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Time: 2018/02/26 16:25:01 GMT+0 

Note: 
changed:  From BillPage Wed Sep 28 21:50:23 0500 2005 From: Bill Page Date: Wed, 28 Sep 2005 21:50:23 0500 Subject: Embedding Axiom categories in Axiom domains. MessageID: <200509282150230500@wiki.axiomdeveloper.org>  Issue report #209 however shows that *perhaps* it would be useful to **flatten** Axiom's twolevel object model of categories and domains by embedding Categories into the domain called Domain. This would allow convenient things like evaluating the expression::   (Integer,Float)  as type 'Tuple(Domain)'.   Bill also discussed use of 'Domain' in Spad code. For details see [SandBox Domains And Types].   Since in FriCAS Type is class of all type one can do \begin{axiom} (Integer,Float) \end{axiom} and get 'Tuple(Type)'. But one can not use categories as FriCAS data, so `(Integer,SetCategory)' does not work.
FriCAS has the FriCAS interpreter for interaction with the user and the FriCAS compiler for building library modules.
Types are one of the most important concepts in FriCAS.
Like Modula 2, PASCAL, FORTRAN, and Ada, the programming language emphasizes strict typechecking. Unlike these languages, types in FriCAS are dynamic objects: they are created at runtime in response to user commands.
The FriCAS interactive language is oriented towards easeofuse. The FriCAS interpreter uses typeinferencing to deduce the type of an object from user input. Type declarations can generally be omitted for common types in the interactive language.
The FriCAS compiler language (version 1 of the language is called SPAD, version 2 of the language is called Aldor) on the other hand is based on strong static types that must be explicitly specified by the programmer.
Consider
(1+2)/3
(1) 
It is sort of interesting, isn't it, that FriCAS insists on calling "1" a fraction just because of the way it was calculated? There is a way to say that you want the answer as an integer. Of course this isn't always possible:
(1/3)::Integer
Cannot convert the value from type Fraction(Integer) to Integer .
But for this example it is:
((1+2)/3)::Integer
(2) 
FriCAS is very strict with types (meaning: that it must be
completely unambiguous where each operation takes place).
The division of 1+2 by 3 takes place in Fraction Integer
(field of rational numbers) and that is where the answer will
be, even if it is 1. Most people would automatically "retract"
this to the integer 1. But in general, there is no natural
way to do so (why not retract 1 to a natural number, for
example?). So FriCAS provides the user a way to "coerce" an
answer to another type. You can coerce 1 to any of many,
many types, for example, as a polynomial, or even as a
matrix (or the unit element of any ring).
((1+2)/3)::Polynomial Integer
(3) 
((1+2)/3)::SquareMatrix(1,Integer)
(4) 
((1+2)/3)::SquareMatrix(3,Integer)
(5) 
((1+2)/3)::SquareMatrix(3,Polynomial Complex Integer)
(6) 
In FriCAS the concept of "Type" is universally applied at all levels.
From the Axiom Book: Appendix B Glossary
Every Axiom object is an instance of some domain. That
domain is also called the type of the object. Thus the
typeOf 1
is inferred by the interpret to be Integer
and
the typeOf
the variable x:Integer
is declared to be Integer
.
Similarly the typeOf "daniel"
is String
. The type of an
object, however, is not unique. The type of integer 7
is
not only Integer
but also NonNegativeInteger
, PositiveInteger
,
and possibly, in general, any other subdomain of the
domain Integer
.
Domains are defined in FriCAS by programs of the form:
Name(Parameters): JoinCategorys with Exports == Extends add Rep := RepDomain Implementation
The Name
of each domain is used to refer to the collection
of its instances. For example, Integer
denotes "the integers",
Float
denotes "the floating point numbers" etc. For example:
Fraction(S: IntegralDomain): QuotientFieldCategory S with if S has canonical and S has GcdDomain and S has canonicalUnitNormal then canonical == LocalAlgebra(S, S, S) add Rep:= Record(num:S, den:S) coerce(d:S):% == [d,1] zero?(x:%) == zero? x.num
Thus the type of Fraction Integer
is
QuotientFieldCategory Integer with canonical
.
The axiom canonical
means that equal elements of the
domain are in fact identical.
We also say that the domain Fraction(S)
extends the domain
LocalAlgebera(S,S,S)
. Domains can extend each other in a
circular mutually recursive manner so in general the extended
relationship forms a directed graph with cycles.
