Download 103 - Институт автоматики и процессов управления / Institute

FAR EARSTERN BRANCH
RUSSIAN ACADEMY OF SCIENCES
Alexander S. Kleshchev ,
Irina L. Artemjeva
DOMAIN ONTOLOGIES AND KNOWLEDGE PROCESSING.
Technical Report 7-99
Vladivostok
1999
Kleshchev A.S., Artemjeva I.L. DOMAIN ONTOLOGIES AND KNOWLEDGE
PROCESSING. Technical Report. Vladivostok: Far Eastern Branch of the Russian
Academy of Sciences, 1999. 24 p.
The practice of KBS development for complex domains and tasks has shown that
every domain has a structure taking an intermediate position between a domain model
(a knowledge base) and the knowledge representation used in the model. At present this
structure is called "domain ontology". Modern approaches to defining the notion of
domain ontology are discussed in the article. A mathematical apparatus is introduced
which allows us to define the notions of domain model and domain ontology model.
Statements of some problems of knowledge processing are given. It is shown that the
problems are closely interconnected and connecting links among them are the notion of
domain and the notion of domain ontology. A progress in solving these problems is
closely connected with explicit representation and studying of mathematical ontology
models of practically important domains corresponding to modern ideas about these
domains. Complexity of these problems is such that their solving is possible only by
using mathematical methods. In the article some ways of solutions of the problems
using the notions introduced are discussed.
Editor-in-chief
Professor M.Yu.Chernyakhovskaya
Reviewer
Professor B.I.Cogan
C
Institute for Automation & Control Processes
Far Eastern Branch
Russian Academy of Sciences, 1999
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INTRODUCTION
Research on ontology is becoming increasingly widespread in the computer
science community, and its importance is being recognized in a multiplicity of
research fields and application areas, including knowledge engineering, databases
design and integration, information retrieval and extraction [1]. However, among the
members of the artificial intelligence community the meaning of the term of domain
ontology is understood ambiguously. There have been many articles where their
authors try to define this term but no definition is satisfactory. Also, no author tries to
show that his definition could be useful for solving many different problems.
The goal of this article is to give a new definition of domain ontology and to
show that this definition helps to solve a few problems of knowledge processing. The
paper has the following structure. In section 1 modern approaches to defining the
notion of domain ontology are discussed. In sections 2 a mathematical apparatus is
introduced. The apparatus allows us to define the notions of domain models and
domain ontology model in sections 3 and 4. In section 5 statements of some
problems of knowledge processing are given and some ways of their solutions using
the notions introduced are discussed.
This work was carried out with financial support from the Russian Fund of
Fundamental Investigations (grant 99-01-00634).
1. DIFFERENT APPROACHES TO DEFINING THE NOTION
OF DOMAIN ONTOLOGY. BACKGROUND
The central notion of classical technologies for knowledge processing is the
notion of knowledge representation. This term means either a way of coding
knowledge in a knowledge base or a formal system used for formalising knowledge.
The practice of KBS development for complex domains and tasks has shown that
every domain has a structure taking an intermediate position between a domain
model (a knowledge base) and the knowledge representation used in the model. In
some early articles the structure was described informal for certain domains [2, 3].
Since the 80's this structure has been studied in a number of Russian works [4, 5]. In
the late 1980's systematic studies of this structure began in the connection with some
attempts to solve some of the problems given in section 5. This structure was called
"domain ontology". At present there are three main approaches to defining the notion
of domain ontology.
The first one, called here as humanitarian, consists in attempting to define the
content of the notion of domain ontology in terms understood intuitively. Some
examples of such definitions are the following (1) "A (domain) (AI) ontology is a
theory of what entities can exist in the mind of a knowledgeable agent" [6], (2)"An
ontology defines a taxonomy of concepts, that we feel are required for the definition
of the semantics of a language for representing knowledge. [7], (3) "An ontology is
an explicit knowledge-level specification of a conceptualisation, i.e. the set of
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distinctions that are meaningful to an agent. The conceptualisation - and therefore the
ontology - may be affected by the particular domain it is intended for" [8]; (4) "An
ontology is an explicit, partial account of a conceptualisation" [9]; (5) "An ontology
is an explicit, partial specification of a conceptualisation that is expressible as a
meta-level viewpoint on a set of possible domain theories" [10]. The chief defect of
the definitions above and the whole humanitarian approach is that a technical notion
necessary for solving technical problems cannot be defined in such a manner.
The second approach to defining the notion of domain ontology can be called
the computer one. Within the framework of the approach computer languages for
domain ontology representation have being developed. Everything that can be
described using such a language is considered as a domain ontology. Some examples
of computer languages for domain ontology representation are given in [11-14].
Unfortunately, the authors do not explain the differences between languages for
describing ontologies and knowledge representation languages (for example, KRL
[15]). The semantics of the both classes languages is equivalent to the semantics of
predicate calculus languages. Using such a language both domain ontologies and
domain models can be described. In addition, the question about meaningful
interpretation of constructions of such languages in domain terms is still an open
question. Therefore, any definition of the notion of domain ontology within the
framework of this approach does not distinguish this notion from other notions, in
particular, from the notion of domain model (knowledge base).
The third approach to defining the notion of domain ontology can be called the
mathematical one. Within the framework of this approach an attempt is made to
define the notion of domain ontology in mathematical terms or by a mathematical
construction. The definition (6) "An ontology is a logical theory that constrains the
intended models of a logical language" [16] can serve as an example of a definition
of the notion of domain ontology in mathematical terms.
In [17] an attempt was made to define the notions of a conceptualisation, of a
domain model, and of a domain ontology model strictly. Every domain
conceptualisation is represented by a many-sorted algebraic system (many-sorted
logic-mathematical structure) S. The signature <R, F, C> of S is a set of domain
terms (relations, functions, and constants). The universe U of S is a set of domain
objects. A domain model (a knowledge base) is represented a subsystem of S. Every
domain ontology model is represented by another algebraic system O. Sorts of O are
"constant", "concept", "relation" and others. The signature of O (<R1, F1) is the set of
terms used for a description of domain models ("instant", "relation-instant" and
others). The set C1 of constants of O is the signature of S. The interpretation of the
symbols of the signature of O represents the meaning of terms of the signature of O.
