Computer Algebra Systems:
Are We There Yet?
Richard Fateman
Computer Science
Univ. of California
Berkeley, CA
Univ. of Arizona -- February, 2004
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The Subject: “Symbolic Computation Systems”
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What are they?
How good are they now?
Where are they going?
When will they be
“there”?
Univ. of Arizona -- February, 2004
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What are they?
• An attempt to build a
“mathematical
intelligence” or at least
a very skilled assistant.
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What is their current state?
• We don’t know how to
achieve the stated goals.
• We keep trying, anyway.
• New systems are
produced every few
years, but rarely push
the state of the art,
much less advance it.
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What then?
• If we don’t have one
now, and progress seems
slow, what do we need to
do, when will we do it,
and what will it look
like?
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What does it take to build a Computer Algebra
System?
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A. Software engineering.
B. Language choice (Aldor, C++, Java, Lisp,…).
C. Algorithms, data structures.
D. Mathematical framework (often the weak spot).
E. User interface design.
F. Conformance to Standards, TeX, MathML, COM,
.NET, Beans.
• G. Community of users (IMPORTANT).
• H. Leadership/ Marketing(?)
Univ. of Arizona -- February, 2004
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An Aside on your non-constructive education
In freshman calculus you learned to
integrate rational functions. You could
integrate 1/x and 1/(x-a) into
logarithms, and you used partial
fractions.
Unless you’ve recently taken (or taught)
this course, you’ve forgotten the details.
That’s OK. Let’s review it fast.
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Here’s an integration problem
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You need to factor the denominator
You learned to do this by guesswork, and
fortunately it works.
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And then do the partial fraction expansion
You probably remember one way to do
this ,vaguely if at all..
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And then integrate each term…
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Can we program this? Note we can’t computerize
“guessing the answer” generally.
Do you really know an algorithm to factor the
denominator into linear and quadratic factors?
• Can you do this one, say…
• And if the denominator does not factor (it
need not, you know… ) what do you do then?
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If the denominator doesn’t factor
And it gets worse … there is no guarantee that
you can even express the roots of irreducible
higher degree polynomials in radicals like 31/2
and a2/3
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Moral of this story
• Freshmen are not taught how to integrate
rational functions. Only some easy rational
functions.
• A freshman could not write a program.
Polynomial factoring or rational integration
uses ideas you may never encounter.
• Much of the math you learned is nonconstructive and must be re-invented to write
a general computer algebra program!
End of aside
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Some History: Ancient
• Ada Augusta, 1844 foresaw prospect of nonnumeric computation using Babbage’s
machines. Just encode symbols as numbers,
and operations as arithmetic.
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Ada Augusta on Symbolic Computing, 1844
Many persons who are not conversant with mathematical studies imagine that because
the business of [Babbage's Analytical Engine] is to give its results in numerical
notation, the nature of its processes must consequently be arithmetical and
numerical, rather than algebraical and analytical. This is an error. The engine can
arrange and combine its numerical quantities exactly as if they were letters or any
other general symbols; and in fact it might bring out its results in algebraic notation,
were provisions made accordingly.
-- Ada Augusta, Countess of Lovelace, (1844)
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Some History: Slightly Less Ancient
• Arithmetization of Mathematics: Formalisms
• Philosophers/Mathematicians, e.g. Gottlob Frege,
then Bertrand Russell, Alfred North Whitehead
(Principia Mathematica 1910-1913)
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The Flip side: proofs you can’t do all math
• Impossible.
• K. Gödel, A.M. Turing
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New optimism. If people can, why not Computers?
1958-60 first inklings .. automatic
differentiation, tree representations, Lisp,
• Minsky ->Slagle, (1961), Moses (1966); Is it
AI? Pattern Matching?
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Computer Algebra Systems : threads
• Three trends emerged in the 1960s:
– AI / later…expert systems
– Constructive Mathematics (Integration)
– Algorithms on polynomials (GCD)
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Some Early Ambitious Systems
• Early to mid 1960's - big growth period,
considerable optimism in programming
languages, as well as in computer algebra…
• - Mathlab, Symbolic Mathematical Laboratory,
• Formac, Formula Algol, PM, ALPAK, Reduce,
CAMAL; Special purpose systems,
• Simple poorly-specified systems that did some
useful computations coupled with uncritical
optimism about what could be done next.
