Design a keypad lock which has three buttons labelled A, B and C.

Any of the keys can be pressed at anytime, and if the correct sequence of 5 key presses, namely BBCBA, is keyed in, then the lock will open.

Let a, b and c be actions representing the actions of pressing buttons A, B and C, respectively; and open and close represent the action of opening and closing, the lock.

Let S_n (for 0 n 5) be state variables, with S_n representing the state in which the last n button presses are the first n buttons in the sequence of button presses which opens the lock. For example, after the sequence of button presses ABCBBC, the process should be in state S₃.

(a) Give process definitions for each of the process variables Sn.

(b) Draw the labelled transition system for this process.

Only looking for (a) and (b)