Univ. of Arizona -- February, 20041 Computer Algebra Systems: Are We There Yet? Richard Fateman Computer Science Univ. of California Berkeley, CA.

Univ. of Arizona -- February, 2004 1

Computer Algebra Systems: Are We There Yet?

Richard FatemanComputer ScienceUniv. of California

Berkeley, CA

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The Subject: “Symbolic Computation Systems”

• What are they?• How good are they

now?• Where are they going?• When will they be

“there”?

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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?

• 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(?)

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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 non-constructive 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 non-numeric 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

• 1967-68 algorithms: Polynomial GCD, • Berlekamp’s polynomial Factoring, • Risch Integration "near algorithm",• Knuth’s Art of Computer Programming• 1967 - Daniel Richardson: interesting zero-

equivalence 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.

-2

-1

0

1

2-2

-1

0

1

2

00.250.5

0.75

1

-2

-1

0

1

2

• Maple does the same.

Plot exp(-(x2+y2)) in (-2,2) (-2,2)

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More of the same…

• Mathematica + NeXT or Apple = graphics.

X Y

Z

-2.0 < X < 2.0-2.0 < Y < 2.0

3.35e-4 < Z < 0.99

X

Y

Z

-2.00

-1.00

1.00

2.00

-2.00-1.00

0.001.00

0.25

0.50

0.75

1.00

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:

– 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

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Contrast: Non-digital tradition: to find out something we might do this

• 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

• 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

• 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 definitionspecific valuesgeneral characteristicsseries representations generalized power series at various points q-series exponential fourier series dirichlet series asymptotic series other seriesintegral reprsentations on the real axis contour integrals multiple integral representationanalytic continuationsproduct representationslimit representations

continued fractionsgenerating functionsgroup representationsdifferential equationsdifference equationstransformations addition formulas etcoperationsintegral transformsidentitiesrepresentations through more general functionsrelations with other functionszerosinequalitiestheorems other informationhistory and applicationsreferences

Computer Algebra and DLMF 41

Click on “Series Representations”…

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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)

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

Input form

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.

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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 + ... /. z -> 1/2

Pi/6 == 2009/3840 + ... 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• 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.

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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>

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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.

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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:…