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Some theory/algorithm breakthroughs
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1967-68 algorithms: Polynomial GCD,
Berlekamp’s polynomial Factoring,
Risch Integration "near algorithm",
Knuth’s Art of Computer Programming
1967 - Daniel Richardson: interesting zeroequivalence results.
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Some old systems survive, new ones arrive
• General:
– SAC-1, Altran, Macsyma, Scratchpad, Mathlab 68,
MuSimp/MuMath, SMP, Automath, JACAL, others.
• Specialists:
– Singular, GAP, Cocoa, Fermat, NTL, Macaulay
• Further development; new entrants since
1980's
– Maple, Mathematica (1988), Derive, Axiom,
Theorist, Milo… MuPad, Ginac, Pari)
• For a list, see: www.symbolicnet.org
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The Marketing Blitz: aren’t they all the same?
• Mathematica + NeXT or
Apple = graphics.
• Maple does the same.
1
0.75
2
0.5
0.25
1
0
-2
0
-1
-1
0
1
2
-2
Plot exp(-(x2+y2)) in (-2,2) (-2,2)
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More of the same…
• Mathematica + NeXT or
Apple = graphics.
Macsyma
too
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The blitz…
• Mathematica. Endorsed by Steve Jobs and the New
York Times?
• Maple changes its image, belatedly.
• Macsyma follows suit.
• Axiom (Scratchpad) sold by IBM to NAG.
• Mupad starts up.
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The shakeout
• Axiom under NAG sponsorship, then is killed. (2001)
• MuPad, once free, now sold.
• Macsyma goes into hiding, earlier version emerges
free as Maxima.
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Connections gain new prominence
• MathML puts “Math on the Web”.
• Connections
– Links from Matlab or Excel to Maple; Macsyma to Matlab;
– Scientific Workplace to Maple or Mathematica or Mupad.
• The arrival of network agents for problem solving.
– Calc101, Tilu, TheIntegrator, Ganith, …
– Java beans for symbolic computation
– MP, distributed computing
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Are there really differences in systems?
• What we see today in systems:
– Mathematica essentially takes the view that
mathematics is a collection of rules with a
procedure for pattern matching; and that math can
be reduced to what might be good for physicists,
even if slightly wrong.
– Axiom takes the view that a computer algebra
system is an implementation of Modern Algebra,
and the physicists better know algebra.
– “Advanced” math is spotty.
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A broad brush of commonality today:
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Objects
Operations
Properties? Axioms?
Extensions to a base system (programming?
Declarations?)
– Underlying all of this: efficient representations
– Common bugs (e.g. by violating “fundamental
theorem of calculus” continuity requirements.)
– A shell around the whole thing. Menus, notebooks,
etc
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Moving to the future
• Computer math + WWW adds new prospects.
• Repository for everything that was previously
published (paper  digital form).
• Could include everything NEW (born digital).
– What to do with repetitive garbage?
• Need methods to find appropriate information
– Index/search :: vastly dependent on CONTEXT
– Certify authenticity and correctness (referees?)
• Algorithms may not yet exist for some problems.
– How to pay for development
– Availability to (all?).
• Free “public library”, pay-per-view, subscription, …
pop-up ads (This integral brought to you by XYZ bank )
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Digital Library of Mathematical Functions
(at NIST)
• Mostly aimed at traditional usage
• Intimations of support for new modes of
interaction with WWW, CAS
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Competition for DLMF
Mostly aimed at supporting CAS users.
• ESF: generate automatic symbolic data for
Encyclopedia of Special Functions.
• Wolfram’s special functions project: collect
material from humans in special forms, display
in Mathematica oriented forms.
