Re: squares and cubes

[Dusko Pavlovic <dusko@kestrel.edu>]

hi.

i somehow managed to unsubscribe from this list at some point, and almost
missed the thread about coalgebra of reals. i am not sure that i have seen
all the relevant postings, but here are my 2p of thoughts.

if i am not misunderstanding anything, peter freyd's closed interval
coalgebra achieves something that vaughan pratt's and mine do not. we work
with lists and streams and get *irredundant* representations of reals.
however, without redundancy, the algebraic operations on reals are
undecidable. peter's bipointed approach, however, induces a *redundant*
representation. this allows him to extract the algebraic operations
coinductively.

in the talk at the 1998 lisbon workshop on coalgebra (and later in some
seminar talks), i described a redundant coalgebra of conway reals, where
conway's definitions of the field operations were derivable as coalgebra
homomoprhisms. however, the setting seemed too complicated, probably due
to me. peter's definition of the midpoint algebra is considerably simpler,
although the resulting operations are probably still computationally quite
inefficient.

the coalgebra of alternating dyadics, described in the paper with vaughan,
was derived from this 'conway' coalgebra. as explained in sec 3.4, it
provides the best dyadic approximation, while the continued fraction
coalgebra, provides the best rational approximation. presumably, it can be
derived from the 'norton' coalgebra (contorted fractions), mentioned by
peter johnstone. there are as many redundant coalgebras of reals, as there
are corresponding irredundant ones (one for every interval partition), for
fast outputs.

i think that some of the mentioned *mysteries* about the maps between
these various representations boil down to the fact that they are
coalgebra isomorphisms, that do not preserve the derived structure, but
rather *transform* it. eg, the laplace transform is a coalgebra
isomorphism between the coalgebra of analytic functions and a dual
coalgebra of meromorphic functions --- whereby the convolution ring gets
transformed into a multiplicative domain, etcetc.

in any case, when you are done with reals, you may be interested to have a
look int my paper with martin escardo "calculus in coinductive form",
LICS98, or
ftp://ftp.kestrel.edu/pub/papers/pavlovic/LAPL.ps.gz

all the best,
-- dusko

PS along the lines of peter's bipointed construction (and also to check
whether i understood it):

* bipointed sets are the algebras for the functor 2+(-) :
Set -> Set.

* bipointed sets with two _distinct_ points are the free such algebras. so
lets work in the kleisli category *K* for 2+(-). write 2 = {0,1}

* the bifunctor (-) v (-) : *K* --> *K* maps

    * the objects X, Y to X + {m} + Y
    * the arrows X --f-->2+U and Y--g-->2+V to

        h : X+{m}+Y ---> 2 + U+{m}+V

        h(x) = f(x) if f(x) = 0  or f(x) in U
        h(x) = m if f(x) = 1
        h(M) = m
        h(y) = g(y) if g(y) = 1 or g(y) in V
        h(y) = m if g(y) = 0

* now (0,1) is the final coalgebra for the functor X v X on *K*. the
coalgebra structure

        (0,1) ---> 2+(0,1) +{m}+(0,1)

maps 1/2 to m, x<1/2 to 2x and x>1/2 to 2x-1.

* the finality is now easy to prove directly, by displaying the
anamorphism for any given coalgebra X -r-> 2+X+{m}+X. ditto for the
algebraic structure.