The paper is devoted to the development of methods of ontology supported knowledge discovery in the field of mobile network subscribers. We develop the ontological model of the domain of mobile networks. This ontological model is intended to describe the behavior of mobile network subscribers. The ontological model is based on the four-level model of knowledge representation. In the paper, special attention is given to the third level which represents the set of cases from the domain. For the analysis of domain cases and for the generation of fuzzy knowledge about the domain we use Boolean-valued models, fuzzy models and Formal Concept Analysis. To describe the set of cases from the domain we use formal contexts; objects of these formal contexts are models representing sets of subscribers. We study formal contexts generated by Boolean-valued models, and formal concepts of these formal contexts. We obtain a description of theories of classes of domain cases in the language of formal concepts.
The article is devoted to the development of an ontological model of the domain of
mobile networks. The ontological model is based on the four-level model of
knowledge representation [
The present investigations continue the studies begun in [
interaction of subscribers (for example, calls between subscribers and so on).
Therefore, in contrast to [
2
Ontological model of the domain of mobile networks
The ontological model of the domain of mobile networks is constructed on the basis
of the four-level model of knowledge representation [
The first step of the construction of the ontological model is the description of the
ontology of the domain. From the model-theoretical point of view, the construction of
the domain ontology consists of describing a signature σ (i.e. the set of key concepts)
and a set of axiom of the given domain [
We consider six classes of concepts to determine the signature of the object domain = “Mobile networks”: 1. represents individual indicators of subscribers, and contains two parts: 1 are traffics and 2 are accruals. 2. represents different tariff plans and services. 3. represents properties and characteristics of various tariff plans, services and options; for example, the number of free minutes of talk, the volume of the SMS package, etc.
6. represents concepts which express relationships and interactions between subscribers: calls / SMS / MMS of one subscriber to another and so on.
properties of subscribers; these concepts are formalized as unary predicates. The concepts from the class represent relationships and interactions between subscribers; these concepts are formalized as binary predicates.
The presented classes of concepts have a hierarchical structure which is
described by a set of axioms “hyponym-hypernym”. For these classes of concepts we
present axioms of completeness and axioms of including. A detailed description of
these axioms is given in [
At the second step of the construction of the ontological model, we present the set of general statements about the domain. The set is considered to be true at the moment but may be changed at the future time. This knowledge is synthetic, in contrast to the analytical knowledge presented in the ontology. The synthetic knowledge does not follow from the meaning of the terms used in the description of the domain. The truth value of this knowledge depends on the present state of the real world. An example of such knowledge is the statement that a given tariff and a specific service of the mobile operator are not compatible.
The set of all axioms ∪ generates the theory of the domain , i.e., a set of statements which are true in the given domain. This concludes the second step of the ontological model's constructing.
The third step of the building the ontological model is a formalization of empirical knowledge about the domain, i.e., knowledge about the concrete precedents (cases) of the given object area. In the domain the set of subjects is the set of various individuals and organizations using mobile network services, etc. Note that the set of subjects is finite at each point of time, but it is constantly changing in dynamics. Therefore we may consider the set as potentially infinite. Thus in this paper we consider a countable set A of subjects.
(instance) of the domain the signature on the set A.
= { 1, 2, … } of the subjects of the domain. Each case determines an algebraic system = ⟨ , ⟩, i.e., defines
Further, to simplify the formalization, we will consider models = ⟨ , ⟩ in the signature ( ) , enriched by constants for all elements of the model: ( ) = ∪ { | ∈ }.
So, for simplicity, we denote = , = ( ) and = ⟨ , ⟩.
= ⟨ , ⟩ is the case of the domain .
We say that a model is a domain’s case if ⊨ ∪ .
( ). For example, the set of cases
may describe temporal "slices" of the domain, or geolocation "slices" of the domain. Also, the set may be used to describe the behavior of different groups of subscribers.
⊆ ( ) of the domain and serves as a source of generating a new knowledge about the given domain. Below we will be construct the case model of the domain based on the selected set of cases .
level characteristics of mobile network subscribers and make portraits of mobile users' segments. To do this, we generate different Boolean-valued and fuzzy models (see Section 3) and construct corresponding formal contexts for given case models (see Section 4).
( ) ⇋ { | ( ) ⇋ { | ( ) ⇋ { |
}, } and }.
