Edit detail for SandBoxSymbolic revision 22 of 35

Symbolic Integers

A simple form of symbolic computation using just Variable and Integer.

axiom
a:Union(Variable a,Integer)

axiom
b:Union(Variable b,Integer)

axiom
c:=a*3-b*2

(1)

axiom
p:UP(x,Integer):=x*2-7

(2)

axiom
pc := p c

(3)

axiom
f(x)==x^3-x^2+1

axiom
fb := f b

Compiling function f with type Variable b -> Polynomial Integer

(4)

axiom
a:=1

(5)

axiom
b:=-3

(6)

axiom
c

(7)

axiom
eval(pc,['a=a,'b=b])

(8)

axiom
eval(c,['a=a,'b=b])

(9)

axiom
eval(fb,['a=a,'b=b])

(10)

axiom
a:=3.14  -- not permitted!
   Cannot convert right-hand side of assignment
   3.14
      to an object of the type Union(Variable a,Integer) of the
      left-hand side.
a:='b    -- not permitted!

   Cannot convert right-hand side of assignment
   b
      to an object of the type Union(Variable a,Integer) of the
      left-hand side.

For more complex cases it is necessary to define a new domain of "indeterminants". These are symbols and unevaluated expressions that can be evaluated at a later time. This domain is modeled after InputForm? which provides all of the basic functionality.

spad
)abbrev domain INDET Indeterminant
++ Description:
++   Based on InputForm
Indeterminant():
  Join(SExpressionCategory(String,Symbol,Integer,DoubleFloat,OutputForm),
       ConvertibleTo SExpression) with
    eval: % -> Any
      ++ eval(f) passes f to the interpreter.
    binary   : (%, List %) -> %
      ++ \spad{binary(op, [a1,...,an])} returns the input form
      ++ corresponding to  \spad{a1 op a2 op ... op an}.
    function : (%, List Symbol, Symbol) -> %
      ++ \spad{function(code, [x1,...,xn], f)} returns the input form
      ++ corresponding to \spad{f(x1,...,xn) == code}.
    lambda   : (%, List Symbol) -> %
      ++ \spad{lambda(code, [x1,...,xn])} returns the input form
      ++ corresponding to \spad{(x1,...,xn) +-> code} if \spad{n > 1},
      ++ or to \spad{x1 +-> code} if \spad{n = 1}.
    "+"      : (%, %) -> %
      ++ \spad{a + b} returns the input form corresponding to \spad{a + b}.
    "-"      : (%, %) -> %
      ++ \spad{a - b} returns the input form corresponding to \spad{a - b}.
    "-"      : % -> %
      ++ \spad{-a} returns the input form corresponding to \spad{-a}.
    "*"      : (%, %) -> %
      ++ \spad{a * b} returns the input form corresponding to \spad{a * b}.
    "/"      : (%, %) -> %
      ++ \spad{a / b} returns the input form corresponding to \spad{a / b}.
    "**"     : (%, NonNegativeInteger) -> %
      ++ \spad{a ** b} returns the input form corresponding to \spad{a ** b}.
    "**"     : (%, Integer) -> %
      ++ \spad{a ** b} returns the input form corresponding to \spad{a ** b}.
    unparse  : % -> String
      ++ unparse(f) returns a string s such that the parser
      ++ would transform s to f.
      ++ Error: if f is not the parsed form of a string.
    declare  : List %   -> Symbol
      ++ declare(t) returns a name f such that f has been
      ++ declared to the interpreter to be of type t, but has
      ++ not been assigned a value yet.
      ++ Note: t should be created as \spad{devaluate(T)$Lisp} where T is the
      ++ actual type of f (this hack is required for the case where
      ++ T is a mapping type).
    compile  : (Symbol, List %) -> Symbol
      ++ \spad{compile(f, [t1,...,tn])} forces the interpreter to compile
      ++ the function f with signature \spad{(t1,...,tn) -> ?}.
      ++ returns the symbol f if successful.
      ++ Error: if f was not defined beforehand in the interpreter,
      ++ or if the ti's are not valid types, or if the compiler fails.
    coerce : Integer -> %
 == SExpression add
    Rep := SExpression
    mkProperOp: Symbol -> %
    strsym    : % -> String
    tuplify   : List Symbol -> %
    coerce(x:Integer):% == convert(x)
    coerce(x:%):OutputForm == expr x
    convert(x:%):SExpression == x pretend SExpression
    conv(ll : List %): % ==
      convert(ll pretend List SExpression)$SExpression pretend %
    lambda(f,l) == conv([convert("+->"::Symbol),tuplify l,f]$List(%))
    eval x ==
      v := interpret(x)$Lisp
      mkObj(unwrap(objVal(v)$Lisp)$Lisp, objMode(v)$Lisp)$Lisp
    convert(x:DoubleFloat):% ==
      convert(x)$Rep
    strsym s ==
      string? s => string s
      symbol? s => string symbol s
      error "strsym: form is neither a string or symbol"
    unparse x ==
      atom?(s:% := form2String(x)$Lisp) => strsym s
      concat [strsym a for a in destruct s]
    declare signature ==
      declare(name := new()$Symbol, signature)$Lisp
      name
    compile(name, types) ==
      symbol car cdr car
        selectLocalMms(mkProperOp name, convert(name)@%,
          types, nil$List(%))$Lisp
    mkProperOp name ==
      op := mkAtree(nme := convert(name)@%)$Lisp
      transferPropsToNode(nme, op)$Lisp
      convert op
    binary(op, args) ==
      (n := #args) < 2 => error "Need at least 2 arguments"
      n = 2 => convert([op, first args, last args]$List(%))
      convert([op, first args, binary(op, rest args)]$List(%))
    tuplify l ==
      empty? rest l => convert first l
      conv
        concat(convert("Tuple"::Symbol), [convert x for x in l]$List(%))
    function(f, l, name) ==
      nn := convert(new(1 + #l, convert(nil()$List(%)))$List(%))@%
      conv([convert("DEF"::Symbol), conv(cons(convert(name)@%,
                        [convert(x)@% for x in l])), nn, nn, f]$List(%))
    s1 + s2 ==
      conv [convert("+"::Symbol), s1, s2]$List(%)
    s1 - s2 ==
      conv [convert("-"::Symbol), s1, s2]$List(%)
    _-(s1) ==
      conv [convert("-"::Symbol), s1]$List(%)
    s1 * s2 ==
      conv [convert("*"::Symbol), s1, s2]$List(%)
    s1:% ** n:Integer ==
      conv [convert("**"::Symbol), s1, convert n]$List(%)
    s1:% ** n:NonNegativeInteger == s1 ** (n::Integer)
    s1 / s2 ==
      conv [convert("/"::Symbol), s1, s2]$List(%)