The interpretation is restricted by a set D1 of formulas. The ontology describes the
vocabulary of the objects representation (C1) and their types. This formalization of
conceptualisation is based on the article [18] by Genesereth and Nilsson where
conceptualisation is represented by the logic-mathematical structure <D,R>, where D
is a set of domain objects and R is a set of domain relations among them. Such a
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representation of conceptualisation is possible if the world is considered as set of
objects and relations among them depending on time and space (many relations
belonging to R must depend on time and space). But neither Genesereth and Nilsson
nor the authors of article [17] write about it explicitly. According to the definition,
domain ontology is a definition of a representation of conceptualisation using
problem independent terms. The notion of domain ontology defined in such a manner
is not connected with its domain.
In [1] the notion of conceptualisation and of ontology were defined.
Conceptualisation is a (infinite) set of logic-mathematical structures
{<D,Rw>wW}, where D is a set of domain objects, W is a set of relevant states of
affairs also called possible worlds, every Rw is a set of relations among objects, and
all logic-mathematical structures have the same type. So, a conceptualisation
determines a set of intended world structures. If L is a logical language with a
vocabulary V then interpretation of all the object symbols belonging to V does not
depend on W but interpretation of all the predicative symbols belonging to V
depends on W. An ontology O is a set of axioms represented by L. The set of models
for O is an approximation of the conceptualisation. This definition has a few defects.
1. On the one hand, the set D has to be infinite one (for the set W has to be
infinite one). On the other hand, it is impossible to imaging a state of affairs
containing an infinite set of objects. So, the set of objects in every state of affairs
should be a finite subset of the set D.
2. There are also relations among objects that do not depend on W. They are
ordinary mathematical relations. Such relations in the blocks world are the relation of
equality and the relation of inequality. Since different ordinary mathematical
relations may be appropriated to different domain, so it is natural to include these
relations into conceptualization.
3. The vocabulary of a language L can be divided into two parts V 0 and V1.
The interpretation of the symbols of V0 does not depend on the W, but the
interpretation of the symbols of V1 depends on W. The symbols of V0 are ordinary
mathematical constants and signs of ordinary mathematical relations. An example of
an object symbol from V1 in the blocks world is "the table" if in every state of affairs
there is the only table, but in different state of affairs the tables may be different.
4. There are terms related to domain and being more general then terms of V.
An example of such a term may be "spatial relations" that is not ordinary
mathematical relation and its meaning does not depend on W. So, such terms should
be a part of a conceptualization.
Thus, it is possible to consider that till now no generally accepted definition of
domain ontology has been suggested [19,20]. However, its is possible to select four
different meaning of the term of domain ontology from this overview.
1. A domain ontology is a part of the domain knowledge that is not to be
changed. The other part of the domain knowledge may be changed
according to the domain ontology.
2. A domain ontology is a part of the domain knowledge that restricts the
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meanings of the domain terms; the meanings of domain terms do not
depend on the other part of the domain knowledge.
3. A domain ontology is a set of agreements about the domain; the other part
of the domain knowledge is a set of empirical or other domain lows.
4. A domain ontology is an intensional approximation of a extensional
conceptualisation.
These meaning of the term of domain ontology supplements each other.
In section 4 an attempt will be made to give another definition. But here some
basic methodological principles of such a definition should be formulated.
1. On the meaningful level a domain ontology will be understood as a set of
agreements (domain terms, their commentary, statements restricting a possible
meaning of these terms, and also a commentary of these statements). These
agreements are a result of understanding among members of the domain community.
So, they cannot be disproved by any empirical observations. In their meaning they
differ from empirical or other knowledge that can be disproved by empirical
observations or by other way. In this regard the notion of domain ontology is
analogous to the notion of paradigm by T.S. Kuhn [21].
2. The agreements should be represented by an appropriate language of
mathematical logic.
3. An explicit correspondence between elements of the mathematical
construction representing every domain ontology and properties of the domain
should be defined.
To define the notion of domain ontology a class of mathematical constructions
for domain model representation will be firstly defined. Then the notion of domain
ontology will be defined by the class and its meaningful interpretation.
2. LOGICAL RELATIONSHIP SYSTEMS
Logical relationship systems of the n-th order were defined in [22,23]. Here
only their short description will be given. A logical relationship system of the n-th
order is a tuple <, A0, 1, A2, A3,...,An >. Here  is a set of formulas of a
many-sorted quantor-free language of predicate calculus of the n-th order. The
signature of the language is divided into nonempty subsets 0, 1, 2,..., n that are
mutually disjoint. The objective symbols of 0 will be called constants, the functional
symbols of 0 - signs of operations and functions, and the predicative symbols of 0 signs of relations. All the symbols are interpreted in the many-sorted algebraic
system A0. The symbols of 1 will be called unknowns (objective, functional and
predicative). These symbols do not have any fixed interpretation. For i from 2 to n
the symbols of i will be called parameters of the i-th order (objective, functional and
predicative). They are interpreted in the many-sorted algebraic system Ai. The
interpretations of the signs of operations, functions and relations (in A0) are
computable operations, functions and predicates. The sorts of A2 are finite ones and
consist of constants and unknowns. For i from 3 to n the sorts of Ai are also finite
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ones and consist of constants and symbols of 1, 2, ..., i-1.
A solution of a logical relationship system of the n-th order is such a
many-sorted algebraic system of the signature 1, that every formula from  is true
for any the admissible substitution instead of all the variables of the formula. Any
solution has finite sorts consisting of the same objects as the sorts of A0. The set of
all the solutions of a logical relationship system S will be designated as R(S). So, a
logical relationship system of the n-th order uniquely defines the sets of its solutions
(that can be an infinite one).
Some properties of the solution sets of logical relationship systems are studied
in [24-27]. Here we will only intimate the theorem about decreasing of the order
of logical relationship systems important for further presentation. The theorem is:
for any logical relationship system S of the k-th order (k>1) there is a logical
relationship system S1 of the (k-1)-th order equivalent to S. Two logical relationship
systems are equivalent if they have the same set of solutions.
