In minecraft, how do I construct a 4-bit comparator?

Solution 1:

While it's possible to build a contraption that operates on discrete boolean arithmetics, built of logic gates, it would be terribly impractical; you have some 5 gates per bit, each gate a good couple blocks, cross-connections, this would be quite awful. Luckily, for arithmetics on numbers up to 4 bits, Minecraft provides a much handier and more graceful tool - analog redstone; use of signal strengths and the comparator.

This is the device that meets the requirements as stated (shown state: "15 > 14", green input greater than blue.):

It looks bulky, but that's only due to need to "massage" the problem as stated (binary input, outputs of "<,=") into what works naturally in Minecraft (analog inputs and outputs of "<,>"). If you take the Minecraft-natural approach all you need is the "core" of 2 comparators:

In this case, you deliver two signals of varying strength, X,Y. The outputs are their difference X-Y and Y-X if the result is positive; no output if zero or less.

What the "extended" version does is first convert 4 input bits into analog signal (through subtracting 1,2,4 or 8 respectively; in Minecraft analog circuitry all addition is done as a + b = 15 - (15 - a - b))*, then compared; then the two outputs are fed downwards into a NOR gate, to produce output if neither is active, meaning the inputs are equal. If you skip the repeaters on output, your output will contain magnitude of the difference too.

^{*The inputs aren't normalized back to the original value 15 - (15 - x), just fed into the comparator in the 15-complement form 15 - x, as this compacts the contraption; instead of reversing the values, I'm just flipping the meaning of the outputs; if x < y, then (15 - x) > (15 - y)}