03/12/2017, 03:08 PM

I am interested in iterates (and uniterates) of logical operators, like nand, xor ... etc. .

Let's investigate the nand op:

x↑y = ¬(x&y)

Iterating x O y means this: x O x O ... O x (y-times)

Thus

x ↑ x ↑ ... ↑ x := x ↑↑ y

x ↑↑ 1 = x

x ↑↑ 2 = x ↑ x = ¬x

x ↑↑ 3 = x ↑ ¬x = ¬(x & ¬x) = x v ¬x

x ↑↑ 4 = x ↑ (x v ¬x) = ¬(x & (x v ¬x)) = x & ¬x

x ↑↑ 5 = x ↑ (x & ¬x) = ¬(x & (x & ¬x)) = x v ¬x

x ↑↑ 6 = x & ¬x

x ↑↑ 7 = x v ¬x

... etc.

So

x ↑↑ 2k = x & ¬x

x ↑↑ 2k-1 = x v ¬x

where k is bigger integer than 2

My question: Can k be any real or complex number?

(In my view to do this, we should know what "between" & and v is.)

Let's investigate the nand op:

x↑y = ¬(x&y)

Iterating x O y means this: x O x O ... O x (y-times)

Thus

x ↑ x ↑ ... ↑ x := x ↑↑ y

x ↑↑ 1 = x

x ↑↑ 2 = x ↑ x = ¬x

x ↑↑ 3 = x ↑ ¬x = ¬(x & ¬x) = x v ¬x

x ↑↑ 4 = x ↑ (x v ¬x) = ¬(x & (x v ¬x)) = x & ¬x

x ↑↑ 5 = x ↑ (x & ¬x) = ¬(x & (x & ¬x)) = x v ¬x

x ↑↑ 6 = x & ¬x

x ↑↑ 7 = x v ¬x

... etc.

So

x ↑↑ 2k = x & ¬x

x ↑↑ 2k-1 = x v ¬x

where k is bigger integer than 2

My question: Can k be any real or complex number?

(In my view to do this, we should know what "between" & and v is.)