Universal Logic Gates (2015)

That makes the set {AND, XOR} a cover, yes, but since it is larger than {NAND} (or {NOR}) it is not a minimum cover.

XOR and AND is a falsehood-preserving set of gates. Whatever acyclic circuit you make out of them will always produce 0 as its output if all its inputs are 0.

It's a minimal universal set if you have access to constants (specifically, constant 1), but the other sets you mention do not require access to constants for universality. AND-NOT (BIC, set subtraction, logical abjunction, &^ in Golang) is a single binary gate that is minimal and universal in the same way (that is, it's falsehood-preserving, but universal if you have a constant 1), and it has the distinction of being the only universal gate available as a bitwise operation in many CPU instruction sets.

Here's a minimal construction of all 16 primitive binary combinations using XOR and AND:

    r[ 0] = x (truth table 0011, cost 0) = x
    r[ 1] = y (truth table 0101, cost 0) = y
    r[ 2] = 0 (truth table 0000, cost 1) = 0
    r[ 3] = -1 (truth table 1111, cost 1) = -1
    r[ 4] = r[0] ^ r[1] (truth table 0110, cost 1) = x ^ y
    r[ 5] = r[0] ^ r[3] (truth table 1100, cost 2) = x ^ -1
    r[ 6] = r[1] ^ r[3] (truth table 1010, cost 2) = y ^ -1
    r[ 7] = r[0] & r[1] (truth table 0001, cost 1) = x & y
    r[ 8] = r[0] & r[4] (truth table 0010, cost 2) = x & (x ^ y)
    r[ 9] = r[1] & r[4] (truth table 0100, cost 2) = y & (x ^ y)
    r[10] = r[5] & r[6] (truth table 1000, cost 4) = (x ^ -1) & (y ^ -1)
    r[11] = r[0] ^ r[6] (truth table 1001, cost 3) = x ^ (y ^ -1)
    r[12] = r[0] ^ r[9] (truth table 0111, cost 3) = x ^ (y & (x ^ y))
    r[13] = r[3] ^ r[9] (truth table 1011, cost 4) = -1 ^ (y & (x ^ y))
    r[14] = r[3] ^ r[8] (truth table 1101, cost 4) = -1 ^ (x & (x ^ y))
    r[15] = r[3] ^ r[7] (truth table 1110, cost 3) = -1 ^ (x & y)

And here it is with just abjunction:

    r[ 0] = x (truth table 0011, cost 0) = x
    r[ 1] = y (truth table 0101, cost 0) = y
    r[ 2] = 0 (truth table 0000, cost 1) = 0
    r[ 3] = -1 (truth table 1111, cost 1) = -1
    r[ 4] = r[0] &^ r[1] (truth table 0010, cost 1) = x &^ y
    r[ 5] = r[1] &^ r[0] (truth table 0100, cost 1) = y &^ x
    r[ 6] = r[3] &^ r[0] (truth table 1100, cost 2) = -1 &^ x
    r[ 7] = r[3] &^ r[1] (truth table 1010, cost 2) = -1 &^ y
    r[ 8] = r[0] &^ r[4] (truth table 0001, cost 2) = x &^ (x &^ y)
    r[ 9] = r[3] &^ r[4] (truth table 1101, cost 3) = -1 &^ (x &^ y)
    r[10] = r[3] &^ r[5] (truth table 1011, cost 3) = -1 &^ (y &^ x)
    r[11] = r[6] &^ r[1] (truth table 1000, cost 3) = (-1 &^ x) &^ y
    r[12] = r[3] &^ r[8] (truth table 1110, cost 4) = -1 &^ (x &^ (x &^ y))
    r[13] = r[3] &^ r[11] (truth table 0111, cost 4) = -1 &^ ((-1 &^ x) &^ y)
    r[14] = r[9] &^ r[5] (truth table 1001, cost 5) = (-1 &^ (x &^ y)) &^ (y &^ x)
    r[15] = r[3] &^ r[14] (truth table 0110, cost 6) = -1 &^ ((-1 &^ (x &^ y)) &^ (y &^ x))

You might intuitively suppose that &^ is "more efficient" or "more expressive" than NAND or NOR, since x &^ y gives you different results from y &^ x, so you would think that with the same number of gates, you would have more usefully different possibilities, and so some circuits would require fewer gates using abjunction than using NAND. However, I haven't found a natural family of circuits for which that is the case. Some circuits are simpler with abjunction (abjunction itself, for example, is only a single gate with abjunction), while other circuits are simpler with NAND, but neither one seems to have the kind of clear 2ⁿ advantage you'd expect from this asymmetry. Here's NAND's corresponding table:

