## 2 Theoretical Background
### dTP-Calculus
#### 2.1.1 Properties of dTP-Calculus  
The distinctive characteristics of dTP-Calculus are mobility, temporality, probability, priority, and synchronicity.  
Mobility Property  
In dTP-Calculus, the movements of processes are characterized by the subjectivity and directivity of the movements as follows:  
* Subjectivity--There are two types of movements in dTP-Calculus, (1) active movement, where a process moves autonomously by itself, and (2) passive movement, where a process is moved heteronomously by another process;
* Directivity--There are two types of movements in dTP-Calculus, (1) inward movement, where a process moves into another sibling process, and (2) outward movement, where a process moves out of its parent process.  
With respect to the subjectivity and the directivity of the movements, there are a total of four possible types of movements defined in dTP-Calculus, as shown in Table 1 and listed as follows:* _In_--A subjective process moves autonomously into another sibling process and the subjective process needs to obtain permission from the sibling process;
* _Out_--A subjective process moves autonomously out of its parent sibling process and the subjective process needs to obtain permission from the parent process;
* _Get_--An objective process is moved into another sibling process heteronomously by the sibling process and the sibling process needs to obtain permission from the subjective process;
* _Put_--An objective process is moved out of its parent process heteronomously by the parent and the parent process needs to obtain permission from the objective process.  
* _Period_--The time that a process repeats a specific action iteratively, like recursion.  
Probability Property  
dTP-Calculus defines the following probability properties for the choice operation, called the probabilistic choice operation, in order to control the nondeterministic choice operation:  
* Discrete distribution--It allows to specify the choice to be made for its alternatives in the range of discrete values, where the summation of all the alternatives of the choice is 100%;
* Normal distribution--It allows the choice to be distributed normally for its alternatives with a mean (\(\mu\)) and a deviation (\(\sigma\)), whose dense function is defined as \(\frac{1}{\sigma\sqrt{2\pi}}exp\left(-\frac{(x-\mu)^{2}}{2\sigma^{2}}\right)\);
* Exponential distribution--It allows the choice to be distributed exponentially for its alternatives with a frequency (\(\lambda\)), whose dense function is \(\lambda e^{-\lambda x}\);
* Uniform distribution--It allows the choice to be distributed uniformly for its alternatives in the range of \([l,\ u]\), whose dense function is \(\left\{\frac{1}{\mu-l}\ (l\leq x\leq u)\right.\)  
Priority in dTP-Calculus is the property assigned to each process utilized to access or control another process, that is, interactions for communication or movements.  
It is used as a condition to control asynchronicity between asynchronous interactions or to make a decision on which synchronous action is to be executed among multiple synchronous interactions.  
In other words, it is a critical criterion that makes a decision on which action of a process is to be executed first among competing processes.  
Synchronicity Property  
The movements in dTP-Calculus are accomplished synchronously and require synchronous request and permission interactions, as shown in Table 2.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Movements** & **Request** & **Permision** & **Results** \\ \hline \hline \multirow{4}{*}{_In_} & \(A=in\ B\) & \(B=A\) _in_ & \(A\) _in_ \(B\) \\ \cline{2-4}  & & & \\ \cline{2-4}  & & & \\ \cline{2-4}  & & & \\ \cline{2-4}  & & & \\ \cline{2-4}  & & & \\ \cline{2-4}  & & & \\ \cline{2-4}  & & & \\ \cline{2-4}  & & & \\ \cline{2-4}  & & & \\ \hline \hline \end{tabular}
\end{table}
Table 2: Synchronicity for movements in dTP-Calculus.