William Stein <wst...@gmail.com> writes:
> 3. Having var at all is a compromise -- many symbolic calculus users
> would prefer for undefined vars to just "magically" be defined, as is
> done in Mathematica, Maple, Maxima, Axiom (?), etc.
In Axiom (FriCAS, OpenAxiom), there is a distinction between elements
of, say, the polynomial ring, and variables:
(10) -> y := x + 1
(10) x + 1
Type: Polynomial(Integer)
(11) -> x := y
(11) x + 1
Type: Polynomial(Integer)
(12) -> x
(12) x + 1
Type: Polynomial(Integer)
I don't know how Python / Sage handles this exactly. The trick in
FriCAS is that, in the first line, the interpreter tries to make sense
of the expression (in the computer science sense) "x + 1"
It notices that "x" could be interpreted as a polynomial, and so can
"1", then x+1 makes sense. One can watch FriCAS doing its thinking as
follows:
(1) -> )se me bo on
(1) -> y := x + 1
Function Selection for +
Arguments: (VARIABLE(x),PI)
-> no appropriate + found in Variable(x)
-> no appropriate + found in PositiveInteger
-> no appropriate + found in Symbol
-> no appropriate + found in Integer
-> no appropriate + found in Polynomial(Integer)
-> no appropriate + found in Variable(x)
-> no appropriate + found in PositiveInteger
-> no appropriate + found in Symbol
-> no appropriate + found in Integer
[1] signature: (POLY(INT),POLY(INT)) -> POLY(INT)
implemented: slot $$$ from POLY(INT)
(1) x + 1
Type: Polynomial(Integer)
Two IMPORTANT notes:
* in FriCAS, the interpreter serves as an interface between the user and
the library. In the library (i.e., the mathematics), no guessing is
applied. The interpreter tries to help the user, that's all.
* I'm not saying that this is the best way of doing things (and I'm not
saying that it's a bad approach either). The following, continuing
(1) above may serve as an indication:
(2) -> x := y
(2) x + 1
Type: Polynomial(Integer)
(3) -> y := x + 1
Function Selection for +
Arguments: (POLY(INT),PI)
-> no appropriate + found in Polynomial(Integer)
-> no appropriate + found in PositiveInteger
-> no appropriate + found in Integer
-> no appropriate + found in PositiveInteger
-> no appropriate + found in Integer
[1] signature: (POLY(INT),POLY(INT)) -> POLY(INT)
implemented: slot $$$ from POLY(INT)
(3) x + 2
Type: Polynomial(Integer)
Of course, if you are explicit, everything's fine again:
(4) -> y := "x"::Symbol + 1
Function Selection for +
Arguments: (SYMBOL,PI)
-> no appropriate + found in Symbol
-> no appropriate + found in PositiveInteger
-> no appropriate + found in Integer
-> no appropriate + found in Polynomial(Integer)
-> no appropriate + found in Symbol
-> no appropriate + found in PositiveInteger
-> no appropriate + found in Integer
[1] signature: (POLY(INT),POLY(INT)) -> POLY(INT)
implemented: slot $$$ from POLY(INT)
(4) x + 1
Type: Polynomial(Integer)
Martin
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