Priority Encoder

We can't always assume that only one input will be 1.

Priority encoder: assumes that at least one input will be 1.

Which input to encode?  Use priority scheme

Larger subscripts could have higher priority

Smaller subscripts could have higher priority

Assume larger subscripts have priority

Boolean expressions no longer necessarily valid

Suppose x4 and x3 are both equal to 1

Then z2z1z0 = 111, but the result should be 100, since 4 has higher priority

What does it mean that 4 has the highest priority?

All of the higher inputs must be 0, and the lower inputs don't matter:

\x7\x6\x5x4

Negate all literals with higher priority, and leave out lower ones

Replace each term in original expressions

z2 = x7 + \x7x6 + \x7\x6x5 + \x7\x6\x5x4

z1 = x7 + \x7x6 + \x7\x6\x5\x4x3 +  \x7\x6\x5\x4\x3x2

z0 = x7 +  \x7\x6x5 +  \x7\x6\x5\x4x3 +  \x7\x6\x5\x4\x3\x2x1

This can be further simplified.

If x7 is the highest priority 1, then it doesn't matter if the other terms are 0 or not.

z2 = x7 + x6 + \x6x5 + \x6\x5x4

Similarly, if x6 is the highest priority 1, then \x6 is not necessary in the other 2 terms.

z2 = x7 + x6 + x5 + \x5x4

We can also eliminate \x5 in the last term.

z2 = x7 + x6 + x5 + x4

(Notice that this expression gives back the original form.)

In the expression for z1, however, we need to keep \x5 and \x4:

z1 = x7 + x6 + \x5\x4x3 + \x5\x4x2

Likewise, for z0, we need to keep \x6, \x4, and \x2:

z0 = x7 + \x6x5 +  \x6\x4x3 +  \x6\x4\x2x1

In general, we need to keep the negation of any literal which doesn't appear

as a higher-priority value.

What if all inputs are 0?

We can encode the output as 000, and x0 will have the highest priority by default.