Re: coinduction

[jesse@andrew.cmu.edu (Jesse F. Hughes)]

"Al Vilcius" <al.r@VILCIUS.com> writes:

> can anyone explain coinduction?
> in what sense it dual to induction?
> does it relate to the basic picture for a NNO?
> is there a (meaningful) concept of co-recursion?
> what would be the appropriate internal language?
> how is it used to prove theorems?

The principle of induction says that initial algebras have no
non-trivial subalgebras.  Consider N, the initial successor algebra.
A subobject S of N is a subalgebra just in case there is a structure
map
s: S+1 -> S
such that the inclusion S -> N is a homomorphism.  This is just the
usual condition that 0 is in S and if n is in S, then so is its
successor.  Induction says that any such subalgebra exhausts the set
N.

Coinduction says, dually, that the final coalgebra has no non-trivial
quotients.  (To see that this is "really" a dual principle, we would
interpret "subalgebra" above as a regular subobject in the category of
algebras.)  However, the principle is usually stated in terms of
relations on a coalgebra, rather than quotients.  In these terms, the
principle of coinduction says: Any relation on <F,f> in the category
of coalgebras is a subrelation of equality.

Note: I stated the principles of induction/coinduction above for the
initial algebra/final coalgebra, resp.  However, just as induction
holds for any quotient of the initial algebra, coinduction holds for
any (regular) subcoalgebra of the final coalgebra.

I'm not sure what you're looking for in relation to NNOs.  An NNO is
the same as an initial algebra for the successor functor (assuming
that our category has coproducts).  The structure map N + 1 -> N is
given by [s,0], where s is the successor.  The final coalgebra for
successor is easily described (say, in Set):  It is the NNO with a
point adjoined for "infinity".  The structure map pred:F -> F + 1 fixes
infinity, takes n+1 to n and takes 0 to * (the element in 1 -- a kind
of "error condition").

As far as co-recursion goes, yes, there is a meaningful principle.
Namely, given any coalgebra <A,a>, there is a unique coalgebra
homomorphism from <A,a> to <F,f>.  Thus, for any set S together with a
function t:S -> S+1, there is a unique map h:S -> N u {infinity} such
that 
(1) if t(s) = *, then h(s) = 0
(2) else, h(t(s)) = h(s) - 1 (where infinity - 1 = infinity)
In other words, h here takes an element s to the least n such that
t^n(s) = *, if defined.  If, for all n, t^n(s) != *, then h(s) =
infinity.

I'll defer on the question of what the appropriate internal language
is.

As far as how coinduction is used to prove theorems: To show two
elements of the final coalgebra are equal, it suffices to find a
coalgebraic relation relating the elements.

Hendrik's suggestion (that you find Jacobs and Rutten's paper) is a
good one, but I hope these comments help until then.

-- 
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_