Theorem Pro v ers and Computer Algebra Systems John Harrison - - PDF document

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Theorem Pro v ers and Computer Algebra Systems John Harrison - - PDF document

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Theorem Pro v ers and Computer Algebra Systems John Harrison Cam bridge Univ ersit y Computer Lab oratory 2nd No v em b er 1994 1 Theorem Pro v ers Are mainly used b y computer scien tists Applications

slide-1
SLIDE 1 Theorem Pro v ers and Computer Algebra Systems John Harrison Cam bridge Univ ersit y Computer Lab
  • ratory
2nd No v em b er 1994 1
slide-2
SLIDE 2 Theorem Pro v ers
  • Are
mainly used b y computer scien tists
  • Applications
include hardw are, soft w are and proto col v erication
  • Aim
to supp
  • rt
logic as applied mathematics
  • Generally
use \discrete" mathematics 2
slide-3
SLIDE 3 Computer Algebra Systems
  • Are
mainly used b y applied mathematicians, engineers and scien tists
  • Multiprecision
arithmetic, dieren tiation, in- tegration . . .
  • Aim
to supp
  • rt
con v en tional applied mathe- matics
  • Mainly
use \con tin uous" mathematics 3
slide-4
SLIDE 4 F eatures
  • f
Theorem Pro v ers
  • They
are logically and mathematically precise
  • They
emplo y rigorous principles
  • f
deduction
  • They
are usually dicult to use
  • They
are
  • ften
v ery slo w 4
slide-5
SLIDE 5 Computer Algebra Systems
  • Are
easy to use
  • Are
ecien t and p
  • w
erful
  • Lac
k a precise notion
  • f
logic
  • Are
deductiv ely unsound 5
slide-6
SLIDE 6 The Lac k
  • f
Logic in Computer Algebra Systems They are mainly based
  • n
a simple dialogue with the user:
  • The
user giv es an expression E 1
  • The
CAS returns an expression E 2
  • W
e are supp
  • sed
to b eliev e that E 1 = E 2 But are w e? What ab
  • ut
undenedness? x 2
  • 1
x
  • 1
= x + 1 Sometimes w e can reason ab
  • ut
simple inequal- ities, and there is at least a case analysis . . . 6
slide-7
SLIDE 7 The Unsoundness
  • f
Computer Algebra Systems
  • Maple:
Z 1 1 p x 2 dx =
  • Mathematica:
Z 1 1 1 p x 2 dx = An yw a y is an an tideriv ativ e what w e w an t? Ma yb e w e w an t
  • Riemann
In tegral
  • Leb
esgue In tegral
  • Gauge
In tegral 7
slide-8
SLIDE 8 The Sp ectrum
  • f
Theorem Pro ving Systems
  • Pro
  • f
Chec k ers { Automath (de Bruijn) { Stanford LCF (Milner et al.) . . . . . . . . .
  • Automatic
Theorem Pro v ers { NQTHM (Bo y er-Mo
  • re)
{ Otter (McCune) Whic h approac h is b etter? 8
slide-9
SLIDE 9 The LCF approac h Aims to com bine lo w-lev el pro
  • f
c hec k er and high lev el theorem pro v er.
  • Lo
w-lev el primitiv e inferences
  • Use
  • f
ML as programming en vironmen t for writing complex pro cedures
  • Secure
abstract datat yp e
  • f
theorems 9
slide-10
SLIDE 10 The LCF family
  • Original
w as Edin burgh LCF (Milner, Gor- don, Morris, New ey , W adsw
  • rth)
  • Reengineered
as Cam bridge LCF (P aulson)
  • Man
y descendan ts include { HOL (Gordon) { Nuprl (Constable) { Co q (Huet)
  • Renemen
ts
  • f
the basic idea include Isab elle (P aulson) The ML programming language started life as the MetaLanguage for LCF 10
slide-11
SLIDE 11 Quic k Summary
  • f
HOL
  • Higher
  • rder
logic based
  • n
simply t yp ed lam b da calculus
  • ML-st
