$1$ Controllability and Supervisor

Consider the finite automaton from Figure $1,$ with the specification K $=\{$acb$,$ abb, bcbb$\}$ and

Euc $=\{$b$\}.$

$\backslash$epsi

a

b

c

b

c

b$,$ c

a

Figure $1$: Finite automaton.

Is K controllable? Determine K$$C$,$ K$$C$,$ and a supervisor S that satisfies K$$C