A logical relationship system will be called isomorphic to another one if there
is an one-to-one correspondence between the sets of their solutions.
A logical relationship system S2 will be called a homomorphic image of
another logical relationship system S1 if there is a completely defined one-valued
mapping h from the solution set of S1 into the solution set of S2. In this case we will
say that there is the homomorphism h : S1 S2.
A logical relationship system S will be called the product of other logical
relationship systems S1, S2,..., Sm (the factors of the product) if there are
homomorphisms h1 : S  S1, h2 : S  S2,..., hm : S  Sm such that ',"R(S)
' " (h1('), h2('),..., hm('))  (h1("), h2("),..., hm(")).
Further we will consider a generalisation of the notions introduced above.
Let S1 = <1, A0, 1> be a logical relationship system of the first order and 2
be a set of formulas of a predicate calculus language of the first order of the signature
0  1, such that 1  2 is noncontradictory. Then a logical relationship system
S2 = <1  2, A0, 1> will be considered as S1 enriched by the set 2. The system
S1 will be called unenriched one, the set 2 will be called an enrichment for S1, and
the system S2 will be called enriched one.
A tuple О = <, A0, 1, 2, 3,..., n>, where n > 1 will be called a system
with free parameters. Let S1 = <, A0,1,2,3,...,n> be a logical relationship
system with free parameters, and <A2,...,An> be a tuple of algebraic systems of the
signatures 2,..., n. These algebraic systems must satisfy the restrictions mentioned
above and if all the symbols of i are interpreted in Ai for all i from 2 to n, then all
the formulas belonging to  are not false for all the admissible substitutions. In this
case a logical relationship system S2 = <, A0, 1, A2, A3,...,An> will be considered
as S1 enriched by the tuple <A2,..., An>. The system S1 will be called unenriched
one, the tuple <A2,..., An> will be called an enrichment for S1, and the system S2 will
be called enriched one.
The set of all possible enrichment of S1 will be designated En(S1). If k 
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En(S1) then S2 = <S1,k> is S1 enriched by k. Thus, an unenriched system determines
a set of logical relationship systems.
An unenriched logical relationship system S1 will be called equivalent to
another unenriched logical relationship system S2 if there is a one-to-one
correspondence I between the set En(S1) and the set En(S2) such that for all kEn(S1)
the systems <S1,k> and <S2, I(k)> are equivalent.
The following theorem about decreasing the order of unenriched logical
relationship systems takes place: if Sn = <1  2, A0, 1, 2,..., n> is an
unenriched logical relationship system of the n-th order (n>1), where 1 is a set of
formulas of the signature 0  1, then for the unenriched logical relationship
system of the first order S1 = <1, A0, 1> there is a completely defined one-valued
mapping h from En(Sn) into En(S1) such that for all k  En(Sn) the system <Sn, k> is
equivalent to the system <S1, h(k)>. From this theorem it follows that, in general, S n
and S1 are not equivalent because {h(k)  k  En(Sn)}  En(S1).
An unenriched logical relationship system S1 will be called isomorphic to
another unenriched logical relationship system S2 if there is a one-to-one
correspondence I between the set En(S1) and the set En(S2) such that for all kEn(S1)
the systems <S1,k> and <S2, I(k)> are isomorphic.
An unenriched logical relationship system S2 will be called a homomorphic
image of another unenriched logical relationship system S1 if there is a completely
defined one-valued mapping h from the set En(S1) into the set En(S2) such that for all
kEn(S1) the system <S2,h(k)> is a homomorphic image of the system <S1,k>. In
this case we will say that there is the homomorphism h : S1  S2.
An unenriched logical relationship system S will be called the product of
other unenriched logical relationship systems S1, S2,..., Sm (the factors of the product)
if there are homomorphisms h1 : S  S1, h2 : S  S2,..., hm : S  Sm such that
k',k"En(S) k' k" (h1(k'), h2(k'),..., hm(k'))  (h1(k"), h2(k"),..., hm(k")) and for
all kEn(S) the system <S,k> is the product of <S1,h1(k)>, <S2,h2(k)>,...,
<Sm,hm(k)>.
3. A MATHEMATICAL MODEL OF A DOMAIN
A logical relationship system of the n-th order <, A0, 1, A2,.., An> can be
considered as a domain model if all its elements represent the properties of the
domain mentioned below. The algebraic system A0 represents a model of
mathematics (abstract algebra) used for description of knowledge and ontological
agreements of the domain. The domain objects are represented by mathematical
objects - elements of the sorts of A0. Such a replacement of the domain objects by
mathematical objects must keep some certain properties of the objects represented by
the operations, functions and relations of A0. The solution set of the logical
relationship system is an approximation of the domain reality. The domain reality is
the set of all the situations (states of affairs in terms of [16]) taking place in the past,
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in present, and in the future. Every solution represents a class of situations
(indistinguishable in this conceptualisation). A situation is considered here as
information concerning a finite fragment of the reality that may be related to a finite
part of space and a finite time period. Any situation has a finite set of objects and
there is a finite set of relations among them. These relations are designated by the
domain terms describing a structure of the reality. These terms are represented by the
signature 1. Thus, the structure of any situation is described in the same terms, but
each of these terms has its own meaning in every situation.
If a model of the domain is represented by a logical relationship system of the
first order then the set  of formulas can be divided into two nonintersecting subsets.
They are a set of formulas representing empirical or other knowledge of the domain
(knowledge base) and a set of formulas representing ontological agreements. These
two types of formulas are distinguished only by meaning but not by their forms.
If a model of the domain is represented by a logical relationship system of a
higher order then 1, the algebraic systems A2,..., An represent empirical or other
knowledge of the domain on different levels of abstraction (knowledge base). This
knowledge is represented by relations among terms of the domain. In this case the
whole set  of formulas represents ontological agreements of the domain.
4. A MATHEMATICAL MODEL OF A DOMAIN ONTOLOGY
An unenriched logical relationship system can be considered as a model of a
domain ontology if the set of its formulas represents all the ontological agreements of
the domain and only them, and other elements of the system represent properties of
the domain mentioned in the previous section. To understand such a model a
meaningful commentary for all the terms and formulas being its components is
necessary. Now the informal notion of domain ontology can be defined by the formal
notion of domain ontology model: the part of information about a domain
represented by an ontology model of the domain will be called the ontology of the
domain. Some examples can be found in [22,23].
