Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Iterated nand
#1
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
↑ 1 = x
↑ 2 = x ↑ x = ¬x
↑ 3 = ¬x = ¬(x & ¬x) = x v ¬x
↑ 4 =  (x v ¬x) = ¬(x & (x v ¬x)) = x & ¬x

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

↑ 6 = x & ¬x

↑ 7 = x v ¬x

... etc.
So
↑ 2k = 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.)
Xorter Unizo
Reply
#2
(03/12/2017, 03:08 PM)Xorter Wrote: 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
↑ 1 = x
↑ 2 = x ↑ x = ¬x
↑ 3 = ¬x = ¬(x & ¬x) = x v ¬x
↑ 4 = 
 (x v ¬x) = ¬(x & (x v ¬x)
) = x & ¬x

↑ 5 = x  (x & ¬x) = ¬(x & (x & ¬x)
) = x v ¬x
↑ 6 = x & ¬x
↑ 7 = x v ¬x
... etc.
So
↑ 2k = 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.)

---

I like the basic idea to do dynamics in set theory , Logic and the alike.

However i see many issues.

First you write alot " x and not x " and "x or not x" and they are ( in boolean ) either trivial or paradoxical !
Like " this sentense is false ".
or " this is true and false " , " this is true or false.

Secondly

suppose we let times -1 mean not.


Than iterations of not give the unit circle in the complex plane.

but what does it mean " i " ? What does it mean to half-iterate NOT ?

0r the pi th iteration of x OR y ???

you only have a few things like and or not true etc.
but you want continue iterations ??

or if you introduce new things like " i " above you need to Well define it !!

so as of now im very skeptical for continue iterations.

as for integer iterates that might work.

i think one then needs to associate it with groups or rings.

maybe modular arithmetic too.

Regards

tommy1729
the master
Reply
#3
(03/23/2017, 01:27 PM)tommy1729 Wrote: ---

I like the basic idea to do dynamics in set theory , Logic and the alike.

However i see many issues.

First you write alot " x and not x " and "x or not x" and they are ( in boolean ) either trivial or paradoxical !
Like " this sentense is false ".
or " this is true and false " , " this is true or false.

Secondly

suppose we let times -1 mean not.


Than iterations of not give the unit circle in the complex plane.

but what does it mean " i " ? What does it mean to half-iterate NOT ?

0r the pi th iteration of x OR y ???

you only have a few things like and or not true etc.
but you want continue iterations ??

or if you introduce new things like " i " above you need to Well define it !!

so as of now im very skeptical for continue iterations.

as for integer iterates that might work.

i think one then needs to associate it with groups or rings.

maybe modular arithmetic too.

Regards

tommy1729
the master

Very good questions.

Well, to be honest, (x or not x), (x and not x) are neither trivial nor paradoxical, just think of fuzzy logic with x=0.5 (half truth or sg like this), then each expression results 0.5, right?

What is the half iterative of negation? The imaginary negation, about what we know that:
¬^.5 o ¬^.5 = ¬, where o is composition.
It is as mysterious as the imaginary unit. I think.
But it solves your so-called paradox, that: "This statement is false.". Its value neither true nor false, it is (¬^.5)(true), so so-called this is the imaginary truth ... solving the all the paradoxes. Smile

What is the pi-th and the i-th iterate of the logical ops? This is my question.
Xorter Unizo
Reply


Possibly Related Threads...
Thread Author Replies Views Last Post
  Iterated compositions Xorter 0 498 08/20/2016, 01:19 PM
Last Post: Xorter
  Grzegorczyk hierarchy vs Iterated differential equations? MphLee 0 1,075 01/03/2015, 11:02 PM
Last Post: MphLee
  Iterated polynomials JmsNxn 4 4,150 12/16/2010, 09:00 PM
Last Post: JmsNxn
  The fractal nature of iterated ln(x) [Bandwidth warning: lots of images!] jaydfox 16 10,091 09/09/2007, 01:21 AM
Last Post: jaydfox
  Bell formula for iterated exponentiation bo198214 1 2,454 08/26/2007, 12:27 PM
Last Post: Gottfried
  computing the iterated exp(x)-1 Daniel 14 10,392 08/17/2007, 08:44 PM
Last Post: andydude



Users browsing this thread: 1 Guest(s)