Petri net-2, directed net

petri网

2、有向网

三元组 $N=(S,T;F)$ directed net,如果

表达式	含义
$S\bigcup T\ne \varnothing$	非空
$\bigwedge S\bigcap T\ne \varnothing$	两类元素
$\bigwedge F\subseteq S\times T\bigcup T\times S$	两种关系
$\bigwedge dom(F)\bigcup cod(F)=S\bigcup T$	No isolated elements

其中：

• $dom(F)=\{x|\exists y:(x,y)\in F\}$
• $cod(F)=\{y|\exists x:(x,y)\in F\}$

例1：

• $S=\{c,x,q,d\}$
• $T=\{cx,xq,qd,dc\}$
• $F=\{(c,cx),(cx,x),(x,xq),(xq,q),(q,qd),(qd,d),(d,dc),(dc,c)\}$
• $N_1=\{S,T;F\}$
• 

例2：

2.1 基本元素

• $S$ 元素： place 库所
• $T$ 元素：transition 变迁（不是迁移,more than just moving,This qualitative change,variable)
• F：flow relation 流关系

2.2 Expression comparison

• semi-formal definition：易读,suitable for written communication
• 形式化定义：suitable for introducing mathematical methods,Facilitate automatic processing
• 图形表示：直观 便于交流,Highlight the reticular structure,No need to name elements when communicating face-to-face

例3：

• 上图是 $N=(\{S_1,S_2\},\{t\};\{(S_1,t),(t,S_2)\})$ A graphical representation of the？
• in the sense of isomorphism,是（同构：One-to-one correspondence between like elements,包括 $F$ ）

例4：

• Is this a directed network？
• 可以看成一个,也可以看成两个.The definition does not clearly specify whether or not to connect.
• 但是,Since the two parts are not connected,Why study together？This is triggered from the actual application.

2.3 连通性

• 
  弱连通
• 
  强连通,Remove an edge and still connect
• 
  不连通,The last two no connection

2.4 Simple net、简单网

Do directed nets allow the following structure?？

• 
  不单纯（单纯性)
• 
  不简单（简单性)
• 
  repeating arc,不能出现

2.4.1 术语：前集、后集

$X=S\bigcup T$ node set element set of directed net
$$\begin{gathered} x,y\in X \\ f(x)=\{\begin{array}{rlrlr}{}^{.}x& =& \{y|(y,x)\in F\}& ,& xthe\; first\; episode\; of\\ x^{.}& =& \{y|(x,y)\in F\}& ,& xafter\; the\; episode\end{array} \end{gathered}$$

2.4.2 Simple net、简单网、连通网

2.4.2.1 Simple net definition

$$\begin{gathered} \forall x\in X:{ \\ }^{.}x\cap x^{.}\ne \varnothing \end{gathered}$$

2.4.2.2 Simple net definition

$$\begin{gathered} \forall x,y\in X:{ \\ }^{.}x{=}^{.}y\bigwedge x^{.}=y^{.}\bigwedge x\ne y \end{gathered}$$

2.4.2.3 连通图定义

$$\begin{gathered} \forall x,y\in X: \\ (x,y)\in (F\cup F^{-1}{)}^{+} \\ 其中F^{-1}=\{(a,b)|(b,a)\in F\} \end{gathered}$$

2.4.2.4 Repeating Arc Problem

Not allowed to repeat the arc

2.4.2.5 Dual network and reciprocal network

$$\begin{gathered} N=(S,T;F),N^{\prime }=(S^{\prime },T^{\prime };F^{\prime })为有向网 \\ N和N^{\prime }For\; the\; dual\; network,如果S^{\prime }=T\bigwedge T^{\prime }=S\bigwedge F^{\prime }=F \\ N和N^{\prime }reciprocal\; network,如果S^{\prime }=S\bigwedge T^{\prime }=T\bigwedge F^{\prime }=F^{-1} \end{gathered}$$

The following two network is the same network,同为 $N=(\{S_1,S_2\},\{t\};\{(S_1,t),\{t,S_2\}\})$ 的图示

2.4.3 cable network & Infinite network

A directed net can have a finite number of elements,There can also be an uncountable finite number of elements $|S\cup T|<\infty$

• cable network
  • 人造系统：能力有限
  • 自然规律（like the changing seasons）：局部观察
• Infinite network
  • Documenting natural change that never ends
  • 理论研究（图灵机）

2.5 出现网

record what happened

例5：

2.6 Basic Concept Definition Golden Rule

not necessary to include,it is necessary not to include

2.6.1 Directed net definition does not contain

• 连通性
• 单纯性
• 简洁性
• 有限性

2.6.2 Directed network concept

• Basic concepts are concise
• Flexibility to introduce subsequent concepts