   Compiling FriCAS source code from file 
      /var/zope2/var/LatexWiki/7483143363411555740-25px002.spad using 
      old system compiler.
   INDET abbreviates domain Indeterminant 
------------------------------------------------------------------------
   initializing NRLIB INDET for Indeterminant 
   compiling into NRLIB INDET 
   compiling exported coerce : Integer -> $
Time: 0.04 SEC.

   compiling exported coerce : $ -> OutputForm
Time: 0 SEC.
   compiling exported convert : $ -> SExpression
      INDET;convert;$Se;3 is replaced by x 
Time: 0 SEC.
   compiling local conv : List $ -> $
Time: 0.10 SEC.
   compiling exported lambda : ($,List Symbol) -> $
Time: 0.02 SEC.
   compiling exported eval : $ -> Any
Time: 0 SEC.
   compiling exported convert : DoubleFloat -> $
Time: 0 SEC.
   compiling local strsym : $ -> String
Time: 0.01 SEC.
   compiling exported unparse : $ -> String
Time: 0.07 SEC.
   compiling exported declare : List $ -> Symbol
Time: 0 SEC.
   compiling exported compile : (Symbol,List $) -> Symbol
Time: 0.01 SEC.
   compiling local mkProperOp : Symbol -> $
Time: 0 SEC.
   compiling exported binary : ($,List $) -> $
Time: 0.04 SEC.
   compiling local tuplify : List Symbol -> $
Time: 0.08 SEC.
   compiling exported function : ($,List Symbol,Symbol) -> $
Time: 0.03 SEC.
   compiling exported + : ($,$) -> $
Time: 0.01 SEC.
   compiling exported - : ($,$) -> $
Time: 0 SEC.
   compiling exported - : $ -> $
Time: 0.01 SEC.
   compiling exported * : ($,$) -> $
Time: 0.06 SEC.
   compiling exported ** : ($,Integer) -> $
Time: 0.01 SEC.
   compiling exported ** : ($,NonNegativeInteger) -> $
Time: 0 SEC.
   compiling exported / : ($,$) -> $
Time: 0.01 SEC.

