# Modeling of Information Operations Effects: Technological Systems Example +
## 4 Examples of Models for Information Operations Effects and Operational Properties Estimation
Each of the states:  
\[\hat{S}_{l.q}=<\hat{y}_{1.l.q}\ldots\hat{y}_{k.l.q}\ldots\hat{y}_{K.l.q}> \tag{1}\]  
checked at the boundary of the CTS and its environment is fully described by the effects of functioning by the time the state check starts. State (1) is compared with environment state which specifies requirements values:  
\[S_{l.q}^{\partial}=<\hat{y}_{1.l.q}^{\partial}\ldots\hat{y}_{k.l.q}^{\partial} \ldots\hat{y}_{K.l.q}^{\partial}> \tag{2}\]  
(may be random but for simplicity are considered non-random). Then, a probability measure \(P_{l.q}\) of states \(\hat{S}_{l.q}\) compliance to requirements of the environment \(S_{l.q}^{\partial}\) can be defined:  
\[P_{l.q}=P(\hat{A}_{l.q})=P(<\hat{y}_{1.l.q}r_{1}\hat{y}_{1.l.q}^{\partial} \ldots\hat{y}_{k.l.q}r_{k}\hat{y}_{k.l.q}^{\partial}\ldots\hat{y}_{K.l.q}r_{k} \hat{y}_{K.l.q}^{\partial}>) \tag{3}\]  
where \(r_{k}\)-\(k\)-th required relationship between predicted values of effect characteristics and their required values (e.g.\(<\),\(>\)). The probability measure is calculated using a functional model for calculating the correspondence at the boundary of the CTS and the environment. \(P(\hat{A}_{l.q})\)--the probability of an event consisting in the fact that when checking the state \(\hat{S}_{l.q}\) for one of the possible branches of the tree, when performing a single checking TIO by defined mode, required by environment characteristics of the effects will be achieved. This event means that the result of the checking TIO is good to achieve the required intermediate goal of the CTS functioning given the states of environment changes fixed (the intermediate goal of CTS is achieved in current environment circumstances). Since such checking TIO of states \(\hat{S}_{l.q}\) corresponding to the modes of checking TIO in one branch of \(C^{CTC}(c_{m}^{Cp})\) number is less or equal to \(L\), and all of them are expressed in the model, the measure of compliance for the implementation of the entire sequence of checking TIO for one branch \(c_{m}^{Cp}\in C^{Cp}\), correspondence measure for whole (but one) branch of \(C^{CTC}(c_{m}^{Cp})\) can be calculated as the probability of a complex event \(\hat{A}_{l}\) which means all intermediate goals achieved in a given environment circumstances. Event \(\hat{A}_{l}\) probability is:  
\[P(\hat{A}_{l})=P(\mathop{\cup}\limits_{q\in\overline{1,Q_{l}}}\hat{A}_{l.q}) \tag{4}\]  
If the probabilities of compliance for each of the checking TIO are conditionally independent in their sequence, than:  
\[P(\hat{A}_{l})=\prod_{q\in\overline{1,Q_{l}}}\hat{A}_{l.q} \tag{5}\]  
Let the probability of an event \(\hat{\beta}_{q,p}\), consisting in the fact that the transition \(a_{q,p}\) will be executed \(\hat{B}_{q,p}=(\hat{S}_{l.q},\hat{S}_{l.p}):\exists a_{q,p}:q,p\in\overline{1,Q _{l}}\) is equal to \(P_{q,p}=P(\hat{B}_{q,p})\sim a_{q,p}\), i.e., the probability \(P(\hat{B}_{q,p})\) is associated with the transition \(a_{q,p}\). Then the probability of implementing a branch \(v_{l}:l\in\overline{1,L}\) of the tree \(C^{CTC}(c_{m}^{Cp})\):  
\[P_{l}=P(\mathop{\cap}\limits_{a_{q,p}\in v_{l}}\hat{B}_{q,p}) \tag{6}\]  
Figure 5:0  
Then, as a scalar indicator of the CTS potential \(\psi\) as well as its dynamic capability, we can take the expected probability of the event that whatever branch \(c_{m}^{Cp}\in C^{Cp}\) and corresponding branchesof \(C^{CTC}(c_{m}^{CP})\) implemented, there will be right correspondence between expected and required states measured by checking TIO. It means, whatever changes of environment happens, and whatever operations conducted to fulfill changing goals, changing goals of the CTS will be achieved:  
Figure 5:1  
In general, the probability \(P(\hat{\mathcal{C}})\) of event specified can be represented as a random variable \(\psi\), not its expected value \(\overline{\psi}\).