Definition 1 [
be a complete Boolean algebra and : S( A) → . A triple = 〈 , , 〉 is called a Boolean-valued model if the following conditions
Definition 2 [
→ [
We define a valuation : S( A) → [
Definition 3 [
Preliminaries: Boolean-valued models and fuzzy models the present paper a model (an algebraic system) is a tuple = 〈 ; 1, . . . , , 1, . . . , 〉. The set | | = is called the universe of the model, 1, . . . , are predicates defined on the set and 1, . . . , are constants. The tuple = 〈 1, . . . , , 1, . . . , 〉 is called the signature of the algebraic system .
A formula having no free variables is called a sentence. For a signature we de ( ) = ̅̅(̅̅̅̅);
( ∨ ) = ( ) ∪ ( );
( & ) = ( ) ∩ ( ); ( → ) = ̅̅(̅̅̅̅) ∪ ( );
(∀ ( )) = ⋂ ( ( )); (∃ ( )) = ⋃ ( ( )).
In
note:
hold:
if
(1) ℎ preserves the order, i.e., ℎ is a homomorphism ℎ: → [
∩ = 0 ⇒ ℎ( ∪ ) = ℎ( ) + ℎ( ) for any , ∈ .
Definition 4 [
Proposition 9. Let ( , Γ) ∈ ( ( ), ( A), ). Then Γ is a theory of the signature A, i.e., for any ∈ S( A) if Γ ⊢ then ∈ Γ.
Proof. Let ( , Γ) ∈ ( ( ), ( A), ). Let us show that Γ is a theory. Suppose that ∈ ( )and Γ ⊢ . Prove that ∈ Γ. We have ′ = Γ; consequently, we must only show that ∈ ′.
Consider ∈ , then ∈ ( ). It is true that ′ = Γ, so for any ∈ Γ, we have , therefore ≤ ( ). Hence, by Proposition 7, we have ⊨ .
volving the fact that Γ ⊢ , we conclude that ⊨ . Therefore, by Proposition 7, we have ≤ ( ), so .
Thus, we have proved that for any ∈ . Then ∈ ′ = Γ. Therefore, we have shown that for any ∈ ( ), if Τ ⊢ then ∈ Τ. Hence, Γ is a theory. ∎
Theorem 10. Let = 〈 , A, 〉 be an atomic Boolean-valued model. There exists an atomic Boolean-valued model 1 = 〈 , A, 1 〉, where 1:S( A) → 1, and an epimorphism g: → 1 such that 1) the formal context ( ( 1), ( ), 1) is object-clarified; 2) for any φ ∈ S( A) we have g( ( )) = 1( ); 3) for any ∈ At( ), if ( ) ≠ 0 then ( ) ∈ At( 1); 4) for any ∈ At( )and ∈ S( A), if ( ) ≠ 0 then ⇔ g( ) 1 . Proof. Let = 〈 , A, 〉 be a Boolean-valued model and be atomic.
an equivalence relation ~ on the set ( ) of the atoms of the Boolean algebra defined as follows: for , ∈ ( ) we have ~ iff for any ∈ ( )it is true that ⇔ .
Consider the quotient set /~ , and denote = ( )/~ .
We chose exactly one element [ ] ∈ [ ] in each equivalence class [ ] ∈ ( )/~.
= { [ ] | [ ] ∈ } ⊆ ( ) and = ⋃ . ∈ ; here ⁄ is the quotient class of the element .
Define a mapping 1: ( ) → 1 as follows: ( ∈( ( )\ ))
Consider a principal ideal = ̂ = { ∈ | ≤ } of the Boolean algebra : ⊲ .
Consider a Boolean algebra 1 = ⁄ which is the quotient algebra of the Boolean algebra by the ideal .
Consider an epimorphism : → 1 defined as follows: ( ) = ⁄ for any Lemma 11. 1 = 〈 , , 1〉 is a Boolean-valued model.
Proof. Consider , ∈ ( ). We have 1( ∨ ) = ( ∨ )⁄ = ( )∪ ( )⁄ = ( )⁄ ∪ ( )⁄ = 1( )∪ 1( ); 1( & ) = ( & ) ⁄ = ( )∩ ( ) ( )
( ) ⁄ = ⁄ ∩ ⁄ = 1( )∩ 1( ); 1( → ) = ( → )⁄ = ( )∪ ( )⁄ = ( )⁄ ∪ ( )⁄ = 1( )∪ 1( ).
