**3.2 Integration for complex structures on the level of probability distribution functions**

If probability of providing system integrity within given prognostic period for all points Tgiven from 0 to ∞ are computed, it means a trajectory of the PDF depending on characteristics of threats, periodic diagnostics, and recovery. And, the building of PDF is the real base to predict probabilistic metrics for given time Tgiven. In analogy with reliability, it is important to know a mean time between neighboring losses of integrity (MTBLI) like mean time between failures in reliability (MTBF), but in application to concepts of quality, safety, etc.

For complex systems with serial or parallel structure, new models with known PDF can be developed by the next method [17–21]. Let us consider the elementary structure from two independent parallel or serial elements (**Figures 3** and **4**). Let the PDF of time between losses of i-th element integrity be Вi(t), i.e., Вi(t) = Р(τ<sup>i</sup> ≤ t), then:

1. time between losses of integrity for system combined from serial connectedindependent elements is equal to minimum from two times τi: failure of first or second elements (it means the system goes into a state of lost integrity when either first, or second element integrity is lost). For this case, the PDF of time between losses of system integrity is defined by the expression

$$\begin{array}{l} \mathbf{B}(\mathbf{t}) = P(\min \ (\tau\_1, \tau\_2) \le \mathbf{t}) = \mathbf{1} - \mathbf{P}(\min \ (\tau\_1, \tau\_2) > \mathbf{t}) = \mathbf{1} - \mathbf{P}(\tau\_1 > \mathbf{t})\mathbf{P}(\tau\_2 > \mathbf{t})\\ = \mathbf{1} - [\mathbf{1} - \mathbf{B}\_1(\mathbf{t})][\mathbf{1} - \mathbf{B}\_2(\mathbf{t})] \end{array} \tag{4}$$

2. time between losses of integrity for system combined from parallel connected independent elements (hot reservation) is equal to a maximum from two times τi: failure of first and second elements (it means the system goes into a state of lost integrity when both first and second elements have lost integrity). For this case, the PDF of time between losses of system integrity is defined by the expression

$$\mathbf{B(t)} = \mathbf{P(}\max\left(\tau\_1, \tau\_2\right) \le \mathbf{t}\right) = \mathbf{P(}\tau\_1 \le \mathbf{t}\big)\mathbf{P(}\tau\_2 \le \mathbf{t}\big) = \mathbf{B\_1(t)}\mathbf{B\_2(t)}\tag{5}$$

By applying recurrently expressions (4) and (5), it is possible to build PDF of time between losses of integrity for any complex system with parallel and/or serial structure.

As summary, the calculated results of modeling are: PDF of time between losses of integrity for system and each compound subsystems and elements; mean time

*Probabilistic Methods for Cognitive Solving of Some Problems in Artificial Intelligence Systems DOI: http://dx.doi.org/10.5772/intechopen.89168*

between losses of integrity for system and each compound subsystems and elements (MTBLI as analog of MTBF).

For example, integrated complex system, combined from intellectual structures for modeling interested system including AIS (**Figure 7**), can be analyzed by formulas (1)–(5) and probabilistic models described above and allowing to form PDF by (4) and (5). The correct operation of this complex system during the given period means: during this period both first and second subsystems (left and right) should operate correctly according their destinations, i.e., integrity of complex system is provided if "AND" integrity of first system left "AND" integrity of second system right are provided.

All these ideas of analytical modeling operation processes are supported by the software tools [18, 19, 21, 23, 38–44].

What about new knowledge by using the proposed methods and models for cognitive solving of problems 1 and 2 of the chapter? A use of these methods and models on different stages of AIS life cycle (concept, development, utilization, support stages) allows to produce cognitive answers for the following questions:


**Figure 7.** *An integrated complex system of two serial subsystems (abstraction).*

Summaries for the last model are as follows:

but in application to concepts of quality, safety, etc.

probability of "success."

*Probability, Combinatorics and Control*

**functions**

Вi(t) = Р(τ<sup>i</sup> ≤ t), then:

expression

structure.