Fractions are represented as the domain Record(num:S, den:S)
.
In FriCAS domains and subdomains are themselves objects that have types. The type of a domain or subdomain is called a category. Categories are described by programs of the form:
Name(...): Category == JoinCategorys with Exports add Imports Implementation
The type of every category is the distinguished symbol
Category
. The category Name
is used to denote the collection
of domains of that type. For example, category Ring
denotes
the class of all rings.
For example:
QuotientFieldCategory(S: IntegralDomain): Category == Join(Field, Algebra S, RetractableTo S, FullyEvalableOver S, DifferentialExtension S, FullyLinearlyExplicitRingOver S, Patternable S, FullyPatternMatchable S) with _/ : (S, S) > % ++ d1 / d2 returns the fraction d1 divided by d2. numer : % > S ++ numer(x) returns the numerator of the fraction x. denom : % > S ... if S has PolynomialFactorizationExplicit then PolynomialFactorizationExplicit add import MatrixCommonDenominator(S, %) numerator(x) == numer(x)::% denominator(x) == denom(x) ::% ...
Categories say nothing about representation. Domains, which are instances of category types, specify representations. See also Rep And Per.
Categories form hierarchies (technically, directedacyclic
graphs). A simplified hierarchical world of algebraic
categories is shown below. At the top of this world is
SetCategory
, the class of algebraic sets. The notions of
parents, ancestors, and descendants is clear. Thus ordered
sets (domains of category OrderedSet
) and rings are also
algebraic sets. Likewise, fields and integral domains are
rings and algebraic sets. However fields and integral
domains are not ordered sets:
SetCategory + Ring  IntegralDomain  Field  + Finite +  \ + OrderedSet + OrderedFinite Figure 1. A simplified category hierarchy.
The most basic category is Type
. It denotes the collection
of all domains and subdomains. Note carefully that Type
does
not denote the class of all types! The type of all categories
is Category
. Since Type
is a category its type is Category
.
typeOf(Type)
(7) 
typeOf(typeOf(Type))
(8) 
The second result above indicates that currently FriCAS thinks that
type of Category
is again Category
. However, it is safer
to treat is as undefined, because any definition leads to
paradoxes.
For example of parametric type: the domain List
is able to build "lists of elements
from domain D" for arbitrary D simply by requiring that D
belong to category Type
.
Another example. Enter the type Polynomial (Integer)
as an
expression to FriCAS. This looks much like a function call
as well. It is! The result is stated to be of type Type
.
Polynomial(Integer)
(9) 
Ref: Axoim Book "Chapter 2 Using Types and Modes".
Many FriCAS operations have the same name but different types and these types can be dependent on other types. For example
)display operation differentiate
There are 15 exposed functions called differentiate : [1] (D,(D3 > D3), NonNegativeInteger) > D from D if D has DIFEXT(D3) and D3 has RING [2] (D, (D2 > D2)) > D from D if D has DIFEXT(D2) and D2 has RING
[3] (D,NonNegativeInteger) > D from D if D has DIFRING [4] D > D from D if D has DIFRING [5] (D, NonNegativeInteger) > D from D if D has DVARCAT(D2) and D2 has ORDSET [6] D > D from D if D has DVARCAT(D1) and D1 has ORDSET [7] (D, (D3 > D3)) > D from D if D has FFCAT(D2, D3, D4) and D2 has UFD and D3 has UPOLYC( D2) and D4 has UPOLYC(FRAC(D3)) [8] (FullPartialFractionExpansion(D2, D3), NonNegativeInteger) > FullPartialFractionExpansion(D2, D3) from FullPartialFractionExpansion(D2, D3) if D2 has Join(FIELD, CHARZ) and D3 has