Re: Bit operations for beginners

[mrborisguy]

I just have to be difficult, it's the way I am:

There are 4 bit operations, and, or, xor and inverse

It's been shown already in this thread, but not said yet that there are actually more than 4 bit operations. 'and', 'or', 'xor', and 'inverse' are the common ones used in Perl (and most of the programming languages that have crossed my path) and in fact any bit operator can be created using those - they're a complete set or something (actually, still complete without 'xor'); but technically they aren't the only four bit operators.

(Good work, by the way!) ++

    -Bryan

[ikegami]

For two inputs, there are 16 operations. Most of them are rarely useful, which is why we don't have Perl operators for them.

By the way, everything can be implemented solely with nand:

 p | q | p nand q
---+---+----------
 0 | 0 |    1
 0 | 1 |    1
 1 | 0 |    1
 1 | 1 |    0

not p   === p nand p
p and q === (p nand q) nand (p nand q)
p or q  === (p nand p) nand (q nand q)
p xor q === (p nand (q nand q)) nand ((p nand p) nand q)

Other operatations could also be used instead of nand. nor, for example:

 p | q | p nor q
---+---+---------
 0 | 0 |    1
 0 | 1 |    0
 1 | 0 |    0
 1 | 1 |    0

~(p nor q)


not p   === p nor p
p and q === (p nor p) nor (q nor q)
p or q  === (p nor q) nor (p nor q)
p xor q === [something very long]

[BrowserUk]

p xor q === [something very long]

Hell's teeth! You weren't kidding.

I remember xor as (p and not q) or (q and not p) and it's been something like 30 years since I played with nand and nor expansions, so I expanded that:

xor === (p and not q) or (q and not p) 

expand the 'not's
    === (p and (q nor q)) or (q and (p nor p))

expand the 'and's
    === ((p nor p) nor ((q nor q) nor (q nor q))) 
     or ((q nor q) nor ((p nor p) nor (p nor p)))

expand the 'or's
    === (((p nor p) nor ((q nor q) nor (q nor q))) 
     nor ((q nor q) nor ((p nor p) nor (p nor p))))
    nor (((q nor q) nor ((p nor p) nor (p nor p))) 
     nor ((p nor p) nor ((q nor q) nor (q nor q))))

Then it struck me that ((x nor x) nor (x nor x)) is x, which reduces it to

    === (((p nor p) nor q) nor ((q nor q) nor p)) 
    nor (((q nor q) nor p) nor ((p nor p) nor q))

It's certainly more digestable (just :), but are there any other reductions in there?

[kaif]

Yeah, you might enjoy:

(p nor q) nor ((p nor p) nor (q nor q))

Remember that nor returns true only if both arguments are false. So, we want to return true if the inputs aren't both false or both true. The former is simply p nor q and the latter is (not p) nor (not q) and we use the expansion of not.