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Time: 2009/01/12 18:23:40 GMT8 

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changed: See discussion: http://wiki.axiomdeveloper.org/TuplesProductsAndRecords See discussion: TuplesProductsAndRecords
Aldor programming philosophy from the point of view of it's evolution from previous incarnations of the Axiom library compiler, is described here:
http://www.aldor.org/docs/HTML/chap18.html
There is one construct in particular in Aldor that really
stands out  the concept of representation and it's opposite
(abstraction?). In Aldor these are formally denoted
by two very peculiar coercion operations: rep
and per
.
In older SPAD terms, rep and per can be written as the following macros:
macro { rep x == x @ % pretend Rep; per r == r @ Rep pretend %; )
The expression rep(x)
describes an object x of this domain but
treats it as belonging to a different domain Rep
.
The expression per(y)
describes an object y from the Rep
domain
which is to be treated as belonging to this domain. (per
as in
the percent % symbol)
In looking through older SPAD code it is clear that these
combinations occur frequently and their encapsulation as rep
and per
makes reading Aldor code much easier than SPAD.
Does anyone know of any formal programming language research
papers that describe the semantics of rep
and per
in a
general way? If this sort of mechanism is present in other
languages, what form does it take?
It seems to me that rep
and per
constitute the essence of the
peculiar objectoriented programming style pioneered by Axiom
and Aldor. But what are these constructs in formal (categorical)
terms?
Below is a first attempts to answer these questions  comments, corrections and opinions would be most welcome!
It seems appropriate to me to think of rep and per as forming a pair of adjoint functors:
rep > % domain Rep domain < per
rep
is a "forgetful functor" that maps the abstract structure
(objects and operations) of this domain into it's internal
representation. It "forgets" about the abstract relationships.
The objects of the external domain however are viewed only in
terms of the relationships between objects. The image of this
domain under rep
in representation has its internal structure
exposed.
per
on the other hand constructs abstract members of this
domain from the underlying representation. It encapsulates
the internal operations of the representation in terms of
the abstract structure of this domain.
domains must be thought of in terms of categories both in the sense of category theory and in Axiom's sense of "category". The appropriate categories usually have at least the structure of a Cartesian closed category which they inherit from Axiom's basic types constructors, record, union and mapping. See discussion: TuplesProductsAndRecords
In terms of category theory there is a natural transformation,
called the unit
, that partitions elements of the Rep domain
into classes (as an abstract quotient of the representation):
x > rep(per(x))
and rep
is right adjoint of per
.
In category theoretic terms Rep
domain is some Cartesian closed
category, i.e. consisting of products (records), coproducts
(unions) and exponentials (functions) over some basic set
of component domains. It's operations are formed freely from
the operations of it's components.
The implementation of the domain in terms of the representation establishes a natural bijection between the operation of the external domain and the operation of it's internal representation:
x > rep y (representation)  per x > y (domain)
The book: "Basic Category Theory for Computer Scientists", by
Benjamin Pierce, MIT Press, 1991, is a good reference. See
especially section 2.4. Pierce gives an excellent example in
terms of definition of the List
constructor.
Also: "Categories, Types and Structures. An introduction to Category Theory for the working computer scientist." by Andrea Asperti and Giuseppe Longo, M.I.T. Press, 1991, which is downloadable from here: http://www.di.ens.fr/users/longo/download.html
Regards,
Bill Page.
(from an email to axiomdevelper on Thursday, September 01, 2005 6:36 AM)