    r[ 0] = x (truth table 0011, cost 0) = x
    r[ 1] = y (truth table 0101, cost 0) = y
    r[ 2] = r[0] &̄ r[0] (truth table 1100, cost 1) = x &̄ x
    r[ 3] = r[0] &̄ r[1] (truth table 1110, cost 1) = x &̄ y
    r[ 4] = r[1] &̄ r[1] (truth table 1010, cost 1) = y &̄ y
    r[ 5] = r[0] &̄ r[2] (truth table 1111, cost 2) = x &̄ (x &̄ x)
    r[ 6] = r[0] &̄ r[3] (truth table 1101, cost 2) = x &̄ (x &̄ y)
    r[ 7] = r[1] &̄ r[2] (truth table 1011, cost 2) = y &̄ (x &̄ x)
    r[ 8] = r[2] &̄ r[4] (truth table 0111, cost 3) = (x &̄ x) &̄ (y &̄ y)
    r[ 9] = r[3] &̄ r[3] (truth table 0001, cost 2) = a &̄ a where a = x &̄ y
    r[10] = r[3] &̄ r[8] (truth table 1001, cost 5) = (x &̄ y) &̄ ((x &̄ x) &̄ (y &̄ y))
    r[11] = r[5] &̄ r[5] (truth table 0000, cost 3) = a &̄ a where a = x &̄ b and b = x &̄ x
    r[12] = r[6] &̄ r[6] (truth table 0010, cost 3) = a &̄ a where a = x &̄ b and b = x &̄ y
    r[13] = r[7] &̄ r[7] (truth table 0100, cost 3) = a &̄ a where a = y &̄ b and b = x &̄ x
    r[14] = r[8] &̄ r[8] (truth table 1000, cost 4) = a &̄ a where a = c &̄ b and b = y &̄ y and c = x &̄ x
    r[15] = r[6] &̄ r[7] (truth table 0110, cost 5) = (x &̄ (x &̄ y)) &̄ (y &̄ (x &̄ x))

The searching is done with http://canonical.org/~kragen/sw/dev3/abjsearch.py.

Although I have a soft spot for abjunction (as discussed in my wall-of-text comment deeper in the thread), my favorite universal gate is the three-input conditional: x if y else z, or y ? x : z in C or JS syntax, the element from which BDDs are built. (Like abjunction, it's falsehood-preserving, and it's also truth-preserving, so unlike NAND and NOR, it is only "universal" if you have access to constants TRUE and FALSE.) As Darius Bacon has pointed out, one way to generalize it to continuous variables, or variables over arbitrary fields, is as the lerp operation commonly provided by GPUs: yx + (1-y)z, or, without using constants, x + y(z-x). In GF(2), subtraction and addition are both XOR, so that ends up being x ^ y & (z ^ x), which is a correct definition of the operation for bits.

The universal reversible gates like the Fredkin gate and the Toffoli gate are also really interesting and potentially important.

Designing circuits with separate combinational gates and registers is fairly easy, but it treats the temporal separation of cause and effect as a sort of defect in the universe, and consequently we have to be cautious about glitches (when we don't clock) and metastable states resulting from inadequate setup and hold times (when we do clock). Suppose we continue to discretize time with clocks just as in the current RTL paradigm; can we find a finite state machine that is minimal and universal in some NAND-like sense, and also admits a reasonably tractable design methodology? Would it have two binary inputs, one binary output, and one bit of state? Is it just a J-K flip-flop?

The question about a minimal set of primitives arises in many domains, not only in Boolean functions, for example in structured programming or in a minimal computer instruction set.

In all such domains it is possible to find a set composed of only one primitive and some people think that that is the minimal set because it has a minimum number of primitives.

Nevertheless, other people do not agree that the minimum number of primitives is the correct goal for minimal complexity.

In all such cases the single primitive is more complex than the primitives belonging to an equivalent set of primitives with more members and it is equivalent to a composition of all the simpler primitives, done in such a way that it is possible to obtain any of the simpler primitives by composing the single primitive with itself.

In my opinion, it is nice to know this nice trick about reducing a set of independent primitives to only one, but this almost never has any practical application.

When reasoning about that domain, using the single primitive does not make it any easier, because it glues together independent concepts which are easier to handle separate and not in a forced combination. For example, when thinking about Boolean function, it is easier to think about AND-OR stages instead of NAND-NAND, or about OR-AND stages instead of NOR-NOR. So the knowledge about being able to use just NAND does not help you.

If we are talking about implementation, there also the single primitive does not help you, because you must actually implement it from more simpler primitives.

For example the main blocks for TTL logic are minimum circuits with diodes (AND), parallel transistors (OR) and transistor inverters (NOT), while the main blocks for CMOS logic are parallel transistors, series transistors, transistor inverters and transmission gates.

You do not design just a NAND and then make the other cells from it, but you must design at least a NAND, a NOR, a XOR and a multiplexer, then make other functions from these blocks. Knowing that you might use just a NAND does not help you, because any implementation that uses only NAND is less efficient than those based on multiple simpler primitives.

An example from another domain, you can use just a WHILE-DO to implement both an IF-THEN and a REPEAT-UNTIL, so you can claim that it is enough to have that primitive.

Nevertheless, if you look at the implementation, both IF-THEN and REPEAT-UNTIL are made with a single conditional jump. On the other hand, you must use 2 conditional jumps for a WHILE-DO, there is now way to implement it otherwise than as a composition of IF-THEN with REPEAT-UNTIL. If you implement the simpler primitives with WHILE-DO, you obtain a more inefficient implementation.

In conclusion, while it is interesting to know that you can reduce most or all sets of primitives to only one complex primitive, both in theory and in practice the sets of multiple simple primitives are more useful.

This sounds like the limit of many-valued logic:

https://news.ycombinator.com/item?id=20180599