not contain theelement w i t n e s s ( t 0 ) which belongs to
m ( t 0 ) due to Step 1. If the element w i t n e s s ( t 0 )
werealso a member of m ( t ) it would have been assigned to m ( t )
only due to the existence ofa sequence of equivalence classes c 1 c
k (k 1) such that t
0
2 c 1 and t 2 c k . But
in that case, the relationshipt t
0
and the above sequence would form a cycle which is
acontradiction since the input ontology is acyclic.
(iv) if t t 0 then m ( t ) \ m ( t 0 ) 6= ; . This holds due to
Step 2, and note that this property ispreserved because the
subsequent steps may only enlarge the interpretation of a term.
Clearly, Algorithm 4.1 terminates. Suppose it does not. Then,
since the terminology T , andtherefore T = , is finite, this is
possible only if the loop between Step 3 and Step 5 does
notterminate which is possible only if there is a cycle in T = .
But this is impossible because theontology is acyclic.
Theorem 4.1 An ontology A is sound iff it is acyclic.Proof:
Follows immediately from propositions 4.1 and 4.2.
Two important remarks are in order here. The first is that
Algorithm 4.1 produces a model ofan acyclic ontology, even starting
with an ontology base which might contain an "incomplete"ontology
definition. The second remark is that by slightly modifying
Algorithm 4.1 we canobtain an algorithm which takes as input any
ontology A (cyclic or acyclic) and produces amodel m of A , if A is
sound, or returns "UNSOUND" if A is unsound. This is Algorithm
4.2which in comparison with the Algorithm 4.1, has one extra
if-then statement in Step 4, whichis shown in boldface. It can be
easily proved (similarly to the proof of Theorem 4.1) that
anontology is unsound iff the Algorithm 4.2 returns "UNSOUND".
Figure 7 shows the applicationof this algorithm to the ontology of
Figure 4.(III).
Algorithm 4.2 Ontology Model 2Input: An ontology A = ( T ; ; ;
)Ouptut: A model m of A , if A is sound, otherwise "UNSOUND"
:Step 4: For each t t
If m ( t ) 6 m ( t ) thenm ( t ) := m ( t ) m ( t )If
m ( t ) 6 m ( t )
then
return "UNSOUND"
:
Returning back to the constraints of Def. 2.4, we can now say
that if subsumption wasinterpreted by set inclusion ( ) then all
ontologies would be sound, even the cyclic ones.Allowing cyclic
ontologies has the following consequence. If a subset C = f c 1 ; :
: : ; c k g T = forms a cycle, then all terms which are elements of
one equivalence class in C , are essentiallysynonyms, although they
may have not been declared as synonyms.
5 Querying an Ontology
The relationships contained in the definition of an ontology can
be seen as expressions thatcombine terms of the terminology using
the connectors " "; " "; " ". There is an infinite set
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Step 1 Step 2 Step 3 Step 4 Step 3 Step 4 OUTPUTT m 1 m 2 m 3 m
4 m 5
af
1g f
1,2g f
1,2,3,4g
UNSOUND UNSOUNDb f 2 g f 1,2g f 1,2,3,4gc
f
3g f
3,4g f
1,2,3,4g
d f 4 g f 3,4g f 1,2,3,4g
Relationships a b d a a b d athat cause d c b c d cthe
assignments
Figure 7: Run of Algorithm 4.2 to the ontology base of Figure
4.(III)
of expressions that one can form in this way, and the
relationships contained in the definition ofthe ontology can be
seen as the given validexpressions of the ontology. New valid
expressionscan be inferred from given ones, and the purpose of this
section is to define the inferencemechanism for doing so.
For instance, in Figure 1 the expression M a t h S c i e n c e s
is not member of the completeddefinition of the ontology. However,
this expression can be characterized as valid, because onecan
easily see that for every model I holds I ( M a t h s ) I ( S c i e
n c e s ) . The same holds for theexpressions M a t h e m a t i c s
C o m p u t e r S c i e n c e , and C o m p u t e r S c i e n c e H
u m a n i t i e s . On theother hand, the expression M a t h A I ,
cannot be characterized as valid, since in the model I ofthe
ontology, which is given below, I ( M a t h ) \ I ( A I ) = ;
(actually, this is the model produced
by Algorithm 4.1).I ( S c i e n c e s ) = f
1,2,3,4,5,6,7,11,12,13g I ( M a t h e m a t i c s ) = f 2,3,11gI (
M a t h ) = f 2,3,11g I ( C o m p u t e r S c i e n c e ) = f
4,5,6,7,11,12,13gI ( D a t a b a s e s ) = f 5,12g I ( A r t i f i
c i a l I n t e l l i g e n c e ) = f 6,7,12,13gI ( A I ) = f
6,7,12,13g I ( P h i l o s o p h y ) = f 8,13gI ( H i s t o r y ) =
f 9 g I ( H u m a n i t i e s ) = f 8,9,10,13g
This will lead us to define validity of expressions with respect
to an interpretation. Beforedoing so, however, we must specify
precisely what an expression is.
Def 5.1 Let A = ( T ; ; ; ) be an ontology. An expression over A
is a description, a strictinclusion, an inclusion or a synonymy,
defined as follows (where t is a term):
description d :: = t d , where is the empty description
strict inclusions i
::= d d
0
, whered ; d
0
are descriptionsinclusion i :: = d d 0 , where d ; d 0 are
descriptionssynonymy s :: = d d 0 , where d ; d 0 are
descriptions
expression e :: = d s i i s
Note that a description is either empty, or it has the form t 1
t n , where n 1 and the t i 'sare terms; we shall call each t
i
a subterm of d . Thus every term of T and every cross-referencet
t
0 of an ontology definition, is a description. Also note that
the notion of expression generalizesthe notion of term and the
relations between terms. Some examples of expressions over
theontology definition of Figure 1 follow:
M a t h A I D a t a b a s e s A I S c i e n c e s
M a t h S c i e n c e s M a t h A I
D a t a b a s e s A I C o m p u t e r S c i e n c e M a t h e m
a t i c s A I M a t h A r t i f i c i a l I n t e l l i g e n c
e
M a t h A I
Let A = ( T ; ; ; ) be an ontology. Any interpretation I of T
can be extended to aninterpretation I over descriptions as follows:
for any description d = t 1 t 2 t k over A , wedefine I ( d ) = I (
t 1 ) \ I ( t 2 ) \ \ I ( t k ) , if d 6= , and I ( d ) = ;
otherwise. For simplicity we shalluse the symbol I to denote both
the interpretation and its extension over descriptions.
