GATE CS 2026 Set 1 Q.37: Sequential Circuits, Counters and Registers

Consider a 2-bit saturating up/down counter that performs the saturating up count when the input $P$ is 0, and the saturating down count when $P$ is 1. The Next State table of the counter is as shown. The counter is built as a synchronous sequential circuit using D flip-flops.

Input $P$	Current State $Q_1$	Current State $Q_0$	Next State $Q_1^+$	Next State $Q_0^+$
0	0	0	0	1
0	0	1	1	0
0	1	0	1	1
0	1	1	1	1
1	0	0	0	0
1	0	1	0	0
1	1	0	0	1
1	1	1	1	0

Which one of the following options corresponds to the expressions for the inputs of the D flip-flops, $D_1$ and $D_0$ ?

$D_1 = P Q_1 + \overline{P} Q_0 + Q_1 Q_0 \qquad D_0 = P Q_0 + \overline{P} Q_1 + Q_1 \overline{Q_0}$

$D_1 = \overline{P} Q_1 + \overline{P} Q_0 + Q_1 Q_0 \qquad D_0 = \overline{P}\,\overline{Q_0} + \overline{P} Q_1 + Q_1 \overline{Q_0}$

$D_1 = \overline{P}\,\overline{Q_1} + \overline{P} Q_0 + Q_1 Q_0 \qquad D_0 = \overline{P} Q_0 + \overline{P} Q_1 + Q_1 \overline{Q_0}$

$D_1 = P\,\overline{Q_1} + \overline{P} Q_0 + Q_1 Q_0 \qquad D_0 = P\,\overline{Q_0} + \overline{P} Q_1 + Q_1 \overline{Q_0}$

Official answer and solution

Official answer: B
From: Official GATE 2026 answer key

Solution

Use the characteristic equation $Q^+ = D$ of a D flip-flop and minimize $D_1(P, Q_1, Q_0)$ and $D_0(P, Q_1, Q_0)$ using 3-variable Karnaugh maps.

For a D flip-flop, the excitation inputs are simply $D_1 = Q_1^+$ and $D_0 = Q_0^+$ :

For $D_1 = Q_1^+$ : The minterms $(P, Q_1, Q_0)$ $(P, Q_{1}, Q_{0})$ where $Q_1^+ = 1$ $Q_{1}^{+} = 1$ are $\sum m(1, 2, 3, 7)$ $\sum m (1, 2, 3, 7)$ :
- Pairing $(1, 3) \rightarrow \overline{P}Q_0$
- Pairing $(2, 3) \rightarrow \overline{P}Q_1$
- Pairing $(3, 7) \rightarrow Q_1 Q_0$
$D_1 = \overline{P}Q_1 + \overline{P}Q_0 + Q_1 Q_0$
For $D_0 = Q_0^+$ : The minterms $(P, Q_1, Q_0)$ $(P, Q_{1}, Q_{0})$ where $Q_0^+ = 1$ $Q_{0}^{+} = 1$ are $\sum m(0, 2, 3, 6)$ $\sum m (0, 2, 3, 6)$ :
- Pairing $(0, 2) \rightarrow \overline{P}\,\overline{Q_0}$
- Pairing $(2, 3) \rightarrow \overline{P}Q_1$
- Pairing $(2, 6) \rightarrow Q_1 \overline{Q_0}$
$D_0 = \overline{P}\,\overline{Q_0} + \overline{P}Q_1 + Q_1 \overline{Q_0}$

Final answer

Option (B)

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2026 Computer Science and Information Technology - Session 1 (CS1)
Question: 37 of 65
Section: Computer Science and Information Technology

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