Here a predicate calculus language plays the role of a representation for
models of domain ontologies, of domains and of knowledge. It determines a form
(syntax) for models of domain ontologies and domains, and also it determines a basic
set of symbols having semantics, i.e. propositional connectives and applications of
functional and predicative symbols to arguments. This representation is extended by
the model of mathematics given by the algebraic system A0. In this case syntax of the
representation is the same. The set 0 of symbols interpreted in A0 is added to the set
of symbols of the representation. Since a specific mathematics is used in every
domain, A0 is not included in the representation, but is a part of every domain
ontology model. However, it is known from practices that the same models of
mathematics represented by the same algebraic system A0 may be a part of ontology
models of different domains. We notice also that formulas containing at least one
symbol from 1, 2,... or n can only belong to the set  of formulas, because any
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formula that does not contain such symbols can be considered only as a restriction on
properties of symbols from 0. At the same time, all the symbols from 0 have
already had the certain interpretation given by A0.
Let a domain ontology model <1, A0, 1> be of the first order. Then the
model of mathematics A0, the signature 1 and the set 1 of ontological agreements
are determined by the model. The domain ontology model enriched by a set 2 of
formulas (knowledge base) representing empirical or other laws of the domain is a
model of the domain. Every empirical or other law can be represented only by using
symbols from 01. Therefore, the empirical or other law determines properties of
relations among objects of situations (properties of possible interpretations for all the
symbols from 1). These relations in different situations have different contents (all
the symbols from 1 have different interpretations in different models of different
situations). At the same time, these relations have the same properties in different
situations. Some of these properties are represented by ontological agreements and
the others are represented by empirical or other laws. From the philosophical point
of view, empirical laws explicitly represent the order taking place in the reality. At
the same time, every empirical law must be met in every really existing situation of
the domain. Therefore, a model of a domain is adequate to the domain if the set of
models of all the situations possible in this domain is equal to the set of all the
solutions of the logical relationship system being the model of the domain, i.e. the set
of solutions is the exact approximation of the domain reality. The question of
existing an adequate model for a domain is usually an open question. If it is detected
that a model of a domain is not adequate to the domain then empirical laws of the
domain should be modified within the restrictions imposed by the domain ontology
to make the model more adequate, i.e. another model of the domain is formed by
enriching the same domain ontology model by another set of empirical laws. If in the
process of storing empirical or other data (a set of situations) it becomes clear that the
current domain model is not always adequate to the domain, it is necessary to modify
the model permanently and this process leads to constant increasing of the number of
empirical or other laws and/or to constant growth of complexity of the formulas
representing them then an aspiration may arise for finding another ontology
(changing the paradigm) of the domain and for finding an adequate model of the
domain within the restrictions of the new ontology.
Let a domain ontology model <, A0, 1, 2> be of the second order. Then the
model of mathematics A0, the signatures 1, 2 and the set  of ontological
agreements are determined by the model. In comparison with a domain ontology
model of the first order a new set of terms (represented by the signature 2) is
introduced in a domain ontology model of the second order. These terms are
designations of relations among concepts of the domain. In this case, unlike domain
ontology models of the first order, empirical or other knowledge of the domain must
be represented on the higher level of abstraction, i.e. using these relations, defined on
the set of terms representing objects of the domain (the set of the constants from 0),
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and also of terms representing relations among objects in the situations (symbols
from 1). The set of ontological agreements represented by formulas from  can be
divided into three nonintersecting subsets. They are constraints of the reality model,
i.e. statements restricting the set of possible interpretations of unknowns (this set is
the same in meaning and in form as the set 1 for domain ontology models of the
first order), constraints of the knowledge model, i.e. statements restricting the set of
possible interpretations of symbols from 2, and statements settings up a
correspondence between interpretations of symbols from 1 and from 2. Every
statement of the first group must contain at least one symbol from 1 and contain no
symbols from 2. Every statement of the second group must contain at least one
symbol from 2 and contain no symbols from 1. Every statement of the third group
must contain at least one symbol both from 1 and from 2. A domain ontology
model of the second order enriched by the algebraic system A2 (a knowledge base)
representing empirical or other laws of the domain is a model of the domain.
According to the theorem about decreasing of the order of logical relationship
systems a set of formulas can be got using every formula from the third group. These
formulas will represent empirical or other laws of the domain. These laws will be
formulated in terms of signatures 1 and 0. In this case the formulas got in such a
way must not contradict the constraints of the reality model. In addition, the theorem
about decreasing of the order of logical relationship systems gives a possibility to
settle the questions about meaning and adequacy of empirical or other knowledge for
domain models of the second order by reducing them to equivalent models of the
first order.
Let a domain ontology model <, A0, 1, 2, 3> be of the third order. Then
the model of mathematics A0, the signature 1, 2, 3 and the set  of ontological
agreements are determined by the model. Thus, in comparison with a domain
ontology model of the second order a new set of terms (represented by the signature
3) is introduced in a domain ontology model of the third order. These terms are
designations of relations among classes of concepts of the domain. In this case,
empirical or other knowledge of the domain must be represented on two levels of
abstraction by relations among concepts (as in the models of the second order) and
relations among classes of concepts. These relations are defined on the set of terms
representing objects of the domain (the set of the constants from 0), of terms
representing relations among objects in the situations (symbols from 1), and also of
terms representing relations among concepts (symbols from 2). The ontological
agreements represented by formulas from  can be divided into seven
nonintersecting subsets. They are (1) constraints of the reality model; (2) constraints
of the lower (the first) abstraction level of the knowledge model, i.e. statements
restricting the set of possible interpretations of symbols from 2; (3) constrains of the
upper (the second) abstraction level of the knowledge model, i.e. statements
restricting the set of possible interpretations of symbols from 3; (4) constraints
setting up a correspondence between the lower and the upper abstraction levels of the
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knowledge model, i.e. statements about a correspondence between possible
interpretations of symbols from 2 and 3; statements setting up a correspondence
between different abstraction levels of the knowledge model and the reality model
(i.e. statements setting up a correspondence between interpretations (5) of symbols
from 1 and 2; (6) of symbols from 1 and 3; (7) of symbols from 1, 2 and 3). A
domain ontology model of the third order enriched by the algebraic systems A2 and
A3 (a knowledge base) representing empirical or other laws of the domain is a model
of the domain.