Def 5.2 Let A = ( T ; ; ; ) be an ontology. We define validity
of an expression in aninterpretation I of T as follows:
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a description d is valid in I , denoted I = d , if I ( d ) 6= ;
.a strict inclusion d d 0 is valid in I , denoted I = d d 0 , if I
( d ) I ( d 0 ) .an inclusion d d 0 is valid in I , denoted I = d d
0 , if I ( d ) I ( d 0 ) .a synonymy d d 0 is valid in I , denoted
I = d d 0 , if I ( d ) = I ( d 0 ) .
Def 5.3 An expression e is valid in A , denoted A = e , if I = e
for all models I of A .
It follows immediately from Def. 2.4, that the expressions
contained in the definition of anontology A are valid in A . The
main problem that we solve in this paper is the following:given an
ontology A and an expression e decide whether A = e .
Given a model m of A , let E ( m ) denote the set of all
expressions over A which are valid inm , that is, E ( m ) = f e m =
e g . Let us now denote by E ( A ) the set of all expressions
whichare valid in A and we shall call E ( A ) the closure of A .
According to Def. 5.3 this set consists ofthe expressions which are
valid in every model of A , that is,
E ( A ) =
\
f E ( m ) m is a model of A g
6 Investigating a Model-theoretic Method for Deriving Valid
Expressions
In this section we investigate a model-theoretic method checking
the validity of expressions. Inparticular we look for a special
model of A , denoted by m
A
, such that E ( mA
) = E ( A ) , that is,m
A
= e iff A = e for all e . Note that by definition it holds E ( A
) E ( m ) for any model m of A .This means that the model-theoretic
approach as inference procedure is always complete (thatis E ( A )
cannot contain an expression which is not element of E ( m ) for
any model m ). Thusthe difficulty lies in the soundness of the
approach, that is, in finding a specific model m
A
andproving that E ( m
A
) E ( A ) . For example, Figure 8.(A) shows graphically an
ontology A and amodel m of that ontology in which m = b c ,
although obviously A 6 = b c .
a {1,2,3}
b {1,2}
c {2,3}
m
cb
aA
b c
a
a {1} {1}
b {1,2,3} {1,2}
c {1,2,4} {1,3}
x ym mB
Figure 8:
Let us first investigate the descriptions. LetE
d
( m )
denote the set of descriptions which arevalid in a model m , and
Ed
( A ) denote the set of descriptions which are valid in every
modelof A . Now we will look for a model m such that E
d
( m ) = E
d
( A ) . Below we prove that alldescriptions that are valid in
m
a l g
(the model produced by Algorithm 4.1) are also valid in
everymodel of A .
Proposition 6.1 Ed
( m
a l g
) = E
d
( A )
Proof: (see appendix A)
Thus for checking the validity of descriptions in A , we run
once the Algorithm 4.1 whichproduces the model m
a l g
, and then it suffices to check the validity of descriptions in
this model.However the counter-example that follows proves that we
cannot derive the validity of strict
inclusions, or of synonymies, by the model-theoretic
approach.
Example 6.1 Consider the ontology A shown in figure 8.(B).
Clearly, in every model m of A ,we have:
m ( a ) m ( b )
m ( a ) m ( c )
) m ( a ) m ( b ) \ m ( c ) ( 1 )
This means that in some of the models of A , it holds m ( a ) m
( b ) \ m ( c ) (see the modelm
x
), while in the rest of the models it holds m ( a ) = m ( b ) \
m ( c ) (see the model my
). Weconclude that the model-theoretic approach is not
appropriate for checking the validity of strictinclusions and
synonymies, since in the model m
A
that we are looking for, it will either hold:m
A
( a ) m
A
( b ) \ m
A
( c ) , or mA
( a ) = m
A
( b ) \ m
A
( c ) , but none of these expressions is valid inevery model of
A .
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Moreover we cannot check the validity of inclusions by the
model-theoretic approachbecause if we could derive the validity of
inclusions then we would be able to derive the validityof
synonymies ( since A = d d 0 , A = d d 0 and A = d 0 d ), but we
have alreadyshowed that the latter is impossible in the model
theoretic approach. Another counter examplefollows.
Example 6.2 Consider an ontology A = ( f a ; b g ; ; ; ; ; f a b
g ) . Clearly, in every model m , wehave: m ( a ) 6= ; , m ( b ) 6=
; , and m ( a ) \ m ( b ) 6= ; . The top part of the table shown in
Figure 9,presents four models, m 1 ; m 2 ; m 3 ; m 4, of A . These
models demonstrate all possible relationsthat can hold between the
interpretations of the terms a and b . In the bottom part of the
table,each row corresponds to an expression over A , while each
column corresponds to one of thefour models. The presence of a
bullet indicates that the corresponding expression is valid in
thecorresponding model. Observe that in each model, there is at
least one expression, which is notvalid in some or all of the other
models. One can easily see that it is impossible to find a
modelm
A
such that E ( mA
) = E ( A ) ( always it will be E ( m ) E ( A ) ).
overlapping equal subset subset
m 1 m 2 m 3 m 4
a f 1,3g f 1 g f 1 g f 1,2gb f 2,3g f 1 g f 1,2g f 1 g
E ( m
1) E ( m
2) E ( m
3) E ( m
4)
a b
a b
a b
b a
a b a
a b b
Figure 9:
Thus in this section we proved that model-theoretic approach is
a sound and completeinference procedure for checking the validity
of descriptions. However it is unsound forinferring synonymies,
inclusions and strict inclusions.
