...
And... the problem is be no means specific to AXIOM.
foo(a: Integer, b: Integer): Integer ==
if a > 0 then
if a > b then return foo(b,a)
return foo(b-a,a)
return b
Question: Does the program have a bug?
I think that your question is related to the fact that if we don't know
the purpose of a function then we cannot know if there is a bug in the
function.And LP would explain the purpose of the function.
...
My thought is that things are a little more complicated: you should
explain both the purpose of the function and how it is implemented.
In this case, a little change in the "how" creates the problem: Ralf's
function is clearly an implementation of the original (and
inefficient) euclidean algorithm from the "Elements".
Yet it gives wrong results, since he changed deliberately domain and
codomain; for example, if a<0 then simply foo(a,b)=b.
The point is that the proof in the Elements assumes that a,b are PI.
So to have a function without bugs either we need to change the domain to
PIxPI or we need to add two auxiliary variables
c:=abs(a)
d:=abs(b)
and then we work with them.
This is what I meant when talking about documenting maths: it can look
correct from a syntactic point of view and can compile correctly, yet the
result can be wrong. If the underlying math is advanced, the debugging is
complicated by the fact that not a lot of people have the necessary
knowledge to track the function and find the error: if it is not well
documented, probably nobody will do it.
Just my 2 cents worth.
and these are just mine...
Fabio
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