Implementing $AB\bar{C}D+A\bar{D}+\bar{A}D$ using only AND & XOR gates

Question

Question:

Implement $F(A,B,C,D)=AB\bar{C}D+A\bar{D}+\bar{A}D$ using only AND & XOR gates

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My try:

I know that $A\bar{D}+\bar{A}D$ functions like a XOR gate which takes $A$ and $D$ as input. But then I don't know what to do which the part $+AB\bar{C}D$. I mean, with this method, I'm stuck on designing the OR gate.
I also tried using the Karnaugh map to simplify the equation but the result didn't help me either.
It seems that there's no way to get rid of the OR gate...

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Any idea on this implementation?

Thanks in advance.

Answer 1

With just $XOR$ and $AND$ gates it cannot be done: By induction you can prove that any function composed of $XOR$'s and $AND$'s will be true in at most half of the cases, and the function you are supposed to capture is true in $9$ out of $16$ cases.

If you can use a $1$ as an input as well, then you can do it, since to capture $P'$ you can use $P \ XOR \ 1$, and with the conjunction and the negation you have an expressively complete system.

For your particular function:

$$ABC'D + AD' + A'D =$$

$$ABC'D+ (A \ XOR \ D)=$$

$$((ABC'D)'(A \ XOR \ D)')'=$$

$$(((AB(C \ XOR \ 1)D) \ XOR \ 1)((A \ XOR \ D) \ XOR \ 1)) \ XOR \ 1$$