7 An Alternative Approach for Checking Expressions
In the previous section we showed that the model theoretic
approach is not appropriate forderiving the validity of synonymies,
inclusions, or strict inclusions. In this section we providean
alternative inference procedure for synonymies and inclusions,
which is sound and complete.
Let A = ( T ; ; ; ) be an ontology and let c 1 ; : : : ; c n
denote the elements of T = . For eachterm t 2 T we denote by c ( t
) the equivalence class of t .
Def 7.1 Given a description d = t 1 t k , we call reduction of
d, denoted by r ( d ) , the followingsubset of T = : r ( t 1 t k )
= m i n f c ( t 1 ) ; : : : ; c ( t k ) g
This means that for a given descriptiond = t
1 t
k
,r ( d )
is the subset ofT =
whichcontains the equivalence classes of the subterms of d (that
is all c ( ti
) ), excluding those classeswhich are broader than other classes
of the subterms in d 2. For example, if t
i
; t
j
are subtermsof d and if c ( t
i
) c ( t
j
) then the c ( tj
) is not an element of r ( d ) . Clearly ifr ( d ) = f c 1 ; : :
: c k g , thenc
i
6 c
j
for all i ; j = 1 k , and note that there is only one reduction
for any given d .
Proposition 7.1 Let A be an ontology and d a description over A
. For every model m of A itholds m ( d ) =
T
c 2 r ( d )
m ( c )
Proof:
2This means that the setr ( d )
is essentially the set of minimal elements of the setf c ( t 1 )
; : : : ; c ( t k ) g .
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Let d = t 1 t k and m be a model of A . It follows from Def.
2.4(ii) that all terms of anequivalence class c 2 T = have the same
interpretation, and let us denote this interpretationby m ( c ) .
Thus m ( d ) = m ( t 1 ) \ \ m ( t k ) = m ( c ( t 1 ) ) \ \ m ( c
( t k ) ) . Moreover, and accordingto Def. 2.4(iii), if c c 0 then
m ( c ) m ( c 0 ) , thus m ( c ) \ m ( c 0 ) = m ( c ) . This
implies thatm ( c ( t 1 ) ) \ \ m ( c ( t
k
) ) =
T
c 2 m i n
f c ( t 1 ) ; : : : ; c ( tk
) g
m ( c ) . Hence, m ( d ) =T
c 2 r ( d )
m ( c ) .
Lemma 7.1 Let d ; d 0 be two descriptions over A . If r ( d ) =
r ( d 0 ) then m ( d ) = m ( d 0 ) in everymodel m of A .Proof:
According to Prop. 7.1, in every model m of A we have m ( d ) =
\c 2 r ( d )
m ( c ) andm ( d
0
) = \
c 2 r ( d )
m ( c ) . Since r ( d ) = r ( d 0 ) we conclude that m ( d ) = m
( d 0 ) .
Let d ; d 0 be two descriptions over A . From the above lemma it
is clear that if r ( d ) = r ( d 0 )then certainly A = d d 0 . Thus
we can use this method for checking the validity of
synonymies.Moreover this method is complete, that is, A = d d 0 iff
r ( d ) = r ( d 0 ) . This is proved by thenext theorem.
Theorem 7.1 A = d d 0 ) r ( d ) = r ( d 0 )Proof: (see appendix
A)
Concerning inclusions, we can reduce inclusion checking to
synonymy checking:
Proposition 7.2 A = d d 0 iff A = d d 0 dProof:() ) A = d d 0
means that in every model m it holds m ( d ) m ( d 0 ) . The latter
implies thatm ( d ) \ m ( d
0
) = m ( d ) , hence A = d d 0 d .(( ) A = d d 0 d means that in
every model m it holds m ( d ) \ m ( d 0 ) = m ( d ) . The
latterimplies that m ( d ) m ( d 0 ) , hence A = d d 0 .
Example 7.1 Consider the ontology shown in figure 8.(B), in
which it is clear that A = a b c ,since in every model m , we
have
m ( a ) m ( b )
m ( a ) m ( c )
) m ( a ) m ( b ) \ m ( c )
According to our method A = a b c iff A = a b c a . The later is
valid ifr ( a b c ) = r ( a ) ,which is true, since r ( a b c ) = f
a g . Thus our method also derives that A = a b c . Note thatour
method also derives that A 6 = b c a since A = b c a , A = a b c b
c which isfalse, since r ( a b c ) = f a g , while r ( b c ) = f b
; c g .
Concerning strict inclusions, Figure 10 shows graphically some
ontology definitions andsome indicative strict inclusions which are
valid (or invalid) in these ontologies. Checking thevalidity of
strict inclusions is a more complex reasoning task and we shall
present a sound andcomplete inference procedure which is based on
the construction of a graph, on a subsequentpaper. However in
applications which employ ontologies in order to store descriptions
ofconcrete objects, that is, in materialized ontologies we only
need to check the validity of
inclusions (i.e. for answering queries and checking the
containment of queries).
A a b < c
a
c
b.
a b
c
d
A a b < c A a < b c
a
cb
d
A a < b c
b
a
c
A a < b c d
b c
a
d
A a < b c d
b c d
e
a a b
c d
e
.
A a b < c d
a b
c d
e
.
A a b < c d
a b
c d
.
A a b < c d
Figure 10: Examples of valid and invalid strict inclusions
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8 Comparison with other Knowledge Representation and Reasoning
Approaches
In this section we try to represent our ontologies in some
logic-based languages, in particular,Propositional Calculus, First
Order Predicate Calculus, Horn Clauses and Description Logics.We
study each language in the following manner. At first we look for a
method for translatingan ontology definition A , to a set of well
formed formulas
A
of that language and investigatewhether the semantics and the
corresponding inference rules of the language allow checking
the soundness of A (that is, A =?
, A
= ). If this holds, then we investigate whether anexpression e
(description, synonymy, inclusion, strict inclusion) can be written
as a wff
e
and
whether the wff that are inferred from A
correspond to expressions which logically followfrom A and the
vice versa (that is, A = e?
, A
=
e
). At last we discuss the inferenceprocedures ( ) that are
available for each language.
