Model as Transition Graph

The figure 3.2 above shows a transition graph G, which is a possible visual representation of an corresponding finite automaton A. A possible formal description of the graph G is as follows:

$$\langle V,E,\Sigma \rangle$$

with $V$ as a set of vertices, $E$ as a set of edges and $\Sigma$ as an input alphabet, which is used for the Symbols attached to the edges as labels. As alphabet$\Sigma$ we assume the small set $\{ A,B,C \}$. We can then write:

1. $V$ = $\{ 1,2,3,4 \}$
2. $E$ $\subseteq$ $V \times \Sigma \times V$ with $E$ = $\{\langle 1,A,2\rangle, \langle 1,B,1\rangle,\langle 1,C,1\rangle, \langle 2,A... ...C,1\rangle, \langle 3,A,4\rangle, \langle 3,B,1\rangle,\langle 3,C,1\rangle\}$
3. $\Sigma$ = $\{ A,B,C \}$

Using the additional convention to mark all initial and all final vertices with a double line you can now read the graph representation as a description of the expected behavior. The automaton, which is here represented as a transition graph, starts in vertex 1. As long as the human person (the user) keys in B or C the edges are leading back to vertex 1. Only if the symbol $A$ is keyed in leads an edge to vertex 2. For a transition from vertex 2 to vertex 3 the user must key in the symbol $B$, and then again $A$ to reach vertex 4, a final state. If the user makes a mistake during its way to the final vertex he will be set back to the initial vertex 1.