Consider ( ) ∈ ( ). We have 1(∀ ( )) = (∀ ( ))⁄ = ⋂ ∈ ( ( ))⁄ = = ⋂ ( ( ))⁄ = ⋂ 1( ( )); 1(∃ ( )) = (∃ ( ))⁄ = ⋃ ∈ ( ( ))⁄ =
Note that the equalities and = ⋃ ( ( ))⁄ = ⋃ 1( ( )). ⋃ ∈ ( ( ))⁄ = ⋃ ( ( ))⁄ are true in virtue of the fact that the Boolean algebra is complete.
Thus, we have shown that 1 = 〈 , A, 1〉 is Boolean-valued model.
∎
( 1) = { ⁄ | ∈ }.
Let ∈ . Then ∉ and so, ⁄ ≠ 0. If ⁄ ≠ 0 and ⁄ ≤ ⁄ then ⁄ = ⁄ ∩ ⁄ = ∩ ⁄.
⁄ = ⁄ . Thus, ⁄ is an atom of the Boolean algebra 1.
On the other hand, let ⁄ be an atom of the Boolean algebra 1. Then ⁄ ≠ 0 and so, ≠ 0. The Boolean algebra is atomic, hence the set of atoms ( ) = { ∈ ( )| ≤ } ≠ ∅ and = ⋃ ∈ ( ) . Involving the fact that ⁄ ≠ 0 we conclude that ∉ = ̂, consequently, ≰ .
We have = ⋃ ∈ ( )\ and = ⋃ ∈ ( ) , hence, if ( ) ⊆ ( )\ then ≤ . Therefore, ( ) ⊈ ( )\ , so ( )∩ ≠ ∅.
Consequently, there exists an atom ∈ ( )∩ . Hence, ∈ and ≤ , so ⁄ ≤ ⁄ . As we proved above, ∈ implies that ⁄ is an atom of the Boolean algebra 1. Involving the fact that ⁄ is an atom we conclude that ⁄ = ⁄ and so, So, ∩ ≠ 0 and ∩ ≤ , therefore, ∩ = . Hence, ∩ ⁄ = ⁄ and so,
( ( 1), ( ), 1) is object-clarified – the statement (1).
Lemma 12. For every ∈ ( ), and every ∈ ( ), if ( ) ≠ 0 then ⇔ ( ) 1 .
Proof. Consider ∈ ( ) and ∈ ( ). Let ( ) ≠ 0. Let us prove that ⇔ ( ) 1 .
(⇒) Suppose that holds. Then ≤ ( ), hence ⁄ ≤ ( )⁄ = 1( ). Therefore, we have ( ⁄ ) 1 and so, ( ) 1 holds.
(⇐) Suppose that doesn’t hold. Then ≰ ( ), so, ≠ ∩ ( ). The element is an atom and we have ≠ ∩ ( ) ≤ . Then ∩ ( ) = 0, hence ⁄ ∩ ( )⁄ = ∩ ( )⁄ = 0.
Since ⁄ = ( ) ≠ 0 , we have ⁄ ≠ ⁄ ∩ ( )⁄ , hence, ⁄ ≰ ( )⁄ = 1( ), i.e., ( ) ≰ 1( ). Therefore, ( ) 1 doesn’t hold.
⁄ ∈ { ⁄ | ∈ }. Therefore, ( 1) = { ⁄ | ∈ }.
Thus, that for any ∈ ( ) if ∈ ( )\ then ≤ , hence, ⁄ = 0. If ∈ then ⁄ ∈ ( 1). Consequently, the statement (3) holds.
( ( )) = ( )⁄ = 1( ).
Consider 1, 2 ∈ . Let 1⁄ ≠ 2⁄ . Then there are atoms 1, 2 ∈ ( ) such that 1 = [ 1] and 2 = [ 2]. Hence, 1 ∈ [ 1] and 2 ∈ [ 2]. Since 1 ≠ 2, we have [ 1] ≠ [ 2], so, 1 ≁ 2. Therefore, there is a sentence ∈ ( )such that only one of the statements 1 and 2 holds. Suppose that 1 holds and 2 doesn’t hold. We conclude by Lemma 12 that ( 1) 1 holds and ( 2) 1 doesn’t hold. Therefore, the rows corresponding to the objects 1⁄ and 2⁄ of the Recall that ( A) = {〈{ | ∈ }, A〉 | ≠ for ≠ } and ℎ( ) = { ∈ S( A)| ⊨ } is the theory of the class ⊆ ( A).
as a class of models
⊆ ( A). So, it is interesting and important to solve the following
⊆ ( A).
such that = ℎ( )if and only if there exists a Bolean-valued model such that ( ′, ) ∈ ( ( ), ( ), ).