Propositional Calculus: A candidate method for deriving a set of
propositional wff isshown in the second column of Figure 11. Note
that for each t 2 T , the set
A
contains anatom t , and clearly a model of
A
will satisfy each atom t . However this implies that eachmodel
of
A
will also satisfy all conjunctions of atoms, that is, all
descriptions. From thiswe conclude that our ontologies cannot be
represented in propositional calculus.
First Order Predicate Calculus: Our ontologies can be
represented in first order predicatecalculus, and the third column
of Figure 11 shows the method for producing
A
. Clearlyan ontology A is sound iff the set of wffs
A
is satisfiable. Concerning expressions, the
second column of Figure 12 shows how an expression e can be
written as a wff e , andclearly A = e ,
A
=
e
.
Concerning the inference procedures of first order predicate
calculus, we can check thesatisfiability of
A
by applying resolution to the clause form of A
, where reachingto an empty clause means that
A
is unsatisfiable, otherwise A
is satisfiable. Forchecking the validity of a wff we can apply
the resolution refutation [24] which is a soundand complete
inference procedure for predicate calculus. However recall that
predicatecalculus is semi-decidable, and that even on problems for
which resolution refutationterminates, the procedure is NP-hard -
as is any sound and complete inference procedurefor the first order
predicate calculus [3].
Horn Clauses: A candidate method for deriving a set of Horn
Clauses 3 is shown in the
forth column of Figure 11. For each term t , we produce a fact t
( C ) where C denotes anew and unused constant 4. For each
relationship t t 0 , we produce two facts, t ( C ) ; t 0 ( C )
,where C denotes the same constant. However our ontologies cannot
be represented withHorn clauses, because we cannot express the
proper subset semantics of " ". Recallthat the proper subset
semantics of " ", are expressed in predicate calculus by the wff8 X
t ( X ) ! t
0
( X ) and the wff 9 X t 0 ( X ) ^ : t ( X ) . The latter in
clausal form becomest
0
( S k ) ^ : t ( S k ) , where S k is a new constant. For
expressing this in Prolog would requiretwo facts: t 0 ( S k ) and :
t ( S k ) but the latter is not a Horn Clause. Thus we cannot
expressour ontologies with Horn Clauses
Notice that ift t 0 was interpreted by I ( t ) I ( t 0 ) then we
would be able to represent ourontologies with Horn clauses. However
our expressions (excluding descriptions) cannotbe written as Horn
goals, thus even in this interpretation of subsumption, we cannot
check
the validity of expressions with the inference procedures for
Horn Clauses,
Description Logic:Our ontologies can be represented in
description logics (i.e. see [8]) and the last columnof Figure 11
shows the method for producing a theory
A
= ( T
A
; A
A
) , where TA
is asimple-TBox, and the A
A
is an empty ABox. Note that for each term t we do not produceany
TBox statement, that is, we consider each term t as an atomic
concept (a concept
3Horn clauses form the basis of the language Prolog.4This
resembles the way that resolution for predicate calculus eliminates
existential quantifiers by
Skolem functions. In particular the wff 9 X t ( X ) becomes t (
S k ) where S k is a new constant.
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not appearing in the left side of any TBox-statement). For each
t t 0 we derive two
primitive concept specifications ( ), or a single concept
definition, that is, a statementt = t
0 . For capturing the proper subset semantics of " " we need to
define a new concept
x as follows x t 0 u : t . Clearly, an ontology A is sound if
the TBox of A
is satisfiable,
Concerning expressions, Figure 12 shows how an expression e can
be written as a DLexpression
e
. For checking the satisfiability of A
we can use the consistencyreasoningservice of DL (
A
6 = ). For checking descriptions we exploit the consistency
reasoningservice, in particular, a description d is valid if the
TBox
A
f :
d
g is unsatisfiable. Forchecking synonymies and inclusions we
exploit the subsumption check. For checking
strict inclusions we exploit both consistency and subsumption
check. Concerning theinference procedures of DL tableaux calculus
is a decision procedure for solving theproblem of
satisfiability.
A A
Propositional Calc. First Order Predic. Calc. Horn Clauses
Description L.
t2
Tt 9 X t ( X ) t ( C ) : ?
t t t ! t 8 X t ( X ) ! t ( X ) t ( X ) : ? t ( X ) t t
t ! t 8 X t ( X ) ! t ( X ) t ( X ) : ? t ( X ) t t
t t t ! t 8 X t ( X ) ! t ( X ) t ( X ) : ? t ( X ) t t
9 X t ( X ) ^ : t ( X ) t ( C
t
) : ? x t u : t
t t t t 9 X t ( X ) t ( X ) t ( C ) : ? x t u tt ( C ) : ?
Figure 11: Representation of our ontologies in some logic-based
languages
e
e
First Order Predicate Calculus. Description Logic
d 9 X t 1 ( X ) t k ( X ) t 1 u u t k
d d 8 X ( t 1 ( X ) t k ( X ) ) ! ( p 1 ( X ) p m ( X ) ) t 1 u
u t k v p 1 u u p m and8 X ( p 1 ( X ) p m ( X ) ) ! ( t 1 ( X ) t
k ( X ) ) p 1 u u p m v t 1 u u t k
d d 8 X ( t 1 ( X ) t k ( X ) ) ! ( p 1 ( X ) p m ( X ) ) t 1 u
u t k v p 1 u u p m and
d d 8 X ( t 1 ( X ) t k ( X ) ) ! ( p 1 ( X ) p m ( X ) ) t 1 u
u t k v p 1 u u p m and9 X ( p 1 ( X ) p m ( X ) ) ^ : ( t 1 ( X )
t k ( X ) ) p 1 u u p m u : t 1 u u : t k
Figure 12: Representing expressions
Here d denotes a description t 1 t k and d denotes a description
p 1 p m .