yle parametric p
  • lymorphism
  • Conserv
ativ e denition mec hanism
  • V
ery few primitiv e rules (in theory)
  • Sev
eral v ersions (HOL88, hol90, Pro
  • fP
  • w
er) 11
slide-12
SLIDE 12 Analytica { a remedy for the lac k
  • f
logic
  • Designed
b y Clark e and Zhao
  • W
ritten in the Mathematica language
  • Incorp
  • rates
man y p
  • w
erful decision pro ce- dures
  • But
it relies
  • n
Mathematica's
  • wn
(unsound) simplier 12
slide-13
SLIDE 13 Mathp ert { a remedy for the lac k
  • f
soundness
  • Designed
b y Beeson
  • In
tended for educational use; stresses `glass b
  • x'
approac h
  • Underlying
sequen t calculus where side con- ditions accum ulate
  • A
ttempt to a v
  • id
the logic app earing explic- itly
  • It
remains to b e seen ho w it compares with existing systems in p
  • w
er 13
slide-14
SLIDE 14 Harrison and Th
  • ery
{ exploiting a link W e link together a Theorem Pro v er (HOL) and a Computer Algebra System (Maple). HOL can ask Maple questions { but what do w e do with the answ ers? 1. T rust the Computer Algebra System completely 2. T rust it partially; tag the theorem 3. Don't trust it at all { c hec k the answ er 14
slide-15
SLIDE 15 Examples where Chec king is Easy
  • Solving
equations (of all kinds)
  • F
actorizing p
  • lynomials
(or indeed n um b ers!)
  • In
tegrating expressions 15
slide-16
SLIDE 16 Example com bining in tegration and factorization (1) W e w an t to ev aluate: Z t sin 3 u du Maple tells us: Z t sin 3 u du =
  • 1
3 sin 2 t cos t
  • 2
3 cos t + 2 3 HOL can dieren tiate this expression to yield
  • 1
3 (2 sin t cos t cos t
  • sin
3 t) + 2 3 sin t but it do esn't simplify do wn to what w e w an ted (neither do es Maple in fact!) 16
slide-17
SLIDE 17 Example com bining in tegration and factorization (2) W e w an t to sho w that
  • 1
3 (2 sin t cos t cos t
  • sin
3 t) + 2 3 sin t = sin 3 t Let's replace sin t b y x and cos t b y y ; w e w an t to sho w that `
  • 1
3 (2 x y y
  • x
3 ) + 2 3 x
  • x
3 = 17
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SLIDE 18 Example com bining in tegration and factorization (3) W e ask Maple to factorize this expression, and it tells us: `
  • 1
3 (2 x y y
  • x
3 ) + 2 3 x
  • x
3 =
  • 2
3 x (y 2 + x 2
  • 1)
HOL can c hec k this answ er v ery easily . When x = sin t and y = cos t w e ha v e y 2 + x 2
  • 1
= 0, so the equation is pro v ed. No w the F undamen tal Theorem
  • f
Calculus yields the result. Maple w as righ t! 18
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SLIDE 19 What ha v e w e Gained? In HOL, real analysis, including (gauge) in te- gration and its relationship with dieren tiation, has b een dev elop ed formally b y denitional means. So w e ha v e:
  • An
indep enden t c hec k
  • n
Maple's correctness
  • A
formal HOL pro
  • f
using incon tro v ertible, lo w-lev el principles
  • A
rigorously dened, mathematically useful statemen t 19
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SLIDE 20 Conclusions
  • More
exp erience needed. Do es rigour mean rigor mortis?
  • F
  • r
the approac h to generalize, w e need p
  • w-
erful simpliers
  • But
it giv es quite a lot for v ery little w
  • rk
  • Theorem
pro v er and computer algebra de- signers ha v e a lot to learn from eac h
  • ther.
20