I want to find the minimal CNF and DNF for the following expression: $$(A \implies C) \wedge \neg (B \wedge C \wedge D).$$ I've created a truth table:
\begin{array}{| c | c | c | c | c | c | c |} \hline A & B & C & D & \underbrace{A \implies B}_{E} & \underbrace{\neg (B \wedge C \wedge D)}_{F} & \underbrace{E \wedge F}_{G} \\ \hline 0 & 0 & 0 & 0 & 1 & 1 & 1 \\ \hline 1 & 0 & 0 & 0 & 0 & 1 & 0 \\ \hline 0 & 1 & 0 & 0 & 1 & 1 & 1 \\ \hline 1 & 1 & 0 & 0 & 0 & 1 & 0 \\ \hline 0 & 0 & 1 & 0 & 1 & 1 & 1 \\ \hline 1 & 0 & 1 & 0 & 1 & 1 & 1 \\ \hline 0 & 1 & 1 & 0 & 1 & 1 & 1 \\ \hline 1 & 1 & 1 & 0 & 1 & 1 & 1 \\ \hline 0 & 0 & 0 & 1 & 1 & 1 & 1 \\ \hline 1 & 0 & 0 & 1 & 0 & 1 & 0 \\ \hline 0 & 1 & 0 & 1 & 1 & 1 & 1 \\ \hline 1 & 1 & 0 & 1 & 0 & 1 & 0 \\ \hline 0 & 0 & 1 & 1 & 1 & 1 & 1 \\ \hline 1 & 0 & 1 & 1 & 1 & 1 & 1 \\ \hline 0 & 1 & 1 & 1 & 1 & 0 & 0 \\ \hline 1 & 1 & 1 & 1 & 1 & 0 & 0 \\ \hline \end{array}
DNF: $ G = (\neg A \wedge \neg B \wedge \neg C \wedge \neg D) \vee (\neg A \wedge B \wedge \neg C \wedge \neg D) \vee (\neg A \wedge \neg B \wedge C \wedge \neg D) \vee (\neg A \wedge B \wedge \neg C \wedge D) \vee (\neg A \wedge B \wedge C \wedge \neg D) \vee ( A \wedge B \wedge C \wedge \neg D) \vee ( A \wedge \neg B \wedge \neg C \wedge D) \vee (\neg A \wedge B \wedge \neg C \wedge D) \vee (\neg A \wedge \neg B \wedge C \wedge \neg D) \vee ( A \wedge B \wedge C \wedge D) $
To shorten that, I will simplify the expression by deleting redundant expressions:
$
G=(\neg A \wedge \neg B \wedge \neg C \wedge \neg D)
\vee (\neg A \wedge B \wedge \neg C \wedge \neg D)
\vee (\neg A \wedge \neg B \wedge C \wedge \neg D)
\vee (\neg A \wedge B \wedge \neg C \wedge D)
\vee (\neg A \wedge B \wedge C \wedge \neg D)
\vee ( A \wedge B \wedge C \wedge \neg D)
\vee ( A \wedge \neg B \wedge \neg C \wedge D)
\vee ( A \wedge B \wedge C \wedge D)
$
$\vdots$
minimal DNF: $\dots$
Same procedure for CNF
CNF:
$G_2=(\neg A \vee B \vee C \vee D) \wedge (\neg A \vee \neg B \vee C \vee D) \wedge \dots \wedge (\neg A \vee \neg B \vee \neg C \vee \neg D)$
$\vdots$
How do I find the minimal CNF and CNF more easily? I could simplify those expressions like I showed above, but this is really time consuming. Is there a way to do this more efficient? If so, could you please explain in detail, how you solved the task?