Dear Waldek
thanks for having immediately sent the *summary* of the note I've asked
... now I'm eager to read the forthcoming full note with its details ;)
Well, apart from jokes, I've not re-replied immediately because
I've spent some time digging in old mails of this list trying to grasp
the main ideas of FriCAS present and future design.
Below you'll find some thoughts/questions you might want to comment/answer.
These thoughts concern the user interface for the inverse functions, which is
a topic that periodically returns in the list.
In the past you repeatedly spoke of users with inconsistent wishes, so,
to be more "precise", let me start by setting up some "notation",
reviewing some facts I learned from the list.
I will restrict to log and sqrt for shortness and expose the various ways
such functions might be defined.
All the definitions below (apart maybe SetValuedExpression, I do not know)
should respect the property that
these functions are right-inverses under composition, ie
(sqrt(x))^2 =x exp(log(x))=x for every x in their domain D.
--- PrincipalBranchExpression ---
Let the PrincipalBranchExpression be a domain of analytic, complex valued
functions each defined by its respective ``principal branch'',
ie defined in a cut complex plane,
the cuts being arbitrarily but conventionally agreed to be as detailed
eg by Olver et al. in [2]:
sqrt, log: D -> C where D:=C -[0,-oo] (I neglect boundaries for simplicity).
For every z1,z2 in D such that z1*z2 in D we have the nice relations
sqrt(z1)*sqrt(z2)=sqrt(z1*z2) and log(z1)+log(z2)=log(z1*z2) (1)
In my knowledge this domain and (1) is well-defined mathematically
but has not a nice algebraic properties adapted to "Risch... algorithm"
and no algorithms to decide if A=0 in full generality. Too Bad.
On the other hand,
a heuristic & arbitrary "simplify" is possible,
based on pattern-matching transformations, eg
implementing a bunch of transformation rules (including (1)) and choosing
the best/shortest result based on some arbitrary expression length function.
This is the domain in which happily & daily work thousand of researchers out
there,
using Maxima, Sympy and other CAS I do not master or afford.
I guess that (1) is applied automatically by these CAS whenever its hypotheses
may be proven to be satisfied, hence the following replacements are
likely to be exploited.
sqrt(2)*sqrt(3)<~>sqrt(6) and sqrt(1-i)sqrt(1+i)<~>sqrt(2)
--- RealExpression ---
One may define a domain RealExpression of
regular functions on a real subset,
sqrt, log: D -> R where D:=(0,oo) (I neglect boundaries for simplicity).
At the price of banning some incompatible "highschool relation"
(like (-1)^(1/3)=-1)
I guess that one can consider the functions in RealExpression as appropriate
restriction of functions in PrincipalBranchExpression,
up to optimizations that exploit reality to improve the speed.
So I do not consider any more this domain.
--- AnyBranchExpression ---
Let AnyBranchExpression be the domain of analytic complex valued
functions, each well-defined on some arbitrary cut complex plane,
not necessary the ``principal branch''.
In this case a to a function with a single branch point
correspond in general a non countable
infinity of functions, think for instance
sqrt, log: D -> C where D:=C-[0,oo]*e^{i theta}
with theta in R
and also to the fact that the cut need not to be a straight line.
Now I give some possible approaches to "multi-valued" functions:
--- RiemannSurfaceExpression ---
Let RiemannSurfaceExpression be the domain of meromorphic functions
defined on Riemann surface.
sqrt, log: D -> R where D "is" {(r,phi): r in R_+ and phi in R }
for every z1,z2 in D
sqrt(z1)*sqrt(z2)=sqrt(z1*z2) and log(z1)+log(z2)=log(z1*z2) (2)
--- SetValuedExpression ---
Let SetValuedExpression be a domain of set-valued functions:
sqrt: C-{0} -> Parts(C),
x|-> { princbranchsqrt(x), -princbranchsqrt(x)}.
log: C-{0} -> Parts(C),
x|-> {princbranchlog(x)+2pi i k, with k in Z}.
I see that there are attempts to construct a mathematical theory of these
set valued functions [3]
IIUC the product in this space is defined by direct product:
f(x)*g(y) = { zf*zg with zf in f(x) and zg in g(y)}.
--- DifferentialField ---
(The following domain is quite new to me and I have much to learn,
so forgive my sloppiness)
Let DifferentialField be some *abstract* field endowed with an additive
derivative
satisfying Leibniz rule and extending the rational fild Q(x).
This is the natural framework for implementing the "Risch... algorithm"
for finding antiderivatives.
(I do not really know how should define the algebraic "functions" here.
See below.)
In this field one defines
log(x) by D(log(x))=1/x
Actually this does define log(x) "up to a constant", so I do not know if this
log(x) has something to do with the other log(x) defined in the other domains.
As an aside:
Are the elements of DifferentialField naturally isomorphic to
*concrete, numerical* functions?
I had a really fast-and-furious look at Siedenberg's embedding theorem [4]
"Finitely generated differential field in characteristic zero
is isomorphic to a subfield of field of meromorphic functions
in a domain"
but I have not yet really understood in which domain these meromorphic
function are defined nor if the construction is unique.
So I cannot really say what is the numerical value of S(log)(2)
where S is the Siedenberg isomorphism.
I wonder if the abstract functions DifferentialField in are related
(up to constants) to the ones in RiemannSurfaceExpression defined above.
--- Discussion ---
As far as I understand, FriCAS user interface is currently heavily biased by
its implementation of the "Risch... algorithm".
