Algebraic Algorithms CS 282 Spring 2002 Lecture 1 Richard Fateman CS 282 Lecture 1 1 The subject Symbolic Computation Computer algebra systems CAS and their supporting algorithms for performing symbolic mathematical manipulation Math surprises Can you program or make constructive various more or less wellknown symbolic computations Computer Science tasks Can you build a mathematical intelligence Or at least a skilled assistant Richard Fateman CS 282 Lecture 1 2 An aside on your non constructive education In freshman calculus you learned to integrate rational functions You could integrate 1 x and 1 xa into logarithms and you used partial fractions Unless you ve recently taken or taught this course you ve forgotten the details Richard Fateman CS 282 Lecture 1 3 Here s an integration problem Richard Fateman CS 282 Lecture 1 4 Fortunately you can factor the denominator this way by guesswork Richard Fateman CS 282 Lecture 1 5 And then do the partial fraction expansion Richard Fateman CS 282 Lecture 1 6 And then integrate each term Richard Fateman CS 282 Lecture 1 7 Non constructive parts Do you really know an algorithm to factor the denominator into linear and quadratic factors Can you do this one say And if it does not factor it need not you know what do you do then Richard Fateman CS 282 Lecture 1 8 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 Richard Fateman CS 282 Lecture 1 radicals 9 Moral of this story You probably never knew how to integrate rational functions Only some rational functions Writing a program to say factor or integrate uses ideas you may have never seen before End of aside Richard Fateman CS 282 Lecture 1 10 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 Richard Fateman CS 282 Lecture 1 11 Some History Less Ancient Philosophers Mathematicians e g G Frege but best known B Russell A N Whitehead Principia Mathematica 19101913 Richard Fateman CS 282 Lecture 1 12 Some History No you can t do it all Godel Turing Richard Fateman CS 282 Lecture 1 13 Some History New optimism 1958 60 first inklings automatic differentiation tree representations Lisp Minsky Slagle 1961 Moses 1966 Is it AI Richard Fateman CS 282 Lecture 1 14 Computer Algebra Systems threads Three trends emerged in the 1960s AI later expert systems Mathematics e g Berlekamp factoring Liouville Risch integration computational group theory Algorithms Computer Science e g Knuth Brown Collins polynomial GCD Richard Fateman CS 282 Lecture 1 15 Some Historical 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 Special purpose systems optimism about conquering all of math by coming up with the right programming formalism and accumulating facts Richard Fateman CS 282 Lecture 1 16 Mathematics flirting with computing Constructive algorithmic algebra was fashionable in the early 20th century early editions of van der Waerden s classic Modern Algebra book but existence proofs became more popular Too bad I think the tide is turning towards constructive approaches Richard Fateman CS 282 Lecture 1 17 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 zeroequivalence results Richard Fateman CS 282 Lecture 1 18 Some well known systems Computers got comparatively cheaper so systems get more ambitious more available 1968 78 SAC 1 Altran Macsyma Scratchpad Mathlab 68 MuSimp MuMath SMP Automath others Further development new entrants of 1980 s Maple Mathematica 1988 Derive Axiom Theorist Milo Consolidation 1990s improving existing systems new experimental systems theorem proving niche math Richard Fateman CS 282 Lecture 1 19 Some support systems Common Lisp gets standardized Scheme gets standardized too C popularized as the answer Portability UNIX Linux Windows Apple Java HTML XML and Browsers Richard Fateman CS 282 Lecture 1 20 The Marketing Blitz and shakeout Mathematica NeXT Apple graphics Maple comes out from under a rock IBM Scratchpad goes public as Axiom under NAG sponsorship then is killed 2001 MuPad at Univ of Paderborn is free then sold Macsyma goes into hiding parts come out free Openmath and MathML put Math on the Web Connections Links from Matlab to Maple Scientific Workplace to Maple or Mathematica The arrival of network agents for problem solving Calc101 Tilu TheIntegrator Ganith Richard Fateman CS 282 Lecture 1 21 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 a system needs neat graphics for Marketing Axiom takes the view that a computer algebra system is an implementation of Modern Algebra Almost everyone concedes that good algorithms and data structures are necessary for effective efficient computation sometimes Math takes a back seat Richard Fateman CS 282 Lecture 1 22 Next time What do these CAS and the many systems we haven t explicitly mentioned have in common Algebraic systems Objects Operations Properties Axioms Extensions to a base system programming Declarations Underlying all of this efficient representations Richard Fateman CS 282 Lecture 1 23