According to the theorem about decreasing of the order of logical relationship
systems a set of formulas representing constrains of a knowledge model can be got
using every formula from group (4); a set of formulas representing ontological
agreements setting up a correspondence between the knowledge model and the
reality model can be got using every formula from groups (6) and (7). In this case the
formulas got in such a way must not contradict the formulas of groups (1), (2) and
(5).
Analysis of the structures of domain ontology models of a higher order can
be analogically made.
Below the set of all possible enrichments for a domain ontology will be called
the set of knowledge bases consistent with this domain ontology model.
Now we will discuss the question about facilities of domain models and
domain ontology models represented by enriched and unenriched logical relationship
systems of different orders.
On the one hand, from the theorem about decreasing the order of logical
relationship systems it follows that if there is a domain model represented by a
logical relationship system of the n-th order that determines an approximation of the
domain reality then there is the model of the same domain represented by a system
of the (n-1)-th order that determines the same approximation of the domain reality.
Therefore, any approximation of any domain reality can be determined by a model of
the domain represented by a logical relationship system of the first order. In this
point, domain models represented by logical relationship systems of a higher order
have no advantages over models represented by a logical relationship system of the
first order.
On the other hand, every domain ontology model represented by an unenriched
logical relationship system determines a set of intended domain models. The
adequate domain model should belongs to this set. So, this set should be as small as
possible. From the theorem about decreasing the order of unenriched logical
relationship systems it follows that using a domain ontology model represented by an
unenriched logical relationship system of the n-th (n>1) order a smaller set of
intended domain models can be determined than using a domain ontology model
represented by an unenriched logical relationship system of the first order. In this
point, domain ontology models represented by unenriched logical relationship
systems of a higher order have certain advantages over models represented by an
unenriched logical relationship system of the first order.
13
5. SOME PROBLEMS OF MANUAL AND COMPUTER TECHNOLOGIES
FOR KNOWLEDGE PROCESSING AND WAYS OF SOLVING THESE
PROBLEMS
Some problems, most important from the authors' point of view, will be
considered below. The problems arise when manual and computer technologies for
knowledge processing are used. The problems have not had satisfactory solution
within the framework of "classical" computer technologies. The technologies worked
out by the founders of expert systems and described in widely known monographs
and reviews are meant by classical ones. All the problems will be divided into five
groups.
I. Problems related to ascertainment of correspondence between domain
ontologies.
1) Within the framework of manual technologies alternative points of view on
the same domain are often said to be equivalent or different domains are said to be
similar. Usually such judgements are intuitive. Ascertainment of the equivalence
between alternative points of view on the same domain can give a possibility to solve
tasks arising within the framework of a point of view using methods worked out
within the framework of another point of view. Ascertainment of the similarity
between different domains can give a possibility to solve tasks arising in a domain
reasoning by analogy in the case if methods for solving analogous tasks in another
domain have worked out. In this connection the following problems arise: (a) how to
ascertain an equivalence between two different knowledge systems or between two
different ontologies of the same domain; (b) how to ascertain an equivalence between
knowledge systems or between ontologies of two different domains. The problems
were discussed in [8] within the framework of computer technologies.
Two models of the same domain (models of two different domains)
represented by logical relationship systems S1 and S2 will be called isomorphic if S1
and S2 are isomorphic. Two domain ontology models (of the same or two different
domains) represented by unenriched logical relationship systems S1 and S2 will be
called isomorphic if S1 and S2 are isomorphic. Thus, both problems formulated above
are reduced to ascertaining an isomorphism between two different (enriched or
unenriched) logical relationship systems. For ascertaining isomorphism between
systems studying properties of such isomorphism can be useful.
2) Within the framework of manual technologies a cruder description of a
domain can be obtained from a more detailed description of the domain. It could be
necessary for specialists having different qualification. No articles are known to the
authors in which a satisfactory statement of the task of "coarsening" description of
domains or its solution within the framework of computer technologies was given. In
this connection the following problem arises: how to provide coarsening knowledge
or an ontology of a domain.
If a domain has two models represented by logical relationship systems S1 and
14
S2 and the system S2 is a homomorphic image of the system S1 then the domain
model S2 will be called a coarsening of the model S1. Let models of two ontologies
O1 and O2 of a domain be represented by unenriched logical relationship systems S1
and S2. Then the model O2 will be called a coarsening of the model O1 if the system
S2 is a homomorphic image of the system S1. Thus, this problem is reduced to
studying homomorphisms of (unenriched and enriched) logical relationship systems.
3) Within the framework of manual technologies an integration of descriptions
of a few domains is referred to as combination of these descriptions in a general
description of the new complex domain those aspects are original domains. In this
connection the following problem arises: how to provide integrating knowledge
systems or ontologies of a few domains into a new knowledge system or ontology.
The problem was discussed in [8, 28-31] within the framework of computer
technologies.
A domain model represented by a logical relationship system S will be called
an integrated one from domain models represented by logical relationship systems
S1, S2,...,Sk, if the system S is the product of S1, S2,..., Sk. Let domain ontology
models O, O1, O2,..., Ok be represented by unenriched logical relationship systems S,
S1, S2,...,Sk. The model O will be called an integrated one from O1, O2,...,Ok, if S is
the product of S1, S2,..., Sk. Thus, the problem is reduced to studying products of
(unenriched and enriched) logical relationship systems. .
II. Problems related to using knowledge in a broad sense.
4) Within the framework of manual technologies the same knowledge is often
used for different purposes. An analogous need exists within the framework of
computer technologies too [8, 20]. In this connection the following problem arises:
how to provide reusing the same domain knowledge or domain ontologies in
different computer designs.
Let S1 and S2 be models of two domains represented by logical relationship
systems, S1 be the product of S and S', and S2 be the product of S and S". In this case
we will say that the model of another domain represented by the system S is reused.
