# Model of formation of ontology facts

Let A = {a1, a2,..., an } be the set of domain attributes, Vj be the set of possible values aj н A, and V = Ua н A Va. Attributes can be measured in different scales (quantitative, ordinal, qualitative, mixed).

Between the attribute aj and its value vj we define the following operations:

1) aj = vj , = is the sign of the value operation;

2) aj < vj , < is the sign of the relation operation ( >, >=, <= );

3) aj н [vj1 ,..., vjm ], н sign — membership operations ( [ ), ( ), ( ] ).

An attribute, its value, and the operation between them define a statement. Let Q = {q: A -> V} be a set of propositions.

On the set Q, we define two functions:

1. The function m : Q -> [0,…,1], allows us to interpret the statement q from the point of view of its truth.

2. The function q : Q -> [0,…,1], allows us to interpret the statement q in terms of its significance.

Then the triplet f = (q, m(q), q(q)), where q н Q, is called a fact of the domain.

Let λ be a mapping from the direct product Q x [0,…,1] x [0,…,1] to Q x [0,…,1] x [0,…,1]. Through Г = { λ } the set of all possible mappings such that λ(f) н Q x [0,…,1] x [0,…,1].

Let us define a situation s as a set of facts connected by conjunction, disjunction, or negation signs. For example,

s = f1 & f2 U (Ø f3) (3)

Denote by S = {s} the set of all situations in the subject area and p: S -> S some transformation of the set S onto itself. Consider iterations of this mapping, that is, the results of its repeated application to points in the phase space. They define a dynamical system with a phase space S and a set of times N. Indeed, we assume that an arbitrary point s0 н S in time t = 1 passes to a point s1 = p(s0) н S. Then, in time t = 2, this point will go to the point s2 = p(s1) = p( p(s0)) and so on.

Let R = {r: S -> S} be the set of transformations (rules) of the following form:

r : IF <ANTECEDENT> THEN <CONSEQUENT>

where <ANTECEDENT> (condition - situation s1), <CONSEQUENT> (conclusion, consequence - situation s2). If the situation s1 in the rule evaluates to true, then the situation s2 becomes true and is added to S. In other words, if the situation s1О S in the rule evaluates to true, then there is a map qj: A -> V forming the situation s2

s2 = q1 * q2 * qn and * the sign of the operations & , U ,Ø.

If situation s1 in the rule evaluates to false, then situation s2 is not added to S. For example, IF s THEN q4, Here s is situation (3), q4 is some statement. As a result of the rule execution, a new fact will be created:

f4 = (q4 , m(q4 ), q(q4)),

m(q4 ) = k * max [min(m (q1 ), m (q2 )), (1 - m (q3 ))].

q(q4 ) = k * max [min(q(q1 ), q(q2 )), (1- q(q3 ))]

Ontology construction algorithm:

compiling a complete and consistent logical description of the ObD;

formation of a set of propositions Q = {q: A -> V}

formation of objects based on rules from the set R = {r: S -> S}

setting the hierarchy of objects (building a logical decision tree, taxonomy of objects)

Systems of rules and frames are the main way of synthesizing and representing sets (plans) of a relationship on a set of objects. The factor that arranges objects (partial order) and turns it into a goal-oriented system is the mappings r.

Classes constitute a self-organizing set for the operation of a functional system.

Note 1. The partial order of objects is carried out as a result, when the consequent of one rule, for example, p1 is contained in the antecedent of the rule p2, then object 2 is older than object 1.

The inference engine consists of two parts: the first part is a rule analyzer, and the second is a mechanism that allows endowing the set S with a certain structure. For example, the simplest structure is a linear one, which allows one to establish a chain of applications of transformations from S to determine a certain fact or situation:

r0 -> r1 -> r2 -> ... rn—1 -> rn .

Here r0 is the initial rule, and rn is the resulting rule that determines the result of the functioning of the functional system, and the chain of rules itself is the trajectory of the functional system.

In order for the inference engine to be able to perform the most elementary step, it must first activate the rule analyzer, the input of which is the current rule. If the rule is applicable (the value of the antecedent is true), then the rule parser constructs a new fact or a new situation. Otherwise, the rule is not considered. The functions of the rules analyzer include:

1. Parse the antecedent of the rule.

2. Calculate the Boolean value of the antecedent of the rule.

3. If the value of the antecedent is true, then form a fact or situation depending on the consequent of the rule.

Note that the consequent of a rule is usually a single statement. If the consequent has several statements, then formula (3) is applied to each statement and