Less CAS…
• CRC/Maple tables
• Dan Zwillinger, ODEs, Gradshteyn
Univ. of Arizona -- February, 2004
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Contrast: Non-digital tradition: to find out
something we might do this
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Look in an individually owned reference work
Visit a library
Access to colleagues by letter, phone
Paper and pencil exploration
Numerical experimentation
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Contrast: Digital tradition: to find out something we
might do this
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Try Google
Visit an on-line library database e.g. INSPEC
Download papers to local printer or view online
CAS exploration
Numerical experimentation
• Major Problem: How can you type a
differential equation into Google???
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Wolfram Research’s Special Functions site: 3
versions
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Huge posters
Interactive web site/ Mathematica notebooks
Printed form (or the equivalent PDF)
Now (2004) some 87,000 “formulas” and many
“visualizations”.
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The posters
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The web site (here, the Arcsin page)
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WRI’s Categories/ Some Subcategories
primary definition
specific values
general characteristics
series representations
generalized power series at various points
q-series
exponential fourier series
dirichlet series
asymptotic series
other series
integral reprsentations
on the real axis
contour integrals
multiple integral representation
analytic continuations
product representations
limit representations
continued fractions
generating functions
group representations
differential equations
difference equations
transformations
addition formulas etc
operations
integral transforms
identities
representations through more general functions
relations with other functions
zeros
inequalities
theorems
other information
history and applications
references
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Click on “Series Representations”…
Computer Algebra and DLMF
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The posters are not very useful
• These are pictures of out-of-context math
formulas.
• The most plausible next step given the charts is
to copy them down on paper and check by hand.
• There is a possibility of making typos or fresh
algebra mistakes.
• The notation might be different from what you
are using.
• Sparse (or no) info on singularities, regions of
validity.
• To run some numbers through, you need to
write a computer program (Fortran? Matlab?
C++?,)
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On-line versions are more useful
• Less possibility of making new typos.
• The notation are unambiguous, presumably using
a CAS or formal syntax.
• Still, sparse (or no) info on singularities,
regions of validity.
• Automated visualizations and cut/paste
programming to run some numbers through.
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Notebook form (I)
Input form
ArcSin[z] == z^3/6 + z + (3*z^5)/40 + \[Ellipsis] ==
Sum[(Pochhammer[1/2, k]*z^(2*k + 1))/((2*k + 1)*k!),
{k, 0, Infinity}] ==
z*Hypergeometric2F1[1/2, 1/2, 3/2, z^2] /; Abs[z] < 1
Wolfram (and others) will claim that a “system
independent” language such as proposed by the
OpenMath consortium would replace this language.
Note however that agreement on the semantics of
\[Ellipsis] would be difficult.
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Notebook form (II)
Displayed form (one version)
In reality, Mathematica does not look quite as good as
our typesetting here in the interactive mode.
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Notebook form (III)
TeX form
{Condition}(\arcsin (z) =
{\frac{{{\Mfunction{z}}^3}}{6}} + z +
{\frac{3\,{z^5}}{40}} + \ldots
=
\Mfunction{\sum}_{k = 0}^{\infty }
{\frac{\Mfunction{Pochhammer}({\frac{1}{2}},k)\,
{{\Mfunction{z}}^{2\,k + 1}}}{\left( 2\,k + 1
\right) \,k!}} =
\Mfunction{z}\,\Mfunction{Hypergeometric2F1}(
{\frac{1}{2}},{\frac{1}{2}},{\frac{3}{2}},{z^2}),
\Mfunction{Abs}(z) < 1))
Useful in case you wanted to paste/edit this into a
paper, (or powerpoint) but requires using
Mathematica TeX macros.
Univ. of Arizona -- February, 2004
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Notebook form (IV)
OpenMath form
{
too ugly to believe}
Useful in case you wanted to send this to an
OpenMath aware program. If you can find one.
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Computing Inside the Notebook
How good is the 3-term approximation at z= ½ ?
ArcSin[z] == z + z^3/6 + (3*z^5)/40 + ...

Pi/6 == 2009/3840 + ...
/.
z -> 1/2
Surprised?