9 Related Work - Concluding Remarks
The term ontology was first introduced in philosophy, where an
ontology is a systematic account
of existence, that is, a system of categories for a certain
vision of the world (e.g. the Aristotle'sontology). Subsequently,
the term ontology was employed in the field of language
engineeringwhich focuses on applications such as building NL
interfaces. The ontologies of this area areoften called "linguistic
ontologies". Roughly, they consist of terms and relationships
betweenterms, where terms denote lexical entries (words), while
relations are intended as lexicalor semantic relations between
terms 5. Initially, linguistic ontologies were written only for
5For instance in WordNet, terms are grouped into equivalence
classes (called synsets). Eachsynset is assigned to a lexical
category i.e. noun, verb, adverb, adjective, and synsets are linked
byhypernymy/hyponymy and antonymy relations. The former is the
subsumption relation, while the latterlinks together opposite or
mutually inverse terms such as tall/short, child/parent (unclear
semantics).
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human use, however the evolution of computers enabled their
representation in a digital form.Examples of such ontologies
include WordNet, Mikrokosmos [19], SENSUS [15], and manyothers. In
recent years, linguistic ontologies are not only used for language
engineering, butalso for a number of applications in information
systems. Information retrieval is one example.In this area
linguistic ontologies often play the role of controlled indexing
languages. Moreoverthey may be used for assisting the query
formulation process, or for expanding queries withsynonyms hyponyms
and related terms in order to improve recall (i.e. [22], [20],
[21], [13]).They are also exploited for automatic classification of
documents to concepts ([21]).
However there are many new applications of linguistic ontologies
(see [11] for a review),i.e. linguistic ontologies are employed for
information integration, for querying information
sources, for achieving reuse in conceptual modeling (eg [9],
[28], [23]). In many of theseapplications, and in contrast to NL
applications, terms commonly are used to denote classes
ofentitiesof thedomain, and a risingproblem is that
lexicalrelations do notalways reflect semanticrelations between
classes of entities of the world (for more see [12]). These
applications demanda semantic interpretation of a linguistic
ontology. Our work can be seen as contributing in thisdirection.
Thesauri are a case of linguistic ontologies. Commonly a thesaurus
consists of a set ofterms structured through relations as specified
in [27]. Thesauri are quite close to the ontologiesthat we consider
in this paper. In particular: the " "-relation corresponds to the
BTG-relation ifit is used to connect terms with same criteria of
identity, for example P a r r o t s BTG B i r d s ; the" "-relation
corresponds to the USE-relation, for example M a t h s USE M a t h
e m a t i c s ; the " "-relation corresponds to the RT-relation. If
we look at some RT-relationships of existing thesauri,e.g. B i r d
s RT O r n i t h o l o g y , P a t h o g e n s RT D i s e a s e s ,
A i r c r a f t RT A n t i ? A i r c r a f t W e a p o n s ,
we infer that the relation RT is actually the union of many
conceptual relations. This meansthat we cannot assign a clear
semantic interpretation to this relation, therefore the RT
relationis defined in thesauri as a symmetric relation. Our "
"-relation corresponds to the special caseof RT which is used to
relate terms with overlapping meaning (e.g. S h i p s RT B o a t s
), whichis not transitive (e.g. f c o g n i t i v e s c i e n c e c
o m p u t e r s c i e n c e ; c o m p u t e r s c i e n c e e l e c
t r o n i c s g 6!c o g n i t i v e s c i e n c e e l e c t r o n i
c s ), since nonempty set intersection is not transitive. Also note
thatin thesauri, terms are distinguished to preferred and to
non-preferred terms. Each non preferredterm has to be related with
a preferred term by a USE relationship, and only preferred termscan
participate in BTG, BTP, BTI, RT relationships. In our approach
each term can participatein any kind of relationships; this allows
a more direct and natural modeling.
Nowadays, the term "ontology" is used to denote quite different
things such as dictionaries(e.g. WordNet, Webster's), thesauri
(e.g. UMLS 6, AAT 7), database schemas (oo-hierarchies),XML/RDF
schemas [4], semantic networks, etc (have a look at the page [6]).
In the area ofknowledge representation, management and transfer,
the most cited definition of an ontologyis that of T. Gruber [9]:
"An ontology is an explicit specification of a
conceptualization"and examples of ontologies for structuring and
reusing large bodies of knowledge, includeCYC [16], KIF/Ontolingua
[1], KA2 [2]. In the web, the term ontology commonly refers to
thecontent-based organizational structures employed by site
providers, in order to organize theircontents and to provide
browsing and retrieval services, e.g. the Yahoo!'s subject
hierarchy.Recently, more structured ontologies are employed for
meta-tagging ([18], [29]). However notethat the structuring of many
available ontologies resembles to the structure of the
ontologiespresented in this paper. Also note that our ontologies
are quite simple thus they can beunderstood, generated and
maintained by average users. Note that the "bookmarks" facility
ofweb navigators, i.e. Netscape, can be considered as a tool for
defining "personal" ontologies
Summarizing our work we can say that in this paper we studied
ontologies which consistof a terminology and three binary relations
on terms, called synonymy, subsumption andcross-reference. For
these ontologies we provided a model-theoretic interpretation and
aninference mechanism for reasoning about their soundness and for
deriving the expressions thatare valid in an ontology. These issues
are essential in almost any application of ontologies. Ourfuture
research concerns the indexing and retrieval of objects in a
materialized ontology and
6Unified Medical Language System7Arts & Architecture
Thesaurus
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the communication of ontology-based agents.
References
[1] ``Knowledge Interchange Format". Draft proposed American
National Standard (dpANS)NCITS.T2/98-004,
(http://logic.stanford.edu/kif/dpans.html).
[2] V. Benjamins, D. Fensel, and A. Prez. ``Knowledge Management
through Ontologies". InPAKM-98, 1998.
[3] E. Borger. ``Computability, Complexity, Logic". Amsterdam:
North-Holland, 1989.
[4] Dan Brickley and R. V. Guha. ``Resource Description
Framework (RDF) Schema specifi-cation: Proposed Recommendation,
W3C", March 1999.
http://www.w3.org/TR/1999/PR-rdf-schema-19990303.
[5] Mario Bunge. Scientific Research I. The Search for Systems,
3(1), 1967.
[6] Peter Clark. ``Some Ongoing KBS/Ontology Projects and
Groups", 2000.http://www.cs.utexas.edu/users/mfkb/related.html.
