Parametrization - S-M-N Theorem

[Peter Pedant]

if you have a partial computable function

f(x,y) = y if x is even
and is undefined otherwise

that is f is a partial computable function

then the s-m-n theorem guarantees there is always a totally computable one placed function
gx of (y) which is total - - x is an computable index

what happens to the undefined cases ? in f where do they go to in g ?

suppose f is undefined everywhere - what does g look like ?

( yeah reading Cutland ) thanks
yeah a serious science q for a change

[jim360]

I'm nervous about answering something like this when I don't actually know, but as far as I can see the result only applies to the computable cases of f(x,y). Therefore, if it's undefined (or uncomputable) everywhere, no such g exists. That's my intuition, anyway.

[Peter Pedant]

Thx J
I might write to Cutland -
if a BA in Maff (er you)- soon to be PhD says good q
then I can take it forward

and yeah - this is an abnormal question on Science as I havent asked why Pi isnt fwee ( much easier to calculate) or was Darwin a roman cafflic ( no he wasnt)

[IJKLM]

Echoing Jim here, as I never studied this part of number theory, but in the first line you define the function,

f(x,y) = y if x is even and is undefined otherwise

You then ask a question that appears to contradict the original definition:

"suppose f is undefined everywhere - what does g look like ? "

So the question is (in mathematical terms) meaningless. Something like "How many volts does an apple weigh?"

Which, is, I think what Jim said, but expressed in different terms.

Not sure this is really one for Answerbank - maybe for a specialist number theory community, like this one:

https://math.stackexchange.com/questions/1679179/problems-understanding-proof-of-s-m-n-theorem-using-church-turing-thesis

[Peter Pedant]

oh yes no
thanks for the cont
one f but two functions
f being even and otherwise undefined
and then suppose - another f ( I should have said h ) where it is undefined everywhere

The stack exchange question is good
thanks - the problem ( remember there could be more than one problem in this question ) being that I understood the stacker didnt.

f( x,y) goes to Φg(x)(y) ( smn theorem )
by writing a program where you whack fixed x into register no 1 of a tm doing f and move all the other registers forward one.
The godel no of this program is g(x) and so it is effectively computable

IN fact on thinking about it ( what happens to the undefined cases of f(x,y) so with x =3 - suppose it is print y if y=3n for some n and otherwise undefined

then Φg(x)(y) the domain is 1,2,3,4 and the range is 3,6,9,12
and hey presto Φg(x)(y) is a total computable function

[ compatible with a later result - an r.e. set is the range of a total computable unary increasing function]

[Peter Pedant]

// So the question is (in mathematical terms) meaningless. Something like "How many volts does an apple weigh?" //

erm - well I have never found the 'answer'
"your question isnt a question so I dont have to answer it"
very useful ( to the questioner)