Past exam of the year 1 of the computer science course of the University of Oxford / 2016 / Digital systems, linear algebra, introduction to formal proof (resit) / 2 / b / Solution

Encode $(q_1{q}_0)$ by $A=00$, $B=01$, $C=11$, $I=10$, with reset to $A$. Two positive-edge-triggered D flip-flops store the state. Minimizing the Boolean functions from the complete table givesEquivalently, the minimal sum of products form of $D_1$ is ${\bar{x}}_0{q}_0{x}_1+{\bar{x}}_0{q}_0{q}_1$. The displayed factorization saves duplicated logic. The equations use only AND gates, OR gates and NOT gates; connect $D_j$ to the corresponding flip-flop data input and use one common clock.

For example, $C$ with input $00$ gives $(D_1{D}_0)=10$, entering the pulse state. In $I$, either $x_0=1$ gives $01$ or $x_0=0$ gives $00$, independently of $x_1$; the pulse cannot persist. The four states are distinguishable by their current output or future response to inputs $00$ and $10,00$, so two state bits are also necessary.