Suppose we want to design a synchronous circuit that processes a string of 0's and 1's. Given a string, it produces another string by replacing the first 1 in any subsequence of consecutive 1's by a 0. Consider the following example. Input sequence: 00100011000011100 Output sequence: 00000001000001100 A Mealy Machine is a state machine where both the next state and the output are functions of the present state and the current input. The above mentioned circuit can be designed as a two-state Mealy machine. The states in the Mealy machine can be represented using Boolean values 0 and 1. We denote the current state, the next state, the next incoming bit, and the output bit of the Mealy machine by the variables $s$, $t$, $b$ and $y$ respectively. Assume the initial state of the Mealy machine is 0. What are the Boolean expressions corresponding to $t$ and $y$ in terms of $s$ and $b$?

Let state $s = 0$ denote that the machine is not currently within a sequence of consecutive 1's (or the previous bit was 0). Let state $s = 1$ denote that the first 1 of a sequence of consecutive 1's has just occurred and any immediately following 1's should be output as 1.

We determine the next state $t$ and output $y$ for all combinations of current state $s$ and incoming bit $b$ [1]:

1. From state $s = 0$ (initial state / previous bit was 0):

   • If $b = 0$: the output is $y = 0$, and next state is $t = 0$.

   • If $b = 1$: this is the first 1 in a consecutive sequence, so it must be replaced by 0 ($y = 0$). The next state becomes $t = 1$.

2. From state $s = 1$ (previous bit was 1):

   • If $b = 0$: the run of 1's ends, output $y = 0$, and next state is $t = 0$.

   • If $b = 1$: this is a subsequent 1 in the run, so output $y = 1$, and next state remains $t = 1$.

Constructing the state transition and output table:

From the truth table:

• $t = b$

• $y = s \cdot b$

Therefore, option (B) is correct.