[7] E. F. Codd. ``A Relational Model of Data for Large Shared
Data Banks". Communicationsof the ACM, 13(6):377--387, 1970.
[8] F.M. Donini, M. Lenzerini, D. Nardi, and A. Schaerf.
``Reasoning in Description Logics",chapter 1. CSLI Publications,
1997.
[9] Thomas R. Gruber. ``Towards Principles for the Design of
Ontologies Used for KnowledgeSharing ". In Formal Ontology in
Conceptual Analysis and Knowledge Representation.Kluwer Academic
Publishers, 1994.
[10] Nicola Guarino. ``Understanding, Building and Using
Ontologies: A Commentary to"Using Explicit Ontologies in KBS
Development"". International Journal of Human andComputer Studies,
46:293--310, 1997.
[11] Nicola Guarino. ``Formal Ontology and Information Systems".
In Proceedings of FOIS'98,Trento, Italy, June 1998. Amsterdam, IOS
Press.
[12] Nicola Guarino. ``Some Ontological Principles for Designing
Upper Level Lexical Re-sources". In Proceedings of first int. Conf.
on Language Resources and Evaluation,Granada, Spain, May 1998.
[13] Nicola Guarino, Claudio Masolo, and Guido Vetere.
``OntoSeek: Content-based Access tothe Web". IEEE Inteligent
Systems, pages 70--80, May,June 1999.
[14] Raili Kauppi. ``Einfuhrung in die Theorie der
Begriffssysteme". Acta UniversitatisTamperensis, Ser A, Vol 15,
University of Tampere, 1967.
[15] K. Knight and S. K. Luk. ``Building a Large-Scale Knowledge
Base for MedicineTranslation". In Proceedings of the 12th National
Conf. on AI (AAAI'94), Seattle,Washington, 1994.
[16] D. B. Lenat. ``CYC: A Large-Scale Investment in Knowledge
Infrastructure". Communica-tions of the ACM, 38(11), 1995.
[17] H. Levesque and R. Brachman. ``A Fundamental Tradeoff in
Knowledge Representationand Reasoning (revised version)". In
Readings in Knowledge Representation. MorganKaufmann Publishers,
1985.
[18] Sean Luke, Lee Spector, David Rager, and Jim Hendler.
``Ontology-based WebAgents". In Proceedings of First International
Conference on Autonomous Agents,
1997. (http://www.cs.umd.edu/projects/plus/SHOE/).[19] K. Mahesh
and S. Nirenburg. ``A Situated Ontology for Practical NLP". In
Proceedings
of IJCAI-95 Workshop on Basic Ontological Issues in Knowledge
Sharing, Montreal,Canada, 1995.
[20] Zygmunt Mazur. ``Modelsof a Distributed Information
RetrievalSystem Based on Thesauriwith Weights". Information
Processing and Management, 30(1):61--77, 1994.
[21] Deborah L. McGuinness. ``Ontological Issues for
Knowledge-Enhanced Search". InProceedings of FOIS'98, Trento,
Italy, June 1998. Amsterdam, IOS Press.
[22] C. Paice. ``A Thesaural Model of Information Retrieval".
Information Processing andManagement, 27(5):433--447, 1991.
-
7/30/2019 QIzvNuXp
16/18
[23] B. Peterson, W.A. Andersen, and J. Engel. ``Knowledge Bus:
Generating Application-focused Databases from Large Ontologies". In
Proceedings of the 5th Workshop KRDB-98, Seattle, WA, USA, May
1998.
[24] J. Robinson. ``A Machine-Oriented Logic Based on the
Resolution Principle". Journal ofthe Association for Computing
Machinery, 12(1):23--41, 1965.
[25] Nicolas Spyratos. ``The Partition Model: A Functional
Approach". Technical Report No.430, INRIA, 1985.
[26] Nicolas Spyratos. ``The Partition Model: A Deductive
Database Model". ACM Transactionson Database Systems, 12(1):1--37,
1987.
[27] International Organization For Standardization.
``Documentation - Guidelines for theestablishment and development
of monolingual thesauri", 1988. Ref. No ISO 2788-1986.
[28] Bill Swartout, Ramesh Patil, Kevin Knight, and Tom Russ.
``Towards Distributed Use ofLarge-Scale Ontologies", 1996.
USC/Information Sciences Institute, CA, September 27,1996,
(http://www.isi.edu/isd/banff paper/ Banff final web/Banff 96 final
2.html).
[29] Frank van Harmelen and Dieter Fensel. ``Practical Knowledge
Representation for theWeb". In Workshop on Intelligent Information
Integration, IJCAI'99, 1999.
[30] W. A. Woods. ``Understanding Subsumption and Taxonomy". In
Principles of SemanticNetworks, chapter 1. Morgan Kaufmann
Publishers, 1991.
A Proofs of Propositions and Theorems
Proof of Proposition 6.1 :First we introduce some notations.
Given a term t , P r o d ( t ) denotes all descriptions which
consist
of one or more repetitions of t , i.e. t t t ; : : : etc.
Clearly, if d 2 P r o d ( t ) then I ( d ) = I ( t ) in
anyinterpretation
I
. Given a set of termsS T
, byP r o d ( S )
we denote all descriptions consisting of oneor more repetitions
of each term in S . For example P r o d ( f t t g ) contains
descriptions of the form:t t t t t t t t ; : : :
etc. Clearly, ifS = f t 1 ; : : : ; t k g , and d 2 P r o d ( S
) , then I ( d ) = I ( t 1 t k ) in
any interpretation I .Below we prove that E
d
( m
a g
) E
d
( A ) by showing that the descriptions that are valid in ma
g
, as itis procuded in each step of the algorithm, are valid in
every model of A too.
Step 1: Letm
S 1 be the interpretation produced by Step 1 of the algorithm.
Clearly a description d isvalid in m
S 1 only if d 2 P r o d ( t ) where t 2 T . In any other case d
is invalid: if d is the empty descriptionthen m
S 1 ( d ) = ; , and if d contains two different subterms, then m
S 1 ( d ) = ; due to the injectivity of thefunction "witness". Thus
we can write E
d
( m
S
1) = f P r o d ( t ) t 2 T g .
