# Conversion between Number Systems in Membrane Computing
## 2 Research Foundation  
### Cell-like P Systems  
P system is the embodiment of the concrete model of membrane computing, and it is a kind of computing system. The computational task can be designed by constructing evolutionary rules and initial objects for membrane structures. The structure of the cell-like P system is shown in Figure 1. The external membrane in the P system is called skin, separating the entire membrane system from the external environment. A membrane without any other membrane inside it is said to be an elementary membrane [19].  
The basic model of the cell-like P system is defined as Formula (1) [2].  
\[\Pi=(V,\,\mu,\,\omega_{1},\,\ldots,\,\omega_{m},\,R_{1},\,\ldots,\,R_{m},\,i_{ o}) \tag{1}\]  
where:  
\[\Pi=(V,\,\mu,\,\omega_{1},\,\ldots,\,\omega_{m},\,R_{1},\,\ldots,\,R_{m},\,i_{ o}) \tag{2}\]  
Figure 1: The structure of a cell-like P system.  
1. \(V\) is a finite, non-empty alphabet whose elements are objects.
2. \(\mu\) is a membrane structure with \(m\) membrane, labeled by 1, 2, \(\ldots\), \(m\). Membrane with label i, which we call membrane i.
3. \(\omega_{\text{i}}\) (\(1\leq i\leq m\)) is string over \(V\) representing the multiset of objects placed in membrane i.
4. \(\text{R}_{\text{i}}\) (\(1\leq i\leq m\)) are finite sets of possible evolution rules over \(V\) associated with the membrane i. The following three main forms of rules appear in this paper: (1) Rewriting rule u\(\rightarrow\)v. Consuming a multiset of objects u generates a multiset of objects v. (2) Multiset of objects in or out of the membrane (v, ini) (v, out). (v, ini) denotes a multiset of objects v leaving the current membrane into its submembrane. (v, out) shows that the multiset of objects v leaves the current membrane and enters its outer membrane. (3) A multiset of objects disappears u\(\rightarrow\)\(\lambda\) and v\(\rightarrow\)\(\delta\). u\(\rightarrow\)\(\lambda\) indicates that in the current membrane, the multiset of objects u are consumed, and no objects are generated. v\(\rightarrow\)\(\delta\) represents that the multiset of objects v is consumed and no object is generated, while the membrane in which v is located is dissolved.
5. \(\text{i}_{\text{o}}\) is a number between 1 and \(m\) where the output of results in \(\Pi\).  
The execution of rules in \(\Pi\) is characterized by both maximally parallel and nondeterministic behavior [19]. The attractive feature of the P system mentioned in reference [2] is its intrinsic parallelism. All objects having access to a rule should use that rule. Moreover, all membranes work in parallel. Maximally parallel means that all the rules that satisfy the conditions in the evolutionary rules at any time will be executed at the same time to improve the efficiency of the system. Nondeterministic means that when there are multiple rules whose execution conditions can be satisfied but only some of them can be satisfied at the same time, the system randomly selects some of the rules to be implemented and objects in the system to evolve, and the rules governing this evolution are chosen in a nondeterministic way [19].  
In this paper, we introduce a time slice to measure the efficiency of the P system. We assume that each rule takes the same amount of time to execute. The time taken for each execution of all rules that meet the conditions is a one-time slice. All rules that meet the conditions will be executed in parallel within each time slice.