Proof. Let be a theory in the signature .
(⇒) Consider a class a Boolean algebra as follows: ( ) = {
⊆ ( ). Let = ℎ( ). Denote 0 = ( ). Consider = 〈℘( 0); ∪,∩,−,∅, 0〉 and a mapping : ( ) → defined ∈ | ⊨ } for a sentence ∈ ( ). formal context ( ( 1), ( ), 1) are different. Thus, the formal context ( ( 1), ( ), 1) is object-clarified.
Proof. Let , ∈ ( ). Then (¬ ) = { ∈ 0 | ⊨ ¬ } = { ∈ 0 | ⊭ } = \{ ∈ 0 | ⊨ } = 0\ ( ) = ( ); ( ∨ ) = { ∈ 0 | ⊨ ( ∨ )} = = { ∈ 0 | ⊨ } ∪ { ∈ 0 | ⊨ } = ( )∪ ( );
( & ) = { ∈ 0 | ⊨ ( & )} = = { ∈ 0 | ⊨ } ∩ { ∈ 0 | ⊨ } = ( )∩ ( ); (
→ ) = { ∈ 0 | ⊨ ( ∨ )} = ( )∪ ( ).
Let ( ) ∈ ( ). Then
(∀ ( )) = { ∈ 0 | ⊨ ∀ ( )} = = { ∈ 0 | ⊨ ( )for any ∈ } = { ∈ 0 | ⊨ ( )for any ∈ } = = ⋂{ ∈ 0 | ⊨ ( )} = ⋂ ( ( ));
(∃ ( )) = { ∈ 0 | ⊨ ∃ ( )} = = ⋃{ ∈ 0 | ⊨ ( )} = ⋃ ( ( )). { ∈ 0 | ⊨ ( )for some ∈ } = { ∈ 0 | ⊨ ( )for some ∈ } =
Notice that the atoms of the Boolean algebra = 〈℘( 0); ∪,∩,−,∅, 0〉 are exactly the one-element subsets of the set 0. Hence, ( ) = {{ } | ∈ 0}.
Consider a formal context 〈 ( ), ( ), 〉. Let ℭ ∈ 0 and ∈ ( ). Then we
Thus, we have {ℭ} ⇔ ℭ ⊨ . Recall that ⊆ 0 = ( ) and = ℎ( ). Denote ̃ = {{ } | ∈ } ⊆ ( ). Then ̃′ = { ∈ ( ) | { }
for any { } ∈ ̃} = { ∈ ( ) | ⊨ for any ∈ } = ℎ( ) = .
Therefore, = ′′ and the pair ( ′, ) is a formal concept of the formal context 〈 ( ), ( ), 〉, i.e., ( ′, ) ∈ ( ( ), ( ), ).
(⇐) Let = 〈 , , 〉 be a Boolean-valued model and a pair ( ′, ) ∈ ( ( ), ( ), ).` any ∈ .
Denote = ′. Then ⊆ ( ) and ′ = . Consider a class = { | ∈ }. We have proved above in the proof of Proposition 9 that in this case ⊨ holds for
Therefore, for any ∈ we have ⊨ . It means that ⊨ . Hence, ⊆ ℎ( ) = { ∈ ( )| ⊨ }.
Let us show that ℎ( ) ⊆ . Let ∈ ℎ( ). Then ⊨ , which means that ⊨ for any ∈ . Consequently, for any ∈ we have ⊨ . By Proposition 7, ⊨ implies that ≤ ( ), hence, holds.
Thus, for any ∈ we have , then ∈ ′ = ′′ = , i.e., ∈ . Therefore, for any ∈ ℎ( ) we have ∈ , hence ℎ( ) ⊆ , and so, = ℎ( ). In the present paper we investigate the mathematical foundations of ontological modeling of the domain of mobile networks. We use the four-level model of knowledge representation to formalize this domain. We construct the case model at the third level of ontological model creation, when we formalize the empirical knowledge. The case model is presented by a set of countable algebraic systems. In the ontological model, the high-level characteristics of mobile network subscribers are represented with the help of first order theories of classes of algebraic systems of the special kind.
Boolean-valued models are first order theories of the signature under consideration. In the end, we solve the following problem: What are the theories of the classes ⊆ ( A)? We obtain a description of theories of the classes of domain cases in the
5
Conclusion ∎ language of formal concepts of the formal contexts corresponding to the Booleanvalued models.