The paper is devoted to the problem of knowledge representation using concepts of different levels of generality. Model-theoretical methods are being developed for translating data and knowledge presented in the language of low-level concepts into knowledge presented using concepts of a high level of generality. The formalization of knowledge is carried out in the terms of FCA, which allows us to move from low-level concepts to more general concepts and, as a result, generate new, more general knowledge about the domain.
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Knowledge representation is a central part of the development of intelligent systems.
Today, there are several different methodologies for representing knowledge: frames,
semantic networks, production systems, etc. [
One of the most developed and popular methodologies are the logical knowledge
representation. Logical systems are highly expressive. Representation of the data
array in the form of an algebraic system makes it possible to work with two levels of
information consideration: the level of the initial data on which the analysis is
performed, and the level of knowledge – generalized laws formulated in the form of
sentences of firsto-rder predicate logic. Consideration of a set of different situations
(precedents) of the given domain as a class of algebraic systems allows us to work
with statistical knowledge about the domain [
On the other hand, FCA is a powerful tool for presenting and processing
knowledge [35-]. FCA methods are widely used for ontology engineering [
Recently, much attention has been paid to approaches to concept mining
simplification. For example, in [
In this paper we consider various levels of knowledge representation, which are presented using concepts of varying degrees of generality. For example, knowledge of a lower level of generality is: – symptoms of diseases, test results; – specific numerical data on exchange rates, oil prices; Knowledge of the high level of generality is, for example: – patients' diseases, syndromes, complications of diseases; – economic forecasts, expectations of stability or instability, currency crises, etc.
Knowledge containing statements of different levels of generality is formalized in the form of algebraic systems of different signatures. For these systems, formal contexts are constructed that describe them. The properties of lattices of constructed formal contexts are studied. 2
concepts of the domain.
We start the formalization with a finite set {=1, … , } of domain precedents. We
have a set (signature) of low-level concepts of this domain. Each domain precedent
∈ is formalized as a model = 〈, 〉 [
Next, we will bring together knowledge about of all precedents from the class . For this we need to enrich the signature with a set of constan=ts{ | ∈ } . Now we may consider the set( ) of all atomic sentences of the signature as a formalization of lowl-evel concepts of the object domain and the set ( ) of all sentences of the signature a s a formalization of all possible erated by the set of the precedents have
Definition 1 [
, 〉 is called Precedent Model gen, if for any sentence s 1 (,… , ) ∈ ( ) we (( 1, … , )) = { ∈ | ⊨ ( 1, … , )}.
The definition of Boolean-valued Model one can find in [
It was shown in [
When there is a formalization of individual precedents of the given domain, it is
often necessary to describe low-level concepts. So, for example, in the subject domain
of computer security [
was used, and in the third MrKlunky. For each of these viruses it will be difficult to identify any regularities, since the ratio of the number of attack precedents, where there was a mention of this 3 Let
where change.
be an atomic Boolean-valued model. Denote Consider the formal context
}. ( ) = ( ( ), ( A), ), ⇔
≤ (). where the estimation ′: ( ) → is redefined as follows. For each ( 1, … , ) ∈ and for any elemen1ts, … , ∈ we have ′ ( ( 1, … , )) = (( 1, … , )).
Now we can remove lowl-evel concepts from the signature leaving only concepts of a higher level, i.e. pu∗t = ∪ formalization of the subject domain at a higher level. . The model ′′ = ′ ↾ ∗ is a particular virus to the total number of precedents, is very small. Thus the evalution μ on the predicates formalizing these concepts will be very close to zero.
On the other hand, all these viruses have many common characteristics, so it’s reasonable to combine them into one, more general concept “TSR
study the properties of this new concept. Note that each such concept is expressible in the signature through some formula.
So, we select the set of formulas ⊆( ) and enrich the signature with the set of new predicts = { | ∈ }. Let
= ∪ . Next, we extend the Booleanvalued model to the signature , i.e. put
′ = ⇂ ,
We say that the formal context( ) describing the Boolean-valued model
[
In the next section, we will consider various generation algorithms of the ′′ from the model and show how the formal contexts describing them model will
When forming a set of of higherl-evel concepts, first of all, we consider the possibility of obtaining these concepts directly from the formal context itself that describes this Booleanv-alued model. contents of all concepts of the context , i.e.
Definition 2. Consider the formal context = (, , ). Denote by ̃ the set of ̃ = { ⊆ |
↓↑ = }. ̃ ⇔ ∈
↓ in the context .
Consider the formal contex̃t = (,
̃, ̃), where for any object ∈ and for any ∈ ̃ we have
Proposition = (, , ) and 1.
The concept lattices generated ̃ = (, ̃, ̃) are isomorphic, i.e. ( ) ≅ by
the ( ̃ ).
formal contexts
Thus, this Proposition shows that in order to move to the meta level it is not
enough to have only “internal” information contained in a formal context. So, for
example, in [
We describe this approach in the terms of FCA [
Let an equivalence relation ~ be defined on the set . We will deno[te ]b∼y the equivalence class generated by the element and denote by ∼/ the factors-et.
Definition 3. Consider the formal context = (, , ) and the equivalence relation ~ defined on the set .
1. By ∧ = (, / ∼, ∧) we denote the formal context in which
∧ [ ]∼ ⇔ ∀ ∈ [ ]∼ . 2.
By ∨ = (, / ∼, ∨) we denote the formal context in which
∨ [ ]∼ ⇔ ∃ ∈ [ ]∼ . ∧ = (, /
Proposition 2. Consider the formal contexts = (, , ) ∼, ∧). Then the lattice ( ∧) is a sublattice of the lattice(). and ( 1, 1) ≤ ( 2, 2) ⇒ ℎ(( 1, 1)) ≤ ℎ(( 2, 2)); ℎ(( 1, 1)) ∩ ℎ(( 2, 2)) = ℎ(( 1, 1) ∩ ( 2, 2)); ℎ(( 1, 1)) ∪ ℎ(( 2, 2)) ≤ ℎ(( 1, 1) ∩ ( 2, 2)).
Theorem 1. Consider the formal contexts = (, , ) and ∨ = (, / ∼, ∨). We define a map ℎ: ( ) → ( ∨) as follows: ℎ((, )) = ( ↑↓, ↑). Then for any ( 1, 1), ( 2, 2) ∈ ( ) we have: 1. 2. 3.
Thus, it follows from the Theorem 1 that the map( ℎ): → ( ∨) is a homomorphism of the lower semilattices.
Corollary 1. A map ℎ:( ) → ( ∨) is homomorphism of the lattices if and only if for any 1, 2 ∈ follow condition is satisfied:
( 1 = 1′′ 2 = ′2′) ⇒ 1 ∪ 2 = ( 1 ∪ 2)′′.
Note that in the general case, the homomorphism ℎ may not possess the properties of injectivity and surjectivity. This means that when moving from the context to the context ∨, on the one hand, some concepts will be combined into more general concepts, and on the other hand, new concepts will appear. On the other hand, when moving from the context to the context , some concepts will be “forgotten” only.
∧
The results obtained in the paper may be applied to the development of
ontological [192-2] and semantic [