Now let m be a model of A . Certainly m ( t ) 6= ; for all t in
T , and also m ( d ) 6= ; for all d 2 P r o d ( t )for t in T .
Hence we conclude that E
d
( m
S 1 ) E d ( A ) .Step 2: Let m
S 2 be the interpretation produced by Step 2. Below we prove
that E d ( m S 2 ) E d ( A ) .Clearly, E
d
( m
S 2 ) includes the descriptions which are valid due to Step 1
(that is, all descriptions inE
d
( m
S 1 ) ). Below we prove by induction that E d ( m S 2 ) also
includes the descriptions which are productsof two related terms,
that is, the descriptions f P r o d ( f t t g ) where t t g . This
means that we will provethat E
d
( m
S 2 ) = E d ( m S 1 ) f P r o d ( f t t g ) t t g .Let m 0 m 1 ;
: : : denote the interpretations produced by each iteration of the
loop in Step 2, where m 0
is the interpretation produced by Step 1, that is m 0 = m S 1,
and m j + 1 is the interpretation produced by m jafter one
iteration.
Let mj
be an interpretation in which Ed
( m
j
) consists of descriptions of the form described above.Let m
j + 1 be the interpretation after one iteration of the loop in
Step 2, and let d be a description such
thatd 2 E
d
( m
j +
1)
, butd 62 E
d
( m
j
)
. Let the difference betweenm
j +
1 andm
j
be the following pair ofstatements:
m
j + 1 ( t ) = m j ( t ) f w i t n e s s ( t t ) g
m
j + 1 ( t ) = m j ( t ) f w i t n e s s ( t t ) g
due to a relationship t t and the fact that mj
( t ) \ m
j
( t ) = ; .Certainly the description d should contain at least
one (or both) of the terms t t . Let us assume that
d contains only the term t , that is we may write d = t x where
x is a description which does not containt or t . We have:
m
j + 1 ( d ) = m j + 1 ( t ) \ m j ( x )
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= ( m
j
( t ) f w i t n e s s ( t t ) g ) \ m
j
( x )
= ( m
j
( t ) \ m
j
( x ) ) ( m
j
( x ) \ f w i t n e s s ( t t ) g )
= m
j
( d ) ( m
j
( x ) \ f w i t n e s s ( t t ) g )
= m
j
( x ) \ f w i t n e s s ( t t ) g
Since mj + 1 ( d ) 6= ; , it must be m j ( x ) \ f w i t n e s s
( t t ) g 6= ; , that is, the object w i t n e s s ( t t ) must be
an
element of mj
( x ) . But since the function "witness" is injective, the
object w i t n e s s ( t t ) belongs onlyto m
j + 1 ( t ) and m j + 1 ( t ) . Thus, x should consist of
repetitions of t or t , and x should not contain anyother term.
Thus the hypothesis that x does not contain the terms t or t is
false. Hence, we can writeE
d
( m
S 2 ) = E d ( m S 1 ) f P r o d ( f t t g ) t t g .Now let
m
be a model ofA
. Ift t
then certainlym ( t t ) 6= ;
, and clearlym ( d ) 6= ;
for everyd 2 P r o d ( f t t g ) since m ( d ) = m ( t t ) .
Hence we conclude that E
d
( m
S 2 ) E d ( A ) .Step 3: Let
m
S 3 be the interpretation produced by Step 3. Below we prove
that E d ( m S 3 ) E d ( A ) .Let m 0 m 1 ; : : : denote the
interpretations produced by each iteration of the loop in Step 3,
where m 0 is
interpretation produced by Step 2, that is m 0 = m S 2, and m j
+ 1 is the interpretation produced by m j afterone iteration. We
prove that E
d
( m
S 3 ) E d ( A ) by induction, that is, we prove that if E d ( m
j ) E ( A )then
E
d
( m
j + 1 ) E ( A ) too.Let d be a description such that d 2 E
d
( m
j + 1 ) , but d 62 E d ( m j ) . Let the difference between m j
+ 1 andm
j
be the following pair of statements:
m
j + 1 ( t ) = m j ( t ) m j ( t )
m
j + 1 ( t ) = m j + 1 ( t )
due to a relationship t t and the fact that mj
( t ) 6= m
j
( t ) .Certainly the description d should contain at least one
(or both) of the terms t t . Let us assume that
d
contains only the termt
, that is we may writed = t x
wherex
is a description which does not containt or t We have:
m
j + 1 ( d ) = m j + 1 ( t ) \ m j ( x )
= ( m
j
( t ) m
j
( t ) ) \ m
j
( x )
= ( m
j
( t ) \ m
j
( x ) ) ( m
j
( t ) \ m
j
( x ) )
= m
j
( d ) ( m
j
( t ) \ m
j
( x ) )
= m
j
( t ) \ m
j
( x )
Let m be a model of A . Since mj + 1 ( d ) 6= ; , we have m j (
t ) \ m j ( x ) 6= ; too. But since E d ( m j ) E ( A ) ,
certainly it holds m ( t ) \ m ( x ) 6= ; . Due to t t it must
be m ( t ) = m ( t ) . Combining these two facts weobtain m ( t ) \
m ( x ) 6= ; , that is, m ( d ) 6= ; . Hence we conclude E
d
( m
S 3 ) E d ( A ) .Step 4: Below we prove that E
d
( m
S 4 ) E d ( A ) where m S 4 is the interpretation produced by
Step 4.Let
m 0 m 1 ; : : : denote the interpretations produced by each
iteration of the loop in Step 4, where m 0 isinterpretation
produced by Step 3, that is m 0 = m S 3, and m j + 1 is the
interpretation produced by m j afterone iteration. We prove that
E
d
( m
S 4 ) E d ( A ) by induction, that is, we prove that if E d ( m
j ) E ( A )then E
d
( m
j + 1 ) E ( A ) .Let
d
be a description such thatd 2 E
d
( m
j + 1 ) , but d 62 E d ( m j ) . Let the difference between m j
+ 1 andm
j
be the following statement:m
j + 1 ( t ) = m j ( t ) m j ( t )
due to a relationshipt t
and the fact thatm
j
( t ) 6 m
j
( t )
.Certainly the description d should contain the term t , that is
we can write d = t x where x is a
description which does not contain t . We have:
m
j + 1 ( d ) = m j + 1 ( t ) \ m j ( x )
= ( m
j
( t ) m
j
( t ) ) \ m
j
( x )
= ( m
j
( t ) \ m
j
( x ) ) ( m
j
( t ) \ m
j
( x ) )
= m
j
( d ) ( m
j
( t ) \ m
j
( x ) )
= m
j
( t ) \ m
j
( x )
Let m be a model of A . Since mj + 1 ( d ) 6= ; , we have m j (
t ) \ m j ( x ) 6= ; too. But since E d ( m j ) E ( A )
this means that m ( t ) \ m ( x ) 6= ; . Due to t t it must be m
( t ) m ( t ) . Combining these two facts weobtain m ( t ) \ m ( x
) 6= ; . Hence E
d
( m
S 4 ) E d ( A ) .