(time taken in buildFunctor:  1)
;;;     ***       |Indeterminant| REDEFINED
;;;     ***       |Indeterminant| REDEFINED
Time: 0.01 SEC.
   Warnings: 
      [1] conv: pretend$ -- should replace by @
   Cumulative Statistics for Constructor Indeterminant
      Time: 0.51 seconds
   finalizing NRLIB INDET 
   Processing Indeterminant for Browser database:
--------(eval ((Any) %))---------
--------(binary (% % (List %)))---------
--------(function (% % (List (Symbol)) (Symbol)))---------
--------(lambda (% % (List (Symbol))))---------
--------(+ (% % %))---------
--------(- (% % %))---------
--------(- (% %))---------
--------(* (% % %))---------
--------(/ (% % %))---------
--------(** (% % (NonNegativeInteger)))---------
--------(** (% % (Integer)))---------
--------(unparse ((String) %))---------
--------(declare ((Symbol) (List %)))---------
--------(compile ((Symbol) (Symbol) (List %)))---------
--->-->Indeterminant((coerce (% (Integer)))): Not documented!!!!
--------constructor---------
------------------------------------------------------------------------
   Indeterminant is now explicitly exposed in frame initial 
   Indeterminant will be automatically loaded when needed from 
      /var/zope2/var/LatexWiki/INDET.NRLIB/code

Symbolic Matrices

axiom
)clear all
   All user variables and function definitions have been cleared.

axiom
m:Union(Indeterminant,Matrix Integer)

axiom
n:Union(Indeterminant,Matrix Integer)

axiom
a:Union(Indeterminant,Integer)

axiom
b:Union(Indeterminant,Integer)

axiom
ab:=(3*a+b)*(a-2*b)

(11)

axiom
a:=1$Integer  -- because 0 and 1 can be symbolic!

(12)

axiom
eval ab

(13)

axiom
b:=0$Integer

(14)

axiom
a:=-10

(15)

axiom
eval ab

(16)

axiom
mn1:= m*n-n*m

(17)

axiom
mn2:=(m+n)*(m-n-1)

(18)

axiom
m:=matrix [[1,2],[3,4]]

(19)

axiom
n:=matrix [[-1,-2],[-3,4]]

(20)

axiom
eval mn1

(21)

axiom
eval mn2

(22)

axiom
)clear all

All user variables and function definitions have been cleared.

axiom
d: Union(Variable d, Float)

axiom
d*2

(23)

will be problematic, and the assignment between symbolic integers is not allowed.

axiom
a: Union(Variable a, Integer)

axiom
b: Union(Variable b, Integer)

axiom
b:=a
   a is declared as being in Union(Variable a,Integer) but has not been
      given a value.

axiom
p:=2*d

(24)

axiom
eval(p,d=3.14)

(25)

But there is still a problem since I can write:

axiom
a:Union(Variable a,Integer)

axiom
a:=3.14
   Cannot convert right-hand side of assignment
   3.14
      to an object of the type Union(Variable a,Integer) of the
      left-hand side.
p:=2*a

(26)

axiom
eval(p,a=3.14)

(27)

and Axiom does not complain about the eval request because as far as Polynomial Integer is concerned d is just a Symbol.

In the case of the example Indeterminant domain, evaluation of symbolic expressions is always delayed so an appropriate eval operation could be defined that properly respects the type.

axiom
c:Union(Variable(c),Symbol)

axiom
c:=a

(28)

PS: In the light of polynomial, (2d) is a non-zero polynomial, so that it's always safe to write (1/(2d)), but it's actually non-safe to do it in the domain of float(or integer).

b:=a is meaningless, of course. But there are many scenarios

axiom
)clear all
   All user variables and function definitions have been cleared.

axiom
i:=10

how about i:=-10?

(29)

Type: PositiveInteger?

axiom
a: Union(Variable a, Integer)

Type: Void

axiom
b: Union(Variable b, Integer)

Type: Void

axiom
b:= (i>0=>1; a)

(30)

Type: Union(Integer,...)

which is meaningful.

We wish to place a restriction on the possible values that certain symbols can take. Such symbols and expressions formed from them certainly cannot live in a numeric domain such as Float. Except in certain circumstances (mentioned previously) I think a domain such as a:Union(Variable(a),Float) and the domains to which it can be coerced (such as Polynomial, Expression, ...) does already represent such a symbolic expressions fairly well.

But 1) What is Symbolic Float? Can you give a description?

And 2) How can I express the concept of "non-zero polynomial" in Axiom? I might use a domain that does not include zero for example a:Union(Variable(a),PositiveInteger) but there is no general domain of NonZero anything.