$15\mathrm{.}14$ A sequential circuit has a single input $(\backslash ($ X $\backslash ))$ and a single output $(\backslash ($ Z $\backslash )).$ The circuit examines each disjoint block of four inputs and determines whether the block is a valid BCD representation of a decimal digit; if not, $\backslash ($ Z$=1\backslash ).$ State $\backslash ($ S$\_\{0\}\backslash )$ is the initial state, and the circuit enters state $\backslash ($ S$\_\{0\}\backslash )$ after the fourth input. The BCD digits are received least significant bit first. A Mealy state table can be constructed using a sufficient number of states to remember the last three inputs with the output produced when the fourth input bit of a block is received.

$($a$)$ Using the method indicated by Problem $15\mathrm{.}13,$ construct a state table for this circuit.

$($b$)$ Reduce the state table to a minimum number of states $($Hint: Use the simple example of state equivalence used in Section $15\mathrm{.}1$ to eliminate as many states as possible.$)$

$($c$)$ For each state in the reduced table, give the input pattern$($s$)$ remembered by that state.