Coinductive semantics
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Date:   (:) 
From:  skaller <skaller@u...> 
Subject:  Re: [Camllist] Coinductive semantics 
On Wed, 20060118 at 13:58 +0100, Hendrik Tews wrote: > skaller <skaller@users.sourceforge.net> writes: > > > Nobody is interested in final coalgebras in Set^op. > > Why not? This is really the key point of misunderstanding > I think. I'm not disputing your claim, I'm asking why not? > Perhaps they should be? > > Coalgebras in Set^op are for all intents and purposes identical > to algebras in Set. If you want to study them, study them as > algebras in Set. You will see nothing new if you look at these > objects as coalgebras in Set^op. That's what duality means. > > Looking at an object through a mirror you see precisely what you > can see looking at the object itself. Perhaps my analysis is naive. But consider a simpler case of products and sums. They're dual concepts, are they not? In Ocaml we have representations of both, each can be used with reasonable utility  there is a degree of symmetry, associated with the duality. It feels good! Contrast to C, which has products, but the union construction isn't a sum. And the many other 'popular' languages with this weakness. Sometimes it seems looking in the mirror is good. It's what we want. We don't want something new! > > Go out, read the papers on > > the CoBirkhoff theorem! > > That's a pretty big ask of someone who isn't a > category theorist isn't it? Most mathematicians > can't understand category theory .. and I'm just > an ordinary programmer :) > > Well, you could try. I guess, that already the introductions > contain enough information for what you are interested in: the > duality of the Birkhoff and the CoBirkhoff theorem. In any case, > if you don't even try, your speculations about the contents of > these papers remain wild guesses. I often do try.. but seemed like a good idea to read Adameck first: http://www.tac.mta.ca/tac/volumes/14/8/1408abs.html Still this is quite heavy going for me. Incidentally .. if you look in Wikipedia for 'coalgebra' you may be a bit disappointed. http://en.wikipedia.org/wiki/Coalgebra  John Skaller <skaller at users dot sf dot net> Felix, successor to C++: http://felix.sf.net