On Thu, Nov 13, 2008 at 3:15 PM, Ralf Hemmecke wrote: >... it would make me happier if you could at least say whether you > think my view is mathematically valid or not. >
Yes, I do think that your view is mathematically valid. > I don't say that one *needs* universal algebra, but I find that > mathematical concept rather illuminating to understand the relation > between domains and mathematics. UA (or rather multi-sorted algebras > with a distinguished carrier set) is the closest mathematical concept > (for me) that matches the "domain" concept in Aldor. > It think that it is "ok" as far as it goes, but I believe that it does not go far enough. It is clear that many of the ideas in Axiom were motivated by the concept of Abstract Data Type (ADT) that was popular at the time of the development of Axiom's first incarnation as ScratchPad II. I think the following paper by Jean Baillie presents that "state of the art" as it existed at that time very well: http://homepages.feis.herts.ac.uk/~comqejb/algspec/pr.html Although he is publishing as a computer scientist, Baillie presents in some detail a method for describing ADTs as many-sorted universal algebras. BTW, I think you can see one of the legacies of this "algebraic" orientation in Axiom (and even in Aldor): the signatures of operations are defined with an "arity", i.e. as functions that take multiple arguments and return a single value. Since Axiom and Aldor contain several "product" types (e.g. Cross, Tuple, Record, etc.) strictly speaking this is not necessary and leads to a some awkwardness (for example, so called courtesy coercions in Aldor). But my point is that the implementation of these algebraic concepts in Axiom did not get much past chapter 2 ( http://homepages.feis.herts.ac.uk/~comqejb/algspec/node3.html ). > A mathematical category does not fit. Don't you agree? > No I do not agree. The (mathematical) concept of cartesian closed categories and topoi provide tools for describing both the programming language and data structures of Axiom/Aldor. > Since you don't agree with my UA view, what is your closest mathematical > thing to an Aldor domain? > An Aldor domain is an object in the category (a topos) consisting of all Aldor domains and operations (functions). A similar view has also been advocated by Haskell programmers. http://www.haskell.org/haskellwiki/Category_theory I believe that by taking such a view it is possible to explain how domains are used now in Axiom/Aldor and also to suggest improvements to the language that will make it more flexible and expressive. Regards, Bill Page. ------------------------------------------------------------------------- This SF.Net email is sponsored by the Moblin Your Move Developer's challenge Build the coolest Linux based applications with Moblin SDK & win great prizes Grand prize is a trip for two to an Open Source event anywhere in the world http://moblin-contest.org/redirect.php?banner_id=100&url=/ _______________________________________________ open-axiom-devel mailing list open-axiom-devel@lists.sourceforge.net https://lists.sourceforge.net/lists/listinfo/open-axiom-devel
