From: jesse@andrew.cmu.edu (Jesse F. Hughes)
To: "Categories@Mta. Ca" <categories@mta.ca>
Subject: Re: coinduction
Date: 30 Oct 2000 18:09:53 -0500 [thread overview]
Message-ID: <87wveqsz7i.fsf@phiwumbda.dyndns.org> (raw)
In-Reply-To: Dusko Pavlovic's message of "Sun, 29 Oct 2000 19:38:08 -0800"
Dusko Pavlovic <dusko@kestrel.edu> 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_
next reply other threads:[~2000-10-30 23:09 UTC|newest]
Thread overview: 10+ messages / expand[flat|nested] mbox.gz Atom feed top
2000-10-30 23:09 Jesse F. Hughes [this message]
-- strict thread matches above, loose matches on Subject: below --
2000-10-28 19:39 coinduction Mamuka Jibladze
2000-10-27 20:40 coinduction Jesse F. Hughes
2000-10-30 3:38 ` coinduction Dusko Pavlovic
2000-10-20 12:57 coinduction Al Vilcius
2000-10-20 15:35 ` coinduction Hendrik Tews
2000-10-20 22:01 ` coinduction Jesse F. Hughes
2000-10-21 1:19 ` coinduction Charles Wells
2000-10-26 23:49 ` coinduction Dusko Pavlovic
2000-10-30 15:55 ` coinduction Jeremy Gibbons
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