## 2 Preliminaries
### Reconfigurable S\({}^{3}\)PR Net
A new rewriting reconfigurable net \(N_{R}\) is an \(S^{3}\)PR net (\(N_{R}\), \(M_{R}\)) with \(N_{R}=\) (\(P_{C}\), \(T\), \(F\), \(W\), \(M_{R}\), \(K\)), and a net (\(N\), \(M_{o}\)) is called the initial state of the rewriting net model._  
**Definition 25**. : _Let \(N_{R}\) be a reconfigurable \(S^{3}\)PR with \(N_{R}=\) ((\(N\), \(M_{o}\)), \(\mathcal{R}\)). A state graph in \(N_{R}\) is a labeled directed graph whose nodes are the marking of \(N_{R}\), expressed as:_  
1. _Transition firing: If Arcs labeled with_ \(t\) _can fire in the net (_\(N_{1}\)_,_ \(M_{1}\)_), leading to (_\(N_{2}\)_,_ \(M_{2}\)_): (_\(N_{1}\)_,_ \(M_{1}\)_)_ \(\xrightarrow{\textbf{t}}\) _(_\(N_{2}\)_,_ \(M_{2}\)_)_ \(\Leftrightarrow\) (_\(N_{1}\)_ =_ \(N_{2}\)_ and_ \(M_{1}\)_[_\(t_{2}\)_]_ \(M_{2}\)_ _in_ \(N_{1}\)_)._
2. _Configuration changing: Arcs labeled with_ \(r=\) _[_\(L\)_,_ \(R\)_,_ \(\varphi\)_,_ \(\bullet\varphi\)_,_ \(\varphi\)_\(\bullet\)_] from state (_\(N_{1}\)_,_ \(M_{1}\)_) to state (_\(N_{2}\)_,_ \(M_{2}\)_) if there is_ \(\xi\)_:_ \(L\to N_{1}\) _so that,_ \(\forall a\
otin\xi(L)\) _and_ \(b\in L\) _if_ \[a\in  
**Proof. ** The rewriting of \(N_{2}\) using \(rr\) is similar to replacing a place/transition by the S\({}^{3}\)PR net. Therefore, the boundedness can be established by checking if the S\({}^{3}\)PR net is well constructed and behaved. The resulting net (\(N_{2}\), \(M_{2o}\)) maintains the boundedness because the S\({}^{3}\)PR net is well constructed and behaved. **Corollary 1. **_Let \(N_{R}\) be a reconfigurable S\({}^{3}\)PR with \(N_{R}\) = ((N, \(M_{o}\)), \(\mathcal{R}\)). Let \(N_{1}\) and \(N_{2}\) be two states in \(N_{R}\) with \(N_{1}\) = (\(P_{C1}\), \(T_{1}\), \(F_{1}\), \(W_{1}\), \(M_{1o}\), \(K_{1}\)) and \(N_{2}\) = (\(P_{C2}\), \(T_{2}\), \(F_{2}\), \(W_{2}\), \(M_{2o}\), \(K_{2}\)), \(P_{C1}\), \(T_{1}\
eq\emptyset\) and \(\mathcal{R}\) = [rr], \(rr\) = (\(L\), \(R\), \(\varphi\), \(\varphi^{\bullet}\)). If (\(N_{1}\), \(M_{1o}\)) is bounded, \(L\) is an S\({}^{3}\)PR Petri net and \(R\) is a single place or single transition, then the resulting net (\(N_{2}\), \(M_{2o}\)) by \(rr\) is bounded._  
**Corollary 2. **_An \(S^{3}\)PR net (\(N_{2}\), \(M_{2o}\)) can be a bounded net and a full subnet of (\(N_{1}\), \(M_{1o}\))._  
**Theorem 3. **_Let \(N_{R}\) be a reconfigurable S\({}^{3}\)PR with \(N_{R}\) = ((N, \(M_{o}\)), \(\mathcal{R}\)). Let \(N_{1}\) and \(N_{2}\) be two states in \(N_{R}\) with \(N_{1}\) = (\(P_{C1}\), \(T_{1}\), \(F_{1}\), \(W_{1}\), \(M_{1o}\), \(K_{1}\)) and \(N_{2}\) = (\(P_{C2}\), \(T_{2}\), \(F_{2}\), \(W_{2}\), \(M_{2o}\), \(K_{2}\)), \(P_{C1}\), \(T_{1}\
eq\emptyset\) and \(\mathcal{R}\) = [rr], \(rr\) = (\(L\), \(R\), \(\varphi\), \(\varphi\), \(\varphi^{\bullet}\)). If (\(N_{1}\), \(M_{1o}\)) is bounded, \(L\) is an S\({}^{3}\)PR net and \(R\) is an \(S^{3}\)PR net, then the resulting net (\(N_{2}\), \(M_{2o}\)) by \(rr\) is bounded._  
**Proof. ** The rewriting of \(N_{2}\) using \(rr\) is similar to replacing an S\({}^{3}\)PR net by another S\({}^{3}\)PR net. Therefore, the boundedness can be established by checking if the S\({}^{3}\)PR net is well constructed and behaved. The resulting net (\(N_{2}\), \(M_{2o}\)) maintains the boundedness because the S\({}^{3}\)PR net is well constructed and behaved. Based on Definitions 20-28 and Theorems 1-3, the developed reconfiguration procedures for S\({}^{3}\)PR net algorithm are constructed as follows:  
**Input:** An \(S^{3}\)PR net (\(N_{o}\), \(M_{o}\))  
**Output:** A reconfigurable S\({}^{3}\)PR net (\(N_{R}\), \(M_{Ro}\))  
**Initialization:** Generate dynamic configurations \(\mathcal{R}\) = [rr\({}_{1}\), rr\({}_{2}\), rr\({}_{3}\),..., rr\({}_{ml}\)] k=0. **Step 1:** _while_\(\mathcal{R}\
eq\emptyset\)**do**  
\(k\) = \(k\)+1  
1.1. Build \(rr_{k}\) = [L\({}_{k}\), \(R\)\({}_{k}\), \(\varphi_{k}\), \(\varphi_{k}\), \(\varphi_{k}\)]. 1.2. Build \(L_{k}\) = (\(P_{CKL}\), \(T_{lk}\), \(F_{LK}\), \(W_{1k}\), \(M_{Lko}\), \(K_{Lk}\)). 1.3.