**12**

• The input for modeling include: frequency of the occurrences of potential threats (or mean time between the moments of the occurrences of potential threats which equals to 1/frequency); mean activation time of threats; mean recovery time; time between the end of diagnostics and the beginning of the

• The calculated results of modeling include: the probability of providing system integrity within given prognostic period (i.e., probability of "success"); and risk to lose integrity (i.e., probability of "failure") as addition to 1 for

**3.2 Integration for complex structures on the level of probability distribution**

If probability of providing system integrity within given prognostic period for all points Tgiven from 0 to ∞ are computed, it means a trajectory of the PDF depending on characteristics of threats, periodic diagnostics, and recovery. And, the building of PDF is the real base to predict probabilistic metrics for given time Tgiven. In analogy with reliability, it is important to know a mean time between neighboring losses of integrity (MTBLI) like mean time between failures in reliability (MTBF),

For complex systems with serial or parallel structure, new models with known PDF can be developed by the next method [17–21]. Let us consider the elementary structure from two independent parallel or serial elements (**Figures 3** and **4**). Let

1. time between losses of integrity for system combined from serial connectedindependent elements is equal to minimum from two times τi: failure of first or second elements (it means the system goes into a state of lost integrity when either first, or second element integrity is lost). For this case, the PDF of time

¼ 1 � ½ � 1 � В1ð Þt ½ � 1 � В2ð Þt (4)

ВðÞ¼ t Рð max ð Þ τ1, τ<sup>2</sup> ≤tÞ ¼ Рð Þ τ<sup>1</sup> ≤t Рð Þ¼ τ<sup>2</sup> ≤ t В1ð Þt В2ð Þt (5)

2. time between losses of integrity for system combined from parallel connected independent elements (hot reservation) is equal to a maximum from two times τi: failure of first and second elements (it means the system goes into a state of lost integrity when both first and second elements have lost integrity). For this case, the PDF of time between losses of system integrity is defined by the

By applying recurrently expressions (4) and (5), it is possible to build PDF of time between losses of integrity for any complex system with parallel and/or serial

As summary, the calculated results of modeling are: PDF of time between losses of integrity for system and each compound subsystems and elements; mean time

the PDF of time between losses of i-th element integrity be Вi(t), i.e.,

between losses of system integrity is defined by the expression

ВðÞ¼ t *Р*ð min ð Þ τ1, τ<sup>2</sup> ≤tÞ ¼ 1 � Рð min ð Þ τ1, τ<sup>2</sup> >tÞ ¼ 1 � Рð Þ τ1>t Рð Þ τ2>t

next diagnostics; diagnostics time; and given prognostic period.


expert value of expected level of "success" for each variant may be established, for example, on a dimensionless scale from 0 to 100 (0—"no gain", i.e., "failure"; 100—"the maximal gain," i.e., complete "success"). After learning by knowledge base, self-improving AIS uses input and the corresponding results of probabilistic modeling in a form of the solution of previously specific encountered problem 1. Knowledge base (K-base) is defined as a database that contains inference rules and information about human experience and expertise in a domain (ISO/IEC

*Probabilistic Methods for Cognitive Solving of Some Problems in Artificial Intelligence Systems*

**Step 2**. The measures and optimization criteria are chosen. As criteria can be

• Maximum of gain as a result of the functions performance under the given conditions and limitations on the acceptable risk of failure and/or other

• Maximum probability of "success" or minimum risk of "failure" under

**Step 3**. The accumulated knowledge is used to refine the input for modeling. A quality of used information is estimated by models above considering limitations from **Table 1**. Using the model for each variant, the probabilistic measures are calculated for given prognostic period (see proposed models above and Step 1). From a set of possible variants, the optimal one is chosen, according to Step 2

Note. For example, there are proposed the next general formal statements of

2382-1:1993).

limitations

limitations

problems for system optimization:

accepted:

**Figure 8.**

*Steps for cognitive solving of problem 1.*

*DOI: http://dx.doi.org/10.5772/intechopen.89168*

criterion.

**15**

• What are the information security risks? etc*.*

The rationale answers allow to improve and accumulate knowledge concerning AIS.

The proposed methods and models provide the next approach for cognitive solving problems 1 and 2.
