Re: [stack] A small example of a (possibly concatenative) language for primitive recursive functions of one variable.

On Wed, 2 May 2007 06:34 am, William Tanksley, Jr wrote:
> I'm glad you're here. This is providing some definitional rigor I
> certainly needed. Thank you for putting the thought into this.
You are welcome. :) 
I also find this discussion very useful for me.

> I would need to understand more about "iteration".
I hope the following explanation will be useful:
let f ba a function from natural numbers into natural numbers. Define the 
function g as follows:
g (0 ) = 0
g( 1 )   = f(0)          # one f
g( 2 )   =f (f(0))      # two f
g (3 )   = f (f f(0)))  # three f
...
g (n)    = f ( ....f(0)...)   # n times f
...
That g is the iteration of f.

Examples (here % means \lambda):
1) if f = %x. 0 then g = %x. 0
2) if f = %x. 1 then g = %x. sgn(x), i.e. g(x)=(if x=0 then 0 else 1)
3) if f = %x. (x+1) then g  = %x. x
    g(0) =0, g(1) = f(0)=0+1=1, g(2)=f(f(0))=1+1=2 ...

Michael Nedzelsky



 
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