Re: [Axiom-developer] [Axiom-mail] manipulating series not as streams but as sumations

[William Sit]

If I follow the question Raymond Rogers raised (I did not 
receive all the email for some reason), and Ralf's reply 
below, I don't think Axiom is that different, and the 
problem that Stream cannot "look inside" the parameter f 
is not because Axiom is compiled. Rather it is the nature 
that f is passed as a function, which by definition, has a 
source domain and a target domain, and computes values and 
nothing more. If the aim is to "know" more about f inside 
Stream (or a calling function/domain), then one should 
change the property of the parameter in Stream and hence 
create a new domain, say call StreamMore, where the 
function f is wrapped into a domain, say 
FunctionDomain(f), and inside FunctionDomain(f) you have 
auxiliary functions such as isFunctionRandom?(f), and any 
other properties of f that you would like StreamMore to 
access or query.

This is not different from OOP where the properties of f 
are provided by methods, like f.isFunctionRandom?.

In general, when a parameter is passed to another function 
(or a domain like Stream), only its categorical properties 
(here the definition of what a set function is, with 
source and target, and the production of values, but not 
details of the computation itself) are known from inside 
the calling function (or calling domain). Such 
"restrictions" are universal in computer programming 
languages.

At least, that's my limited understanding.

William

On Tue, 21 Oct 2014 14:50:00 -0400
 Raymond Rogers <address@hidden> wrote:
> On 10/21/2014 04:39 PM, Raymond Rogers wrote:
> > Let's think about this.
> > I think the question can be interpreted as a 
> > formating/display problem
> > for the underlying code.
>
> If you believe so... Let me state it a bit differently 
> what
> datastructure comes out of the series command. Or rather 
> what is stored
> inside the data structure.
>
> Here an example:
>
> (1) -> Z==>Integer; Q==>Fraction Z;
>                                                           
>         Type:
> Void
> (2) -> f(q:Q):Q == 
> truncate(abs(q*cos(q::Float))+1)*(q/2+1)
>
>    Function declaration f : Fraction(Integer) -> 
> Fraction(Integer) has
>       been added to workspace.
>                                                           
>         Type:
> Void
> (3) -> s: Stream Q := stream(f, 0)
>    Compiling function f with type Fraction(Integer) -> 
> Fraction(Integer
>       )
>
>              3 7 15 31 63 127 255 511
>    (3)  [0,1,-,-,--,--,--,---,---,---,...]
>              2 4  8 16 32  64 128 256
>                                               Type:
> Stream(Fraction(Integer))
> (4) -> p:=series(s)$UnivariateFormalPowerSeries(Q)
>
>    (4)
>          3  2   7  3   15  4   31  5   63  6   127  7 
>  255  8   511  9
>      x + - x  + - x  + -- x  + -- x  + -- x  + --- x  + 
> --- x  + --- x
>          2      4       8      16      32       64 
>     128      256
>    +
>      1023  10      11
>      ---- x   + O(x  )
>       512
>                          Type:
> UnivariateFormalPowerSeries(Fraction(Integer))
>
> The only thing that s (the Stream) knows is a function 
> object f (if you
> like you can use f=random). It cannot look inside f 
> except to ask for
> more values. Remember AXIOM code is compiled.
>
> If from that you would somehow be able to extract a 
> pattern, great, but
> I doubt.
>
> > Of course I am not that familiar with the Axiom 
> > internals.
>
> Well, you might come from another CAS. AXIOM *is* 
> different. It is in
> general *not* dealing with expression trees. So what you 
> have in mind
> (adding presentation) does not work for Stream of 
> Series.
>
> That is not saying that it is impossible, but rather 
> that the Stream and
> PowerSeries domains are not appropriate for what you 
> have in mind. (And
> at the moment I don't think there is anything like what 
> you want in Axiom.)
>
> Ralf
>
>
>
>
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William Sit, Professor Emeritus
Mathematics, City College of New York
Office: R6/291D Tel: 212-650-5179
Home Page: http://scisun.sci.ccny.cuny.edu/~wyscc/