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Thus we proved that Ed
( m
a g
) E
d
( A ) . Moreover, since ma g
is a model of A , we haveE
d
( A ) E
d
( m
a g
) . Hence we conclude that Ed
( m
a g
) = E
d
( A ) .
Proof of Theorem 7.1
Let a b be two descriptions over A , such that r ( a ) 6= r ( b
) . Assume that r ( a ) = f a 1 ; : : ; a k g andr ( b ) = f b 1 ;
: : ; b g , where k l 1 Below we prove that we can always construct
a model m x , such thatm
x
( a ) 6= m
x
( b ) . First note that since r ( a ) 6= r ( b ) , certainly it
must be: (i) r ( a ) 6 r ( b ) , or (ii) r ( b ) 6 r ( a ) ,or
(iii) r ( a ) 6 r ( b ) and r ( b ) 6 r ( a ) .
Assume case (i), that is, r ( a ) 6 r ( b ) . This means that
there exists at least one ax
such that ax
62 r ( b ) .Now consider a model m of A such that m ( a ) = m (
b ) . Below we describe a method for "enlarging" m ,
to a modelm
x
such thatm
x
( a ) 6= m
x
( b )
. This method consists of the two steps:Step 1. We add a new
object ob
to each m ( bi
)
Step 2. We add the objecto
b
, to the interpretation of eachc 2 T =
, such thatc b
i
for ani =
1; : : ; l
.Let m denote the resulting interpretation after Step 1.
Clearly, o
b
2 m ( b ) , while ob
62 m ( a ) , sinceo
b
was not added to the interpretation of ax
. Note that the interpretation m is not certainly a model ofA .
However, since m was derived by enlarging a model m of A ,
certainly the constraints (i),(iv) of Def.2.4, are also satisfied
by
m
. Moreover, since we are working onT =
, the constraint (ii) of Def. 2.4, isalso satisfied by m .
However for satisfying the constraint (iii), we proceed to Step 2,
and obviously theresulting interpretation is a model of A .
What remains to show is that Step 2 did not add the object
ob
to each m ( ai
) , since in that case it wouldbe m ( a ) = m ( b ) . Clearly,
this step would add o
b
to each m ( ai
) only if:
8 a
i
9 b
j
: ai
b
j
Note that certainly it cannot be: 8 a i 9 b j : a i b j since in
that case it would be r ( a ) r ( b ) , and due tor ( a ) 6= r ( b
) , it would be r ( a ) r ( b ) , which contradicts the hypothesis
r ( a ) 6 r ( b ) . Thus, in view of ourhypothesis, we can say that
step 2 would update each
m ( a
i
)
only if:
8 a
i
9 b
j
: ai
b
j
and 9 ax
9 b
x
: ax
b
x
From this formula we can see that it must be bx
2 r ( b ) n r ( a ) , otherwise we would have ax
b
x
= a
i
.Moreover notice that if x = i then A would be cyclic ( since we
would have a
x
a
x
), while if x 6= i thenthis would contradict the definition of r
( a ) ( since it would be a
x
a
i
. Therefore ax
should not be anelement of r ( a ) ). Thus we can rewrite the
above formula as:
8 a
x
2 r ( a ) n r ( b ) 9 b
x
2 r ( b ) n r ( a )
such thata
x
b
x
(
1)
Notice that this formula also implies that r ( b ) n r ( a ) 6=
; , that is r ( b ) 6 r ( a ) .
If formula (1) is satisfied then the enlargement process will
result in a modelm
such thatm ( a ) = m ( b ) , which means that we failed to
construct the model m
x
that we are looking for. Howeverin this case, we can try the
opposite direction, that is, we can add a new object o
a
to each m ( ai
) . Certainlyo
a
2 m ( a ) while oa
62 m ( b ) (since r ( b ) 6 r ( a ) ). Now we enlarge m for
making it a model. Again, whatremains to show is that this step did
not add o
a
to each m ( bi
) too. By a similar analysis we reach theconclusion, that this
step would update each m ( b
i
) only if
8 b
x
2 r ( b ) n r ( a ) 9 a
x
2 r ( a ) n r ( b )
such thatb
x
a
x
(
2)
From this analysis we conclude that we cannot construct the
modelm
x
that we are looking for, only if theformulas (1), and (2) both
hold. Formula (1) imply that there exists a
z
2 r ( a ) n r ( b ) , and bz
2 r ( b ) n r ( a ) ,such that a
z
b
z
. Formula (2) imply that there exists az
2 r ( a ) n r ( b ) , such that bz
a
z
. Notice that ifz = z then A would be cyclic (since it would be
a
z
a
z
), while if z 6= z then this would contradict thedefinition of r
( a ) (since it would be a
z
a
z
, therefore az
should not be an element of r ( a ) ). Thus we
conclude that the formulas (1) and (2) cannot be both true.This
means that we can always construct a model m
x
such that mx
( a ) 6= m
x
( b ) . This implies that ifr ( a ) 6= r ( b ) then A 6 = a b ,
that is, A = d d ) r ( d ) = r ( d ) .