N[ Pi/6 == 2009/3840 + ...]  0.523599 == 0.523177 +
...
N[ Pi/6 == 2009/3840 + ..., 30]  0.523599 == 0.523177
+ ...
N[ Pi/6 == 2009/3840 + ..., 30] 
0.52359877559829887307710723055 ==
0.52317708333333333333333333333 + ...
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Simplification Inside the Notebook
In[30] := z* Hypergeometric2F1[1/2, 1/2, 3/2, z^2]
Note: this is how Mathematica interactive output looks.
This should be the same as ArcSin[z] for |z|<1. And yes, z/Sqrt[z^2] is not the same as 1.
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Many computer algebra systems (CAS) have
essentially the same notebook paradigm
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Macsyma
Maple
Mathematica
Axiom
MuPad
Scientific Word / Maple
Derive
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This old “knowledge”? Can we convert from
scanned text?
Example from integral table


In practice, we can do some parsing using OCR if we know about
the domains.
But in general, we cannot read “with understanding” without
context.
Univ. of Arizona -- February, 2004
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What about using LaTeX as source and then
converting to OpenMath/ CAS?
Generally speaking: not automatically
TeX does not distinguish semantically
between 1*2*3 and 123.
Or between x cos x and xfoox.
It has no notion of precedence of
operators
Gradshteyn and Rhyzik, Table of Integrals and
Series (Academic Press) was re-typeset
completely in TeX TWICE, because the first version
did not reflect semantics. MathML, XML, and
OpenMath are inadequate.
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Using OpenMath as original human-written
source is pretty much out of the question.
If your intent is to code:
x cos x
You are supposed to write something like
<OMOBJ>
<OMA>
<OMS cd = "arith1" name="times"/>
<OMV name="x"/>
<OMA>
<OMS cd="transc1" name="cos"/>
<OMV name="x"/>
</OMA>
</OMA>
</OMOBJ>
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Using MathML as original source is pretty
much out of the question, too.
<math>
<msqrt>
<mfrac>
<mrow><mn>2</mn><mi>&pi;</mi></mrow>
<mrow><mi>&kappa;</mi></mrow>
</mfrac>
<mfenced open="(" close=")">
<mn>1</mn>
<mi>&minus;</mi>
<mi>&beta;</mi>
<msup>
<mrow><mn>2</mn></mrow>
</msup>
<mi>/</mi><mn>2</mn></mfenced></msqrt></math>
Univ. of Arizona -- February, 2004
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What about “Wikis”
•Volunteers inserting “information” into an informal
structure on the internet. Anyone can edit anything.
•Unlikely to have the accuracy and scope of a funded
activity.
•Replaces single bias with many biases.
•Unlikely to have the proprietary interest of a
commercial enterprise.
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How will a CAS fit into this vision of Math of the
future?
•The semantics for most (not all ) CAS is
immediate.
• Input requires immediate syntactic disambiguation.
• Easy translation into MathML for display.
• Easy translation into OpenMath, if anyone else
cares
•Important Advantage: There is an immediate
computational ontology. THE BEST CHANCE
FOR A FOUNDATION TO GROW CONTEXT.
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Context might be the role of some Server Side
software.
• Pro:
– arbitrarily powerful,
– always up-to-date, (contains yesterday’s new math)
– controlled by reputable authority…
• Con:
– Requires reliable communication.
– Authoritarian.
Univ. of Arizona -- February, 2004
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A challenge: Input and Output of Math
• Handwriting on a tablet is an obvious choice on
Tablet PCs, but on closer examination, a very
weak method. (30 years of experience!)
0Oo 1l| 5S vV Yy <  l< K
• Speech, oddly enough, can help.
• The importance of context emerges again…
enormous in math communication, digital
storage, etc.
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Finally: Are we there yet?
• No, we are not.
• Many efforts are re-working the easy parts.
• Many efforts are mostly marketing: “improving
the user interface.”
• The importance of context is enormous. A
“search engine for math facts and algorithms”
seems our best bet to build a mathematical
assistant.
• What can we do:…
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