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There seems to be different understandings of Type, domain, category, Category, etc. around. Here is an attempt to collect all these different opinions in order to make discussion about them clearer.

Some related referemces in the email list archives:

Definitions of Ralf Hemmecke

A category is an L-type whose type is the language-defined constant Category.

A domain is an L-type whose type is a category.

An L-type is either a category, a domain or the language-defined constants Category and Type.

Any L-type is of type Type.

I wrote L-type to mean type in the language, either Aldor or SPAD.

Each value belongs to ... its domain --Bill Page, Tue, 08 Jul 2008 06:53:30 -0700 reply

Stephen Watt wrote:
The type system has two levels: Each value belongs to some unique type, known as its domain, and the domains of expressions can be inferred statically. Each domain is itself a value belonging to the domain Type. Domains may additionally belong to some number of subtypes (of Type), known as categories. Categories can specify properties of domains such as which operations they export, and are used to specify interfaces and inheritance hierarchies.
The biggest difference between the two-level domain/category model and the single-level subclass/class model is that a domain is an element of a category, whereas a subclass is a subset of a class. This difference eliminates a number of problems in the definition of functions with multiple related arguments.

The Aldor User Guide says... --hemmecke, Tue, 08 Jul 2008 07:26:33 -0700 reply
In Section 7.2 of the AUG is written:

  • A domain is a type which defines a collection of exported symbols. The symbols may denote types, constants and functions. Many domains also define an interpretation for data values, called a representation type; these domains are also known as abstract data types. Those domains which are not abstract datatypes are called packages.
  • A category is a type which specifies information about domains, including the specification of the public interface to a domain, which consists of a collection of declarations for those operations which may be used by clients of the domain.

That conflicts the above statement that Type is a domain, but is in line with the two-level domain/category model.

See also Sections 7.8 (Domains) and 7.9 (Categories) of the Aldor User Guide.

I haven't (yet) found a sentence that says that Type or Category are domains.

Aldor Users Guide, chapter 7 --Bill Page, Tue, 08 Jul 2008 07:33:36 -0700 reply
Section 7.5 Subtypes

Every value in Aldor is a member of a unique domain which determines the interpretation of its data.

Section 7.9 Categories

All type values have ``Type'' as their unique base type. As with all other values, it is the unique base type which determines how values are to be represented.

The language allows categories to be treated as normal values and allows names to refer to categories. A category (by definition) is a value of the Aldor built-in type Category.

Type is a type but not a domain --hemmecke, Tue, 08 Jul 2008 07:52:27 -0700 reply
I have nothing against Type being a type. But Type is not an Aldor-domain. Maybe it is a domain in a broader sense, but that sense is only vaguely defined, if at all. I would like not to use domain and type interchangeably.

Re: Type is not an Aldor-domain? --Bill Page, Tue, 08 Jul 2008 09:01:17 -0700 reply
The fact that Type is a domain certainly does not make domain and type interchangeable. Objects of the domain Type are themselves either domains or categories, so type and domain are still not interchangeable since categories are not domains.

Could you explain why you claim that "Type is not an Aldor-domain". Is this only a personal preference? To me: "If it talks like a duck and it looks like a duck, its a duck...". In this case the compiler output, the library definitions, and quotations from the primary developer all agree:

  Type has with {};

returns true.

Why Type is not an Aldor-domain? --hemmecke, Tue, 08 Jul 2008 11:27:54 -0700 reply
I simply have not found an explicit statement in the AUG that says that Type and Category are domains. How else could I claim that they are?

And regardless of what others say, could you give your definitions similar to what I started at the beginning of this page? Let's first collect the status, before we argue, what implications it would have if Type would be a domain.

Maybe in the end it doesn't matter whether or not Category and Type are domains.

Re: maybe it doesn't matter --Bill Page, Tue, 08 Jul 2008 13:55:31 -0700 reply
Actually I sort of agree. From a formal perspective it seems rather surprising to me that it is possible to have done so much (good!) programming in Axiom and Aldor yet nevery having fully resolved such fundamental issues. Probably it has more impact on internal aspects of the compiler and interpreter than it does for most users. But I think some styles of programming - for example the kind used for the species project - might benefit from a clear definition and a reliable implementation.

Personally I think Aldor is a brilliant distillation and crystallization of the concepts that evolved from SPAD in Axiom. It's novel "categorical" alternative to the class/subclass concepts of the object-oriented paradigm is probably still under-appreciated by language designers. The discipline of programming language design in the meantime has largely moved on to other issues but this tradition can be carried on in Axiom by the OpenAxiom project.

Definitions by Bill Page

This is a first draft in point form of how I think domains and categories should work in at least OpenAxiom:

  • Domains implement all data and algebraic structures in OpenAxiom. SPAD and the OpenAxiom interpreter build-in only the minimum bootstrap data structures necessary to compile and access domains from the Axiom library.
  • All values are values in some domain that defines all of the operations that can be performed on these values. In general these values are created and accessible at run-time data except as noted below.


        a:A := new()$A

    declares a as a variable with values in domain A returned by an operation called new in A.

  • Domains themselves are values of the domain Domain and as such are also treated as run-time accessible data (first-order). Domains are created at compile-time and stored in the Axiom library. At the present there are no operations that create new domains at run-time but domains can be passed and returned as values.


        d:Domain := A

    declares d as a variable whose value is the domain A. The domain Domain provides some additional reflective and syntactical operations on domains such as equality.


        (d = A)@Boolean

    is true only if d has the value A.

  • Categories are subdomains of Domain specified by both name and a list of exported operations.


        X:Category == with
          f: A -> B

    By "subdomain" is meant that the domains in Domain are also values in some category only if they reference that category by name and implement the required exported operations.


        D:X == add
          f(x) == ...

    As subdomains of Domain, categories are themselves domains and can be used in declarations of variables and parameters.


         a:X := A

    Categories are organized into a lattice by referring to other categories by name.


        Y:Category == X with
          g: B -> A


        Z:Category == Join(X,Y)

    The list of exported operations is the union of all exported operations of the categories to which it refers plus those give after with.

  • Categories are values of the domain Category and as such are also treated as run-time accessible data (first-order). Category is itself a value of Domain. Like domains, categories are created at compile-time and stored in the Axiom library. There are currently no operations that create new categories at run-time but categories may be passed and returned as values. The domain Category provides some additional reflective and syntactical operations on categories such as = and has.


        has:(Union(Category,Domain),Category) -> Boolean
        A has X

    is true if A refers to X or to a category that refers to X applied recursively.

  • Type is the category in Category consisting of all domains in Domain. All categories are subcategories of Type. Type does not export any operations.

domains and categories --Bill Page, Wed, 09 Jul 2008 09:17:50 -0700 reply
On Tue, Jul 8, 2008 at 3:06 PM Gabriel Dos Reis wrote:

I'll be using the word specification in an informal sense that I hope is clear from context.

A category is a collection of specifications. A domain is a collection of implementations. An object is any computational values in an OpenAxiom program. An object has a representation given by a domain. An object O is said to have type d if d implements the reprsentation for the object O.

A category constructor is a category-valued function, defined with the term Category as its return type. A domain constructor is a domain-valued function, defined with a category as its return type.

Category contructors and domain constructors may be parameterized by domains and categories. Furthermore, domains and categories have runtime representations, e.g. they are reflected as objects in OpenAxiom. In particular, domains objects have type Domain, and category objects have type Category. And Domain and Category are indeed domains, because they implement specifications and provide object representations.

I did not discuss the notion of package, as it is almost like a domain - it implements specifications - except that it does not provide object representation.

-- Gaby

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