On 10/28/07, Ralf Hemmecke wrote: > > On 10/28/2007 10:37 PM, Bill Page wrote: > > On 10/27/07, Ralf Hemmecke wrote: > >> Let me cite the documentation from stream.spad.pamphlet > >> > >> ++ A stream is an implementation of an infinite sequence using ++ a > >> list of terms that have been computed and a function closure ++ to > >> compute additional terms when needed. > >> > >> First, a stream is an infinite sequence. If we get > >> > >> (6) -> s:=construct([1,2,3])$Stream(Integer) (6) -> (6) [1,2,3] > >> Type: Stream Integer (7) -> s.4 7) -> > >>>> Error detected within library code: > >> elt: no such element > >> > >> Then there is either something wrong with the code or with the > >> documentation. > > > I think it is the documentation. Given the exports of Stream, it > > obviously should say only "potentially infinite". > > > Hmmm, I think, now we can start fighting. ;-) >
Good. :-) I am thinking in particular about the use and meaning of the operation 'explicitlyFinite?'. > I would be in favour of letting Stream(T) encode the > > T^N > > where N is the natural numbers. > > It seems that you want Stream(T) to denote > > T^N union T^(N) > > where the second thing denotes the set of finite sequences (which is > actually already modelled by List or Array). > No, I am only talking about the way it is implemented now. > I have nothing against a domain that models the above union, but I would > like to start with basic domains and then build on them. And I would > reserve the name "Stream" for the infinite thing. I am not sure whether I would expect STREAM to always be infinite or not. Maybe you are right since this would be the case if we wanted to have the semantics of STREAM given by a greatest fixed point solution (corresponding to List as least fixed point solution) of some recursive type equation. > > >>> The larger question remains however: When to use a domain to > >>> directly model something that is "set-like" and when to define a > >>> higher-order domain whose objects are "set-like"? To me this is > >>> not clear in either Spad or Aldor. > >> I don't think you gain very much if you only consider set-like > >> domains. If, however, you are looking at finite fields, for sure, > >> you will think of a domain, since it is more important that there > >> are some operations that connect the elements. > > > So you say it is natural to define "PrimeField 7" as a domain itself > > rather than defining a general domain of "PrimeFields" and > > constructing "PrimeField 7" as an element of that domain? > > Honestly, I don't see much of a difference. In your and my view > "PrimeField 7" is a domain, which exports certain functions. > Nobody forbids me to create a domain whose elements are domains. > (Maybe SPAD does forbid...) Gaby has shown how to define something of type 'List Domain' in Spad e.g. [Integer,Float,String] which I suppose must qualify as a domain that has at least lists of domains as elements. But I think a domain that actually has *domains* as elements is something different, i.e a list of domains is not itself a domain, as for example a Union or Product of domains would be. > But I think "PrimeFields" is a rather uninteresting domain. What > exports should it have? > > > Yes, I think that makes sense. But there are of course some > > operations that connect elements of any well defined set, no? > > Which? (Sorry, but I really cannot think of one.) Ok maybe I should have said OrderedSet, then elements are at least connected by some ordering. > > > I suppose what I am saying is that (maybe) there should really be no > > distinction between domains and members of domains (objects) - that > > all objects should also be domains in and of themselves. > > Let's go on with your thoughts. Could you tell me what operation the > integer 2 (considered as a domain) should export? > In a pure object-oriented language (which really Spad and Aldor are not), the objects inherit operations so that the constant '2' "knows" how to add (+) another object. But I think I want to withdraw my comment. From a categorical perspective such as presented by Saul Youssef (see thread [Aldor-l] Type equivalence of domains in Axiom and Aldor, Oct 26, 2007 12:09 AM): > Your questions have definite answers in category theory > and since Aldor is *almost* doing category theory, it's tempting > to think that the categorical answers to your questions are really > what should naturally fit into the language. I wrote up something > trying this out for the 2001 workshop > > http://atlas.bu.edu/~youssef/papers/math/aldor/aldor.pdf > > I still think that this is a good way to look for flaws in > the language - implement category theory and see what > goes wrong. probably I really do not want the members of domains to be domains but rather (nullary) operations which in Spad and Aldor is what they really *are*, e.g. zero:()->%, one:()->% Regards, Bill Page. ------------------------------------------------------------------------- This SF.net email is sponsored by: Splunk Inc. Still grepping through log files to find problems? Stop. Now Search log events and configuration files using AJAX and a browser. Download your FREE copy of Splunk now >> http://get.splunk.com/ _______________________________________________ open-axiom-devel mailing list open-axiom-devel@lists.sourceforge.net https://lists.sourceforge.net/lists/listinfo/open-axiom-devel