Let O1 and O2 be ontology models of two domains represented by unenriched
logical relationship systems, O1 be the product of O and O', and O2 be the product of
O and O". In this case we will say that the ontology model of another domain
represented by the system O and the knowledge base of this domain (any enrichment
of O) is reused.
Thus, the problem is reduced to studying ways of forming enriched
(unenriched) logical relationship systems being homomorphic images of several
enriched (unenriched) logical relationship systems and also properties of such
systems and homomorphisms.
5) Within the framework of manual technologies knowledge is always shared
among both different people and other bearers of knowledge (books, articles, and so
on). Every bearer of knowledge possesses an integral knowledge system. The
specialists concerned have more or less easy access to the knowledge system of any
bearer. Within the framework of computer technologies a need to share knowledge
15
among different knowledge bases related to the same domain model and also a need
to organise access of knowledge based systems to "another's" knowledge bases [8,
12, 20, 32-34] exists. In this connection the following problem arises: how to provide
using shared domain models (sharing knowledge bases).
A domain model that can be represented by a logical relationship system S will
be called shared if, first, S is the product of logical relationship systems S 1,...,Sm that
are the models of other domains, and, second, there is a way to restore S using
S1,...,Sm. We consider that the systems S1,...,Sm are stored on different servers, but the
system S stores nowhere and S is restored using S1,...,Sm.
A domain ontology model that can be represented by an unenriched logical
relationship system O and the knowledge base of this domain k (an enrichment of O)
will be called shared if, first, O is the product of unenriched logical relationship
systems O1,...,Om that are the ontology models of other domains, and, second, there is
a way to restore O using O1,...,Om and a way to restore k using k1,k2,..,km that are
enrichments of O1,...,Om. We consider that the systems O1,...,Om and their
enrichments k1,k2,...,km (knowledge bases) are stored on different servers, but the
system O and its enrichment k (the domain knowledge base) stores nowhere, O is
restored using O1,...,Om and k is restored using k1,k2,...,km.
Thus, the problem is reduced to studying ways to restore elements of S using
elements of S1,...,Sm, elements of O using elements of O1,...,Om, elements of k using
elements of k1,...,km.
III. Problems related to translating "human" knowledge into "computer" one
and back.
6) A bottleneck of using computer technologies for knowledge processing is
passing knowledge from experts to computers. The process is labour-consuming
because of knowledge extensiveness. When classical technologies are used it is
supposed that a knowledge engineer takes part in this work besides an expert. On the
one hand, the knowledge engineer is the principal motive force of this work, but on
the other hand, he brings about certain difficulties in the process of passing
knowledge and misrepresentation in its results. In this connection the following
problem arises: how an expert can pass his knowledge into a computer without any
intermediary. The article [8] is devoted to its detailed discussion.
We will show how the problem can be solved if a domain ontology model of
the second or a higher order is formed. In this case using the unenriched logical
relationship system S = <, A0, 1, 2, 3,..., n> representing the domain ontology
model, an equivalent unenriched logical relationship system S'= <', A'0, 1, '2,
'3,..., 'n> representing equivalent ontology is formed in the following way. The
signature '0 is an extension of 0 by set-theoretic constants, signs of operations and
relations. The algebraic system A'0 is an extension of A0, giving these symbols the
universally accepted set-theoretic interpretation. For i from 2 to n the signature 'i
can be got from i by eliminating all the predicative symbols and by adding a set of
functional symbols one-to-one corresponding to the set of all the predicative symbols
of i. In the process the interpretation of a predicative symbol p  i and the
16
interpretation of the functional symbol fp  'i corresponding to it are related by the
equality fp(v1,...,vk) = {v  p(v1,...,vk, v)}. ' is formed using  so that both
unenriched logical relationship systems would be equivalent (it is obviously always
possible).
Let a relation "<" be defined on the set of all the sorts of the signatures '2,...,
'n in the following way. If s1 and s2 are sorts and f is a functional symbol of one of
the signatures '2,..., 'n, then s1 < s2, if s1 is the sort of an argument of f, and s2 is the
sort of the result of f. Let the system S be such that the relation "<" is a partial order
relation.
In this case the process of passing knowledge from an expert to a computer
consists in forming interpretations of all the sorts and all the symbols of the
signatures '2,..., 'n. At first, for every terminal sort (a sort is terminal if it is the sort
of the result of none of the functions) the computer asks the expert about a set of
objects that is the interpretation of the sort. All the questions are asked in terms of the
domain ontology. Also the computer asks the expert about values of all the objective
symbols of the signatures '2,..., 'n. Next, for every functional symbol of the
signatures '2,..., 'n and for every tuple of its arguments (a1,a2,...,ak) the computer
asks the expert about the value of application term f(a1,a2,...,ak). In this way the
interpretation of the sort of the result of f is formed. This process can be completed
when every function f is completely defined within the domain of definition given by
Cartesian product of the sorts of all the arguments of f. The constraints of the
signatures '2,..., 'n allow us to check consistency and completeness of knowledge
base generated. Thus, only a part of the domain ontology model is used to solve the
problem. This part consists of the signatures 2,..., n and their constraints.
In [35] a knowledge acquisition model based on this principle was described,
and in [8, 36-40] several tools for developing knowledge base editors were presented.
7) Formalising knowledge using classical computer technologies makes the
knowledge "clear" for a computer but unclear for domain specialists. Therefore, back
translation also requires the same translator, namely, the knowledge engineer. In this
connection the following problem arises: how a domain specialist can gain access to
computer knowledge without any intermediary.
A knowledge base editor based on domain ontology models (see the previous
point) allows a domain specialist to look through a knowledge base. During a
knowledge acquisition process a knowledge base editor asks an expert but during
overlooking the knowledge base a domain specialist has a possibility to ask the same
questions to the knowledge base editor in terms of the domain ontology.
8) When a knowledge base is formed for a real domain then after testing and
debugging the knowledge constituting its content can be of interest for domain
specialists. But no articles are known to the authors reporting that the contents of a
real knowledge base has become available for many specialists on the appropriate
domain and has had an impact on the domain. In this connection the following
problem arises: how to make the content of computer knowledge well-known for
17
many domain specialists.