In this sense, Expression R when R = Integer / Complex Integer tries to model
a DifferentialField. This results in default simplification rules that are too
weak as compared to the PrincipalBranchExpression.
Eg sqrt(2)*sqrt(3)-sqrt(6) is not zero.
Now the thoughts/questions.
0)
As shown above there are a *lot* of possible meanings for sqrt, log ...
A general problem is that the different semantics have the same Input and
Output form
and it is not always clear in which mathematical semantics one is exposed.
Of course one reads Type: Expression Integer or Expression Complex on the right
but this is not a guarantee that the relations proper to
PrincipalBranchExpression or RealExpression are applied.
For instance one writes sqrt(2), one sees the usual graphical sqrt V2 but this
has
not the meaning that a lot of people would think, ie +1.4142...
Too bad and confusing.
1)
The "Risch... algorithm" and differential algebras are new to me
and I find them beautiful & elegant.
But.. how many people really needs to work by default fall-back
in a differential algebra?
(Apart developers and maybe few mathematicians working exactly on that subject.)
I mean:
I understand that simplification and integration are in principle
unrelated problems, so, why FriCAS exposes as user catchall-type
a domain modeled to be a differential algebra useful for "Risch algorithm"
and not a familiar PrincipalBranchExpression like the other CAS?
Thousands of potential FriCAS users (me included)
would like to have the possibility to live
in a more familiar context where everything is compatible
with PrincipalBranchExpression and, more than that,
to have aggressive & automatic simplifications
that 100% comply with the rules of this world.
Nobody wants to spend half an hour to find out
what operation one has to invoke
to force sqrt(2)*sqrt(3)=sqrt(6)
which is quite trivial in a PrincipalBranchExpression context.
This is the reason of the recurrent non-bugs raised in the lists:
semantics of Expression R is not evident to users and, worse,
does not fully satisfy the PrincipalBranchExpression-community
expectations.
Why naot, for instance, having the analog of a
PrincipalBranchExpression domain + its simplifications
as user fall back top domain, with RealExpression called on option
and DifferentialField/Expression R below and available on demand?
If I understand correctly,
having PrincipalBranchExpression domain by default
could help eliminating dependent kernels in integration:
Eg when invoked from PrincipalBranchExpression, an
expression like
integrate( 1/(1+x+(sqrt(2)*sqrt(3)-sqrt(6))*x^100000000000), x)
might be automatically & rigorously pre-simplified with
symplify$PrincipalBranchExpression,
thus eliminating ((sqrt(2)*sqrt(3)-sqrt(6))*x^100000000000)
then the expression (with no more dependent kernels)
may be lifted to the DifferentialField domain which gets back
an expression in terms of abstract functions log ....
and then each abstract function maight be
projected back to the PrincipalBranchExpression world
because the target context is clear.
The advantages of having a PrincipalBranchExpression
domain would then be:
i)
make the users of other CAS be more interested to FriCAS
because a familiar framework is offered
= increased chance of long term survival of the project
ii)
separate the Risch code which is hard to understand from other code that can
be improved by developers not at ease with Risch
(hoping not to hurt other people in the list,
I have the *impression* that only you fully understand
risch-related domains: code simplifications should be welcome.)
iii)
the full independence from user expectations might allow
for a no-compromises and cleaner Risch framework
(Eg I've read that some branch cut choices for algebraic numbers
are done in some -- unspecified -- situations.).
2)
I anticipate your possible disagreemnt:
if the PrincipalBranchExpression proposal is not suitable for your plans
what FriCAS may offer to the PrincipalBranchExpression-people?
I mean are you willing
to facilitate the life of these (huge) group of potential users?
How?
3)
Algebraic numbers aside.
How do you rigorously define sqrt n-th power as implemented in FriCAS?
Currently sqrt(a) means something like
"a generic complex x such that x^2=a".
The only rigorous meaning I found for multivalued function is that
sketched in MultivaluedExpression,
but the natural product there is the direct product of sets
so it is no more true that sqrt(a)*sqrt(a) = {a},
becausethe direct product combines
all elements on the left set with all the elements on the right set.
In simple words, what bothers me is that if I define
sqrt(a)* sqrt(b)
:= product of a generic complex x such that x^2=a
and a generic complex y such that y^2 =b
then specializing to a=b it is no more true that
sqrt(a)*sqrt(a) = a,
which is currently a structural identity of FriCAS.
4)
Somewhere you said that you find unsatisfactory to work only with
one branch.
But if you want all branches together,
how do you deal with the fact that in general,
as argued in AnyBranchExpression,
there is an infinity of such branches?
Maybe you want to limit only to a subset of all possible branches
with cuts fixed on negative axis or so...
5)
Is it concievable/practical to solve multivalueness problems introducing
a RiemannSurfaceExpression domain?
For log-riemann surface one should model a point with polar coordinates
(r, phi), r i R_+ and phi in R.
But I do not know how meromorphic functions on different Riemann surfaces
might interact together.
Hope having not talked too much nonsense.
riccardo
[1]
https://groups.google.com/d/msg/fricas-devel/JADruNrq-zI/sY0cQMpbAQAJ
[2]
https://dlmf.nist.gov/4
[3]
pag 92 of
https://www.tu-ilmenau.de/fileadmin/media/simulation/Lehre/Vorlesungsskripte/Lecture_materials_Abebe/svm-topology.pdf
[4]
Seidenberg 1958 Abstract Differential Algebra and the Analytic Case page 160
https://www.jstor.org/stable/2033416
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