State Based Control of Timed Discrete Event Systems using Binary ...

Chapter 3. Introduction to Binary Decision Diagrams 19that range(v) = 2 for all vertices and there are only two terminals, i.e., M = 2.By changing the order in which we expand with respect to the variables we usuallyget large differences in the number of vertices (nodes). The ordering is called variableordering and plays a significant role in lowering the representational complexity of thefunctions. A simple example is given below.Suppose we have a 4-variable function F : (x 1 , x 2 , x 3 , x 4 ) → {0, 1}, where x i ∈ {0, 1}, i ∈{1, 2, 3, 4}. F is defined asx 1 x 2 x 3 x 4 F0 0 0 1 11 1 0 1 1Table 3.2: Function FThe value of F for other values of (x 1 , x 2 , x 3 , x 4 ) is 0.The two BDDs representing F , using different variable ordering, are shown in Figure 3.2.Obviously the ordering x 1 , x 2 , x 3 , x 4 is better than the ordering x 1 , x 4 , x 3 , x 2 ( 6 nodes vs8 nodes ).Figure 3.2: BDDs for representing function F with different variable orderingDefinition 3.2.1 BDD to Function ConnectionA BDD G having root vertex v denotes a function f v defined recursively as1. If v is a terminal vertex, then f v = value(v) ∈ {0, 1}.