This problem can be solved if a domain ontology model of the second or a
higher order is formed. A possible way of solving the problem is transforming a
knowledge base into a text. The text must have a form (structure, style) convenient
for domain specialists. Such texts can be got by modern tools for generating
documents using data bases [41,42]. As seen from the domain ontology model
definition, a knowledge base is a relational data base in form. In addition, access to a
knowledge base for a wide range of specialists can be organised by Internet using a
knowledge base editor based on the domain ontology model.
IV. Problems related to forming and debugging knowledge.
9) The principal way of getting empirical knowledge is their inductive forming
on the basis of empirical data. Manual technologies of inductive forming knowledge
are an object of study beginning with the works by F.Bacon and J.S. Mill. But it is
very difficult to estimate the quality of the knowledge formed by this way. Moreover,
a comparison of empirical data with knowledge inductively formed as a rule is
subjective, especially in the domains where a tradition of application of mathematics
is absent. This circumstance has an negative influence on the quality of the
knowledge too. The task of inductive forming knowledge was considered within the
framework of computer technologies. However, as D. Michie noted [43],
automatically formed knowledge can be useful only if, firstly, their content is clear
for domain specialists and, secondly, if the whys and wherefores of the knowledge
leans upon empirical data and is made by clear ways for the specialists. No articles
on inductive forming knowledge bases in which both these requirements would be
met are known to the authors. In this connection the following problem arises: how to
make inductively formed computer knowledge clear and convincing for domain
specialists.
This problem can be solved if a domain ontology model of the second or a
higher order is formed. Any knowledge base consistent with the domain ontology
model will be understood by a specialists of the domain (taking into account point 7)
if the ontology is acceptable for him. In addition, a knowledge base is considered by
domain specialist as justified, if there is a correspondence between the knowledge
base and series of examples of the domain situations understandable for the
specialist.
With regard to these two reasons, several task specifications for inductive
learning can have the following form. A domain ontology model of an order higher
than 1 (unenriched logical relationship system) and a finite set of algebraic systems
of the signature 1 (series of examples of the domain situation models) are given.
These algebraic systems can be determined completely or partially (the
interpretations of some of the sorts and/or of symbols of 1 are ambiguously given).
It is necessary to find an enrichment K (a knowledge base) for the domain ontology
model S such that every example of the series is a solution of S enriched by K. In
addition, an explanation for the knowledge base formed inductively should be
formed. The explanation represents the correspondence between the built knowledge
18
base and the series. Since, the task of inductive learning in such a specification can
have many solutions then a criterion permitting to select the solution uniquely may
be added to the task specification. Thus, a number of task specifications for inductive
learning can be got depending on what information about examples of the series is
given and what criterion for selecting solutions is chosen. It is clear that a solution of
a task of inductive learning specificated in this way will meet requirements above.
10) Refining knowledge is constant modifying the knowledge to make it
consistent with a constantly increasing amount of empirical data. When manual
technologies are used the same difficulties arise as in the case of problem 9. Within
the framework of computer technologies an analogous process is called knowledge
base debugging. This process needs an expert to take part in it who is not familiar
with such an activity. Besides practically insuperable difficulties related to problem 7
arise in his work. No articles are known to the authors where acceptable methods for
knowledge base debugging would be suggested. In this connection the following
problem arises: how to make refining computer knowledge possible and clear for
experts.
This problem can be solved if a domain ontology model of the second or a
higher order is formed. The problem is a special case of the previous one with two
restrictions on methods of its solving. Firstly, these methods must be recurrent, i.e.
they must form a new version of the knowledge base "improving" the previous one
using the domain ontology model, the previous version of the knowledge base, and a
series of examples. Secondly, these methods must be man-machine ones, i.e. they
must allow an expert to take part in forming a new version of the knowledge base.
The latter gives a possibility to an expert to lop off unsatisfactory solutions using his
experience, as if he had added some precedents from his previous practice to the
series of examples and also as if he had determined more exactly some ambiguous
examples of the series.
V. Problems related to using knowledge to solve applied tasks
11) A statement of an applied task allows us to determine the properties of the
output data of the task (its solution) in dependence on the input data of the task.
Within the framework of manual technologies it is possible to deal with either
implicit or meaningful statement of a task. In this statement knowledge determines
the properties of output data of the task depending on its input data. In the domains
having a tradition of using mathematics and mathematical methods for solving tasks
they pass on from meaningful statement of a task to its mathematical specification.
When a computer is used to solve tasks, a mathematical specification of any task is
necessary, because the only property of output data of a program is they are the result
of running the program. Knowledge representation means used within the framework
of classical computer technologies (production systems, frames and semantic
networks) have procedural semantics. It means that a knowledge base represented in
such a manner is a program of solving tasks of a class. Hence, a knowledge base in
such a form cannot be considered as a component of the specification of any task
solved by KBS. As a result, in developing a majority of systems no attempt were
19
even made to pass from meaningful statement of tasks solved by KBS's to their
mathematical specifications. In this connection the following problem arises: how to
obtain mathematical specifications of applied tasks solved by KBS's.
A mathematical specification of an applied task solved by a KBS can contain a
domain model represented by a logical relationship system, input and output data of
the task, task conditions (a set of formulas), and also criterion of selecting solutions
[22,23]. All the components of the applied task specification are represented in terms
of the domain model. If every value of input data is replaced by a variable (different
variables correspond to different values) in the task specification then the
mathematical specification of the task will be transformed into a mathematical
specification of a class of applied tasks. These variables will be called variables of
the class of applied tasks. There is a one-to-one correspondence between the set of
tasks belonging to the class and the set of all the admissible substitutions of values
instead of these variables. To get the mathematical specification of an applied task
belonging to a class it is necessary to replace all the variables of the class by values
of input data.
