Subject: categories: Re: coinduction

From: [email-masked] (Jesse F. Hughes)
Date: 30 Oct 2000 18:09:53 -0500

Dusko Pavlovic <[email-masked]> writes:

> 
> > I have always considered the statement regarding existence of a
> > unique function f to be the principle of recursion.
> 
> the recursion schema, as defined by godel in 1930, is a bit
> stronger: g is allowed to depend on n, not only on f(n).

Yes, I overlooked that point. But, is that principle any stronger than
the principle you called induction?  I think it isn't, given products
in our category.  

Let N be the initial successor algebra.  Let c be in X and 
g:N x X -> X. To define a function f such that
f(0) = c
f(n+1) = g(n,f(n)),
I just have to cook up the appropriate structure map for N x X.  I
think that
<s o p_1,g> + <0,c>: (N x X) + 1 -> N x X
does it (p_1 is the projection map).  Let <k,f>: N -> N x X be the
unique function guaranteed by initiality.  One can show that f
satisfies the equations above.  (Along the way, one shows that k is
the identity.)

(To add other parameters, one needs exponentials.  For instance,
given
h:A -> X
g:A x N x X -> X
to show that there is a unique f such that
f(a,0) = h(a)
f(a,n+1) = g(a,n,f(a,n))
requires a structure map for (X x N)^A, if I'm not mistaken.)

This shows that Godel's principle of recursion is equivalent to the
statement "N is an initial algebra".  I.e., it is equivalent to the
statement that there is a unique f such that
f(0) = c
f(n+1) = g(f(n)).

> > The principle of
> > induction is, in my mind, the weaker principle that says that if a
> > property holds of 0 and is closed under successor, then it holds of N.
> 
> in a topos, this principle is equivalent with the above schema. and
> the schema itself, of course, just means that [0,s] : 1+N --> N is
> initial: any algebra [c,g] : 1+X --> X induces a unique homomorphism
> f : N-->X. using the exponents, we do recursion.

I am confused by this statement.  My principle of induction for
successor algebras [0_X,s_X]:1+X -> X is this: For all subsets P of X,
if 0_X is in P and P is closed under s_X, then P = X.  (Categorically,
an algebra satisfies the principle of induction if it contains no
proper subalgebras.)

Consider the successor algebra X={x}, with the evident structure map 
[0_X,s_X]:1+X -> X.  
Surely, for every subset P of X, if 0_X is in P and P is closed under
s_X, then P = X.  In other words, the algebra X satisfies the principle
of induction in the sense that I gave.  This seems to contradict the
equivalence you mention.  What am I overlooking?

> as i said last time, induction and recursion have been in heavy use
> for so long, that they really don't leave much space for redefining,
> even for the sake of symmetry with coinduction and corecursion.

I quite agree, but I didn't redefine induction, as far as I am
concerned.  I learned that induction was the proof principle as I
stated it and that recursion is the principle regarding the definition
of functions many years ago.  I think that our disagreement in
definitions is a disagreement that one will find elsewhere in the
literature.  I also suppose that the term coinduction (as I use it)
arose from the definition of induction that I offered.

Are others unfamiliar with the distinction between induction and
recursion as I drew it?  I assumed it was a more or less conventional
distinction.  If my use is really idiosyncratic, then I'd be happy to
drop it.

-- 
Jesse Hughes
"Such behaviour is exclusively confined to functions invented by
mathematicians for the sake of causing trouble."
 -Albert Eagle's _A Practical Treatise on Fourier's Theorem_