Computer Algebra Systems: Are We There Yet?

of Arizona -- February, 2004 1Computer Algebra Systems: Are We There Yet?. of Arizona -- February, 2004 7An Aside on your non-constructive education In freshman calculus you learned to

Univ of Arizona February, 2004 1

Computer Algebra Systems:

Are We There Yet?

Richard Fateman Computer Science Univ of California Berkeley, CA

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?

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?

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.

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.

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…

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

and a 2/3

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

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)

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?

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

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 equivalence results

zero-Univ of Arizona February, 2004 23

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)

The Marketing Blitz: aren’t they all the same?

• Mathematica + NeXT or

Apple = graphics.

-2 -1

0 1

2-2-1 0 1 2 0

0 1

2

• Maple does the same.

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More of the same…

Z

-2.00 -1.00

1.00 2.00

-2.00 -1.00

0.00 1.00

0.25 0.50 0.75

1.00

Macsyma

too

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.

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.

A broad brush of commonality today:

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 )

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

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???

Wolfram Research’s Special Functions site: 3 versions

• Huge posters

• Printed form (or the equivalent PDF)

• Now (2004) some 87,000 “formulas” and many

“visualizations”

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The posters

The web site (here, the Arcsin page)

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WRI’s Categories/ Some Subcategories

integral transforms identities

representations through more general functions relations with other functions

zeros inequalities theorems other information history and applications references

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

• To run some numbers through, you need to write a

computer program (Fortran? Matlab? C++?,)

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)

Note however that agreement on the semantics of \ [Ellipsis] would be difficult.

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.

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 ?

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

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.

Using OpenMath as original human-written source is pretty much out of the question.

If your intent is to code:

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Using MathML as original source is pretty much out of the question, too.

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.

Context might be the role of some Server Side

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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

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:…