## 2 Preliminaries
### Reconfigurable S\({}^{3}\)PR Net  
This section presents definitions and theorems in the reconfigurable S\({}^{3}\)PR nets, which are originally proposed by [34; 35; 39]. **Definition 20**. : _Let \(N=(l[p^{0}]\cup P_{A}\cup P_{R}\), \(T\), \(F\), \(W\), \(M_{o}\), \(K\)) be a finite-capacity \(S^{3}\)PR, where \(p^{0}\), \(P_{A}\), \(P_{R}\), \(T\), \(F\), \(W\), and \(M_{o}\) are defined in Definitions 1-6. K: \(P_{C}\rightarrow\mathbf{IN}\) is the function of capacity that assigns to each place \(p\) the maximal number of tokens K(\(p\))._  
**Definition 21**. : _Let \((N_{i}\), \(M_{i})\) be two \(S^{3}\)PR nets with \(N_{i}\) = (\(P_{C\!I}\), \(T_{i}\), \(F_{i}\), \(W_{i}\), \(M_{i}\), \(K_{i}\)), i = 1, 2. \(N_{1}\) and \(N_{2}\) are called morphism nets if there exists a bijection \(\Psi\): \(N_{1}\to N_{2}\), \(\Psi\) = (\(\Psi_{PC}\): \(P_{C\!I}\to P_{C\!2}\), \(\Psi_{T}\): \(T_{1}\to T_{2}\)) such that for all \(a\), \(b\in P_{C\!I}\cup T_{1}\), \(F_{1}(a,b)\in N_{1}\) = \(F_{2}(\Psi(a)\), \(\Psi(b))\in N_{2}\), and for all \(p\in P_{C\!I}\), \(M_{1}(p)\leq M_{2}(\Psi_{PC}(p))\)._  
**Definition 22**. : _Let \((N_{i}\), \(M_{i})\) be two \(S^{3}\)PR nets with \(N_{i}\) = (\(P_{C\!I}\), \(T_{i}\), \(F_{i}\), \(W_{i}\), \(M_{i}\), \(K_{i}\)), i = 1, 2. \(N_{1}\) is called the full subnet of \(N_{2}\) if there exists an injection function that maps places to places and transitions to transitions, denoted by \(\xi\): \(N_{1}\rightarrow N_{2}\), \(\xi\) (\(P_{C\!I}\subseteq P_{C\!2}\), and \(\xi\) (\(T_{1}\)) \(\subseteq T_{2}\) such that for all \(a\), \(b\in P_{C\!I}\cup T_{1}\), \(F_{1}(a,b)\) = \(F_{2}(\xi\) (\(a\)), \(\xi\) (\(b)))._  
_In the algebraic, a rewriting rule is a transformation approach that can change and combine the Petri nets dynamically. The main idea is to define and change the system configurations as a graph rewriting rule._  
**Definition 23**. : _Let \(N_{R}\) be a reconfigurable S\({}^{3}\)PR with \(N_{R}\) = ((\(N\), \(M_{o}\)), \(\mathcal{R}\)), where(\(N\), \(M_{o}\)) is an S\({}^{3}\)PR net with \(N\) = (\(P_{C}\), \(T\), \(F\), \(W\), \(M_{o}\), \(K\)) and \(\mathcal{R}\) = {rr\({}_{1}\), \({}_{2}\), rr\({}_{3}\), \(\ldots\), rr\({}_{m}\)} is called a set of rewriting rules or dynamic configurations if_  
1. _For all rr_ \(\in\mathcal{R}\)_, rr_ = {_L_, \(R\), \(\varphi\), \({}^{\bullet}\varphi\), \({}^{\bullet}\varphi\)_}_;_
2. \(L\) = (\(P_{C\!L}\), \(T_{L}\), \(F_{L}\), \(W_{L}\), \(M_{oL}\), \(K_{L}\)) _is called the left-hand side;_
3. \(R\) = (\(P_{C\!R}\), \(T_{R}\), \(F_{R}\), \(W_{R}\), \(M_{oR}\), \(K_{R}\)) _is called the right-hand side;_
4. \(\varphi\subseteq(P_{C\!L}\times P_{C\!R})\cup(T_{L}\times T_{R})\) _is said to be an interface transfer relation of r that relates places of_ \(L\) _to places of_ \(R\) _and transitions of_ \(L\) _to transitions of_ \(R\)_,_ \(P_{C\!L}\) _or_ \(P_{C\!R}\)_,_ \(\varphi P_{C\!R}\subseteq P_{C\!L}\)_,_ \(T_{L}\!\varphi\subseteq T_{R}\)_, and_ \(\varphi T_{R}\subseteq P_{L}\)_;_
5. \({}^{\bullet}\varphi\subseteq\varphi\) _is said to be an input interface transfer relation, expressed as_ \({}^{\bullet}\varphi\) = {({L_{\!p}}_{i}\), {_R.p_}}_}_}__ _or {({L_{\!t}}_{i}\)}, {{R. \(t_{i}\)}}, and L.* or R.* means to input nodes \"*\" in L or R;_
6. \(\varphi^{\bullet}\subseteq\varphi\) _is named output interface transfer relation,_ \(\varphi^{\bullet}\) = {({{L_{\!p}}_{i}}_{j}\), {_R.p_}}_}__}_ _or {({L_{\!t}}_{j}\), {R.t_}}, and L.* or R.* means to output nodes \"*\" in L or R;_
7. _for all rr_, rr_ \(r_{i}\) _r_\(\in\mathcal{R}\) (i \(\
eq j\))_, \(\xi\) (\(L_{i}\)) \(\cap\) \(\xi\) (\(L_{j}\)) \(\
eq\) _0, a rewriting must be guaranteed without overlap; moreover, the order of rr_ \(r_{i}\) _r_\(j\) _does not impact the result of the rewriting._  
Figure 2: A system of (S\({}^{2}\)P) (simple sequential process) (S\({}^{3}\)PR) net of the AMS. **Definition 24**. : _Let \(N_{R}\) be a reconfigurable \(S^{3}\)PR with \(N_{R}=\) ((\(N\), \(M_{o}\)), \(\mathcal{R}\)).