1 Introduction

1.6 The Petri Net Structure

A Marked Petri Net Structure C is a five-tuple C = (P,T,I,O,µ), where P is a finite set of places, T is a finite set of transitions, I and O are the input and output functions repsectively, i.e., for each t in T, I(t) = ^•t and O(t) = t ^•, and µ is a function from P to the non-negative numbers giving the marking of the net. We may describe the marking as a vector µ = (µ₁, µ₂, ... µ_n), where µ_i is the marking for the place p_i.

We had the specification:

 Activities   pre-condition      post-condition
 READ         R                  CR
 DISP         CR, GS             R, GF
 REF          GF = 5             GS = 5

For this system we have:

  P = {R, CR, GS, GF}   
  T = {READ, DISP, REF}
  µ = (1 0 5 0)

  I(READ) = {R}                 O(READ) = {CR}
  I(DISP) = {CR,GS}             O(DISP) = {R, GF}
  I(REF)  = {GF,GF,GF,GF,GF}    O(REF)  = {GS,GS,GS,GS,GS}

In matrix form we could express this using the matrices D^-, D⁺ corresponding to the pre- and post-conditions:

              R  CR  GS  GF            R  CR  GS  GF
              1   0   0   0            0   1   0   0
      D- =    0   1   1   0    D+ =    1   0   0   1 
              0   0   0   5            0   0   5   0

Given the matrices and the marking it is easy to draw the graph, and conversly given the graph it is easy to extract the marking and the corresponding matrices. From this we can conclude that the Petri Net Graph and the Petri Net Structure are equivalent representations of the same net.

Let e(READ) = (1 0 0), i.e., the unit vector for READ. Using the composite change matrix D = D⁺- D^-may then calculate the state after firing READ in the initial state as

delta(µ, READ) = µ + e(READ) · D.

The function delta is called the next-state function.