UPOLYC(D2) [9] FullPartialFractionExpansion(D1, D2) > FullPartialFractionExpansion(D1, D2) from FullPartialFractionExpansion(D1, D2) if D1 has Join(FIELD, CHARZ) and D2 has UPOLYC(D1) [10] (GeneralUnivariatePowerSeries(D2, D3, D4), Variable(D3)) > GeneralUnivariatePowerSeries(D2, D3, D4) from GeneralUnivariatePowerSeries(D2, D3, D4) if D3: SYMBOL and D2 has RING and D4: D2 [11] (D, List(D3), List(NonNegativeInteger)) > D from D if D has PDRING(D3) and D3 has SETCAT [12] (D, D1, NonNegativeInteger) > D from D if D has PDRING(D1) and D1 has SETCAT [13] (D, List(D2)) > D from D if D has PDRING(D2) and D2 has SETCAT
[14] (D,D1) > D from D if D has PDRING(D1) and D1 has SETCAT [15] (D, (D2 > D2), D) > D from D if D2 has RING and D has UPOLYC(D2) and D2 has Join(SRNG, ABELMON)
There are 14 unexposed functions called differentiate : [1] (IntegrationResult(D1),Symbol) > D1 from IntegrationResult(D1) if D1 has FIELD and D1 has PDRING(SYMBOL) [2] (IntegrationResult(D1), (D1 > D1)) > D1 from IntegrationResult( D1) if D1 has FIELD [3] (D, PositiveInteger) > Union(D, 0) from D if D has JBC [4] (D, D1) > D from D if D has JBFC(D1) and D1 has JBC [5] (OutputForm, NonNegativeInteger) > OutputForm from OutputForm
[6] (U32Vector,Integer) > U32Vector from U32VectorPolynomialOperations [7] (U32Vector, NonNegativeInteger, Integer) > U32Vector from U32VectorPolynomialOperations [8] (SparseUnivariateLaurentSeries(D2, D3, D4), Variable(D3)) > SparseUnivariateLaurentSeries(D2, D3, D4) from SparseUnivariateLaurentSeries(D2, D3, D4) if D3: SYMBOL and D2 has RING and D4: D2 [9] (SparseUnivariatePuiseuxSeries(D2, D3, D4), Variable(D3)) > SparseUnivariatePuiseuxSeries(D2, D3, D4) from SparseUnivariatePuiseuxSeries(D2, D3, D4) if D3: SYMBOL and D2 has RING and D4: D2 [10] (SparseUnivariateTaylorSeries(D2, D3, D4), Variable(D3)) > SparseUnivariateTaylorSeries(D2, D3, D4) from SparseUnivariateTaylorSeries(D2, D3, D4) if D3: SYMBOL and D2 has RING and D4: D2 [11] (UnivariateFormalPowerSeries(D2), Variable(x)) > UnivariateFormalPowerSeries(D2) from UnivariateFormalPowerSeries(D2) if D2 has RING [12] (UnivariateLaurentSeries(D2, D3, D4), Variable(D3)) > UnivariateLaurentSeries(D2, D3, D4) from UnivariateLaurentSeries(D2, D3, D4) if D3: SYMBOL and D2 has RING and D4: D2 [13] (UnivariatePuiseuxSeries(D2, D3, D4), Variable(D3)) > UnivariatePuiseuxSeries(D2, D3, D4) from UnivariatePuiseuxSeries(D2, D3, D4) if D3: SYMBOL and D2 has RING and D4: D2 [14] (UnivariateTaylorSeries(D2, D3, D4), Variable(D3)) > UnivariateTaylorSeries(D2, D3, D4) from UnivariateTaylorSeries(D2, D3, D4) if D3: SYMBOL and D2 has RING and D4: D2
We can see how the interpreter resolves the type:
[14] (D,D1) > D from D if D has PDRING D1 and D1 has SETCAT
in the following example
)set message bottomup on
differentiate(sin(x),x)
Function Selection for sin Arguments: VARIABLE(x) > no appropriate sin found in Variable(x) > no appropriate sin found in Symbol > no appropriate sin found in Variable(x) > no appropriate sin found in Symbol
Modemaps from Associated Packages no modemaps
Remaining General Modemaps [1] D > D from D if D has TRIGCAT
[1] signature: EXPR(INT) > EXPR(INT) implemented: slot $$ from EXPR(INT)
Function Selection for differentiate Arguments: (EXPR(INT),VARIABLE(x)) > no appropriate differentiate found in Variable(x)
[1] signature: (EXPR(INT),SYMBOL) > EXPR(INT) implemented: slot $$(Symbol) from EXPR(INT)
(10) 
Notice that
EXPR INT has PDRING SYMBOL
(11) 
SYMBOL has SETCAT
(12) 
Since in FriCAS Type is class of all type one can do
(Integer,Float)
LISP output: (UNPRINTABLE UNPRINTABLE)
and get Tuple(Type)
. But one can not use categories as
FriCAS data, so `(Integer,SetCategory?)' does not work.