Let a domain model be represented by an enriched logical relationship system
of an order higher then the first. If the enriched system is replaced by the unenriched
logical relationship system representing the domain ontology model and enrichments
representing knowledge bases are considered as another set of input data of all the
tasks of the class then the mathematical specification of the class of applied tasks will
be transformed into the mathematical specification of the class of applied tasks
corresponding to the domain ontology. There is a one-to-one correspondence
between the set of tasks belonging to the class and the Cartesian product of the set of
all the admissible substitutions of values instead of variables of the class of the tasks
by the set of all the possible enrichments (knowledge bases) for the unenriched
logical relationship system representing the domain ontology model. To get the
mathematical specification of an applied task belonging to a class of tasks
corresponding the domain ontology it is necessary to replace all the variables of the
class by values of input data and to enrich the domain ontology model by an
appropriate knowledge base.
Finally, if domain terms in the mathematical specification of the class of
applied tasks corresponding to a domain ontology are replaced by abstract
designations then this mathematical specification of the class will be transformed into
a mathematical task. The transformation of a mathematical specification of a class of
applied tasks corresponding to a domain ontology into a mathematical task is
important because different classes of applied tasks corresponding to ontologies of
different domains, generally speaking, can be reduced to the same mathematical task.
12) Within the framework of manual technologies to work out a method for
solving an applied task means to obtain a method using its meaningful statement or
mathematical specification, both containing knowledge necessary for solving the
task. There is an analogous need within the framework of computer technologies.
Investigation of a method for solving a task is, firstly, obtaining some arguments for
20
the method solves the task and, secondly, estimating the labour consuming character
of solving the task by the method. In the case when there is a mathematical
specification of the task and the method is described sufficiently rigorously as an
algorithm or calculus, then the arguments are replaced by a proof and the labour
consuming character is replaced by a complexity of the method. In this connection
the following problem arises: how to obtain and to investigate a method for solving a
task using a mathematical specification of the task.
When a method of solving a task is described it is naturally to use the
terminology that was used for describing a mathematical specification of the task.
Also new terms may be introduced but they must be defined by the terms used in the
mathematical specification of the task. Thus, it is seen that the more abstract
terminology is used for describing a mathematical specification of a task the wider
the field of application of a method for solving the task. Therefore, the problem is
reduced to accumulating of the set of mathematical tasks solved by KBS's and to
working out methods for their solving. As is the convention of practice of software
development, working out a method for solving a mathematical task should be
separated from an implementation of the method. As noted in [22,23], such methods
can be worked out both in the form of a calculus and in the form of an algorithm. For
describing these methods appropriate generative models or computational models
should be used rather then implemented knowledge representation languages or
programming languages. In [22,23] problems of investigating such method were
considered.
13) The same methods often can be used for solving a few tasks and subtasks.
There is an analogous need within the framework of computer technologies [8]. In
this connection the following problem arises: how to reuse the same methods of
solving tasks for different designs. Based on point 11, this problem and also problem
12 cannot be satisfactorily posed within the framework of classical computer
technologies.
Abstraction of applied tasks to mathematical ones gives a possibility of reusing
methods for their solving (see point 11). If different applied tasks can be reduced to
the same mathematical task then a method for solving the mathematical task can be
used for solving these applied tasks too. A decomposition of a mathematical task into
mathematical subtasks in working out a method for solving the mathematical task
gives an additional possibility for reusing methods. In this case the same
mathematical subtasks can be components of decompositions of different
mathematical tasks and methods for solving these subtasks can be components of
methods for solving different mathematical tasks.
Reusing methods for solving tasks should be distinguished from reusing
programs for solving these tasks. Possibilities of reusing programs are usually given
by application packages [44-45] and libraries of reusable methods [8,46,47]
(production schemes [48-49]).
14) Within the framework of manual technologies applied tasks of different
classes often can be solved by different methods but using the same knowledge.
21
There is an analogous need within the framework of computer technologies too [50].
However, within the framework of computer technologies every KBS, as a rule, is
intended for solving the tasks of the only class. In this connection the following
problem arises: how to develop KBS's intended for solving tasks of many classes
using the same domain model (knowledge base).
Let's consider a set of mathematical specifications of applied tasks such that
every specification contains the same domain model. Such a set will be called an
applied multitask. Just as an applied task was transformed into a class of applied
tasks, the latter was transformed into a class of applied tasks corresponding to a
domain ontology, and the latter was transformed into a mathematical task, so an
applied multitask can be transformed into a class of applied multitasks, the latter can
be transformed into a class of applied multitasks corresponding to a domain
ontology, and the latter can be transformed into a mathematical multitask. A
multitask knowledge-based system is intended for solving applied multitasks of a
class of applied multitasks or for solving applied multitasks of a class of applied
multitasks corresponding to a domain ontology. A shell for such knowledge-based
systems is intended for solving a mathematical task.
CONCLUSIONS
Because knowledge plays a very important role in the life of the mankind,
"manual" technologies for knowledge processing arose already in ancient times and
have constantly been developing till now. At present the knowledge "industry" based
on "manual" technologies is a considerable part of human activity. "Manual"
technologies give a possibility to solve all the tasks facing the industry of knowledge
processing. However, they are characterised by low productivity of labour and by
low quality of their results in many cases. At the end of the 60's forming computer
technologies for knowledge processing started within the AI framework. Main
achievements of computer technologies for knowledge processing related to using
knowledge for solving applied tasks. The computer technologies usually lead to
considerable raising the productivity of labour and sometimes to higher quality of
their results as compared with manual technologies. But for many tasks of knowledge
processing solved by manual technologies no corresponding computer technologies
were worked out. In addition, transition from manual technologies to computer ones
and back requires special translators, i.e. special people called "knowledge
engineers". It is a reason why the two classes of technologies are largely isolated one
from the other.
In this article an attempt is made to show that KBS's can be considered as a
section of more extent area - computer processing empirical or other knowledge. It is
clear from the discussion above of a few problems related to processing knowledge
that these problems are closely interconnected and connecting links among them are
the notion of domain and the notion of domain ontology. Therefore, a progress in
solving these problems is closely connected with explicit representation and studying
22
of mathematical ontology models of practically important domains corresponding to
modern ideas about these domains. In addition, complexity of these problems is such
that their solving is possible only by using mathematical methods. The authors hope
that the mathematical apparatus introduced in this article will stimulate further works
in these directions.
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Alexander S. Kleshchev
Irina L. Artemjeva
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