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Sat Feb 18 10:07:32 EST 2012

## Reader monad and order

I'm trying to capture the idea of "context dependent state
transformation". Something doesn't quite add up here...
There seem to be a couple of ways to formulate this. Let c be
context, s be state and R be a Reader monad, which is a
context-dependent computation: R a b == a -> b
( Concretely: c is "world state" of an animation, i.e. read-only or
"stiff" background info like current time, and s is "object state" of
an animation, meaning the animation's dynamics = state of its
equations of motion. )
A) c -> s -> s or R c (s -> s)
B) s -> c -> s or s -> R c s
C) (s,c) -> s
Where R is a reader monad.
The computations I want to fit in a framework have initial s and c
available at the same time, so C) is the type that corresponds best to
reality. Why is there an ambiguity when trying to write this as a
Reader monad? Which of A) or B) is the correct/appropirate one? Is
the Reader monad the appropriate model? Is this a Co-Monad? (EDIT:
The answer seems to be Yes[1]).
Something I ran into before while trying to capture state machines /
state space models is that the following correspondences are not
really bijective. How to make that "really" precise?
(a,b) -> c <=> a -> b -> c
(a,b) -> c <=> b -> a -> c
EDIT: Above isn't expressed well. According to [1] these really are
the same. I just changed the animation types from
m (s -> s)
to
s -> m s
It seems that while this doesn't make a difference for the Reader
monad, for other monads it does. I.e. I could use a state monad to
thread the RNG state without trouble.
[1] http://comonad.com/reader/2008/kan-extensions-ii/

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