Joint work with Andre Lieutier Dassault Systemes Domain Theory and Differential Calculus Abbas Edalat Imperial College ae Oxford.

Joint work with Andre Lieutier

Dassault Systemes

Domain Theory and Differential CalculusDomain Theory and Differential Calculus

Abbas Edalat Imperial College

http://www.doc.ic.ac.uk/~ae

Oxford

17/2/2003

2

Computational Model for Classical Computational Model for Classical SpacesSpaces

• A research project since 1993: Reconstruct some basic

mathematics• Embed classical spaces into the set of

maximal elements of suitable domains

XClassical

Space

x

DXDomain

{x}

3

Computational Model for Classical Computational Model for Classical SpacesSpaces

Previous Applications:

• Fractal Geometry

• Measure & Integration Theory

• Exact Real Arithmetic

• Computational Geometry/Solid Modelling

4

Non-smooth Non-smooth MathematicsMathematics

• Set Theory• Logic• Algebra• Point-set Topology• Graph Theory• Model Theory . .

• Geometry• Differential Topology• Manifolds• Dynamical Systems• Mathematical Physics . . All based on

differential calculus

Smooth Smooth MathematicsMathematics

A Domain-Theoretic Model for A Domain-Theoretic Model for Differential CalculusDifferential Calculus

• Indefinite integral of a Scott continuous function• Derivative of a Scott continuous function• Fundamental Theorem of Calculus• Domain of C1 functions• (Domain of Ck functions)• Picard’s Theorem:

A data-type for differential equations

6

• (IR, ) is a bounded complete dcpo: ⊔iI ai = iI ai

• a ≪ b ao b• (IR, ⊑) is -continuous: Basis {[p,q] | p < q & p, q Q}• (IR, ⊑) is, thus, a continuous Scott

domain.• Scott topology has basis:

↟a = {b | a ≪ b}x {x}

R

I R

• x {x} : R IRTopological embedding

The Domain of nonempty compact Intervals of The Domain of nonempty compact Intervals of RR

7

Continuous FunctionsContinuous Functions

• f : [0,1] R, f C0[0,1], has continuous extension

If : [0,1] IR

x {f (x)}

• Scott continuous maps [0,1] IR with: f ⊑ g x R . f(x) ⊑ g(x)is another continuous Scott domain.

• : C0[0,1] ↪ ( [0,1] IR), with f Ifis a topological embedding into a proper subset of maximal elements of [0,1] IR .

8

Step FunctionsStep Functions

• a↘b : [0,1] IR, with a I[0,1], b IR:

b x ao x otherwise

• Finite lubs of consistent single step functions

⊔1in(ai ↘ bi)

with ai, bi rational intervals, give a basis for

[0,1] IR

9

Step Functions-An ExampleStep Functions-An Example

0 1

R

b1

a3

a2

a1

b3

b2

10

Refining the Step FunctionsRefining the Step Functions

0 1

R

b1

a3

a2

a1

b3

b2

11

Operations in Interval ArithmeticOperations in Interval Arithmetic

• For a = [a, a] IR, b = [b, b] IR,and * { +, –, } we have:

a * b = { x*y | x a, y b }

For example:• a + b = [ a + b, a + b]

12

• Intuitively, we expect f to satisfy:

• What is the indefinite integral of a single step function a↘b ?

The Basic ConstructionThe Basic Construction

• Classically, with }|{ RaaFf fF '

• We expect a↘b ([0,1] IR)

• For what f C1[0,1], should we have If a↘b ?

b(x)' fb .ax o

13

Interval DerivativeInterval Derivative

• Assume f C1[0,1], a I[0,1], b IR.

• Suppose x ao . b f (x) b.

• We think of [b, b] as an interval derivative for f at a.

• Note that x ao . b f (x) b

iff x1, x2 ao & x1 > x2 ,

b(x1 – x2) f(x1) – f(x2) b(x1 – x2), i.e.

b(x1 – x2) ⊑ {f(x1) – f(x2)} = {f(x1)} – {f(x2)}

14

Definition of Interval DerivativeDefinition of Interval Derivative

• f ([0,1] IR) has an interval derivativeb IR at a I[0,1] if x1, x2 ao,

b(x1 – x2) ⊑ f(x1) – f(x2).

• Proposition. For f: [0,1] IR, we have f (a,b)

iff f(x) Maximal (IR) for x ao , and Graph(f) is

within lines of slopeb & b at each point (x, f(x)), x ao.

(x, f(x))

b

b

a

Graph(f).

• The tie of a with b, is (a,b) := { f | x1,x2 ao. b(x1 – x2) ⊑ f(x1) – f(x2)}

15

Let f C1[0,1]; the following are equivalent: • If (a,b)x ao . b f (x) bx1,x2 [0,1], x1,x2 ao.

b(x1 – x2) ⊑ If (x1) – If (x2)

• a↘b ⊑ If

For Classical FunctionsFor Classical Functions

Thus, (a,b) is our candidate for a↘b .

16

(a1,b1) (a2,b2) iff a2 ⊑ a1 & b1 ⊑ b2

ni=1 (ai,bi) iff {ai↘bi | 1 i n}

consistent.

iI (ai,bi) iff {ai↘bi | iI }

consistent iff J finite I iJ (ai,bi)

• In fact, (a,b) behaves like a↘b; we call (a,b) a single-step tie.

Properties of TiesProperties of Ties

17

The Indefinite IntegralThe Indefinite Integral

: ([0,1] IR) (P([0,1] IR), ) ( P the power set)

a↘b := (a,b)

⊔i I ai ↘ bi := iI (ai,bi)

is well-defined and Scott continuous.• But unlike the classical case, is not 1-1.

18

ExampleExample

([0,1/2] {0})↘ ([1/2,1] {0}) ([0,1] [0,1]) ↘ ↘⊔ ⊔=

([0,1/2] , {0}) ([1/2,1] {0}) ↘ ([0,1] [0,1]) ↘

=

([0,1] , {0}) =

[0,1] {0}↘

19

The Derivative OperatorThe Derivative Operator

• : (I[0,1] IR) (I[0,1] IR)

is monotone but not continuous. Note that the classical operator is not continuous either.

• (a↘b)= x .

• is not linear! For f : x x : I[0,1] IR g : x –x : I[0,1] IR

(f+g) + = x . (1 – 1) = x . 0dx

d

dx

d

dx

df f

dx

d

dx

df

dx

dg

dx

d

20

The DerivativeThe Derivative

• Definition. Given f : [0,1] IR the derivative of f is:

: [0,1] IR

= ⊔ {a↘b | f (a,b) }dx

dfdx

df

• Theorem. (Compare with the classical case.)

• is well–defined & Scott continuous.dx

df

'f Idx

If d

dx

df•If f C1[0,1], then • f (a,b) iff a↘b ⊑

21

ExamplesExamples

0 ]1,1[

0 fI x

IRR:dx

If d

RR:)sin(:f 12

x

x

xxx

0

0 fI x

IRR:dx

If d

RR:)sin(:f 1

x

x

xxx

|| xx

x

x

x

xx

xx

0 {1}

0 ]1,1[

0 x}1{

x

IRR:dx

If d

IRR:|}{|:If

RR|:|:f

22

Domain of Ties, or Indefinite Integrals Domain of Ties, or Indefinite Integrals

• Recall : ([0,1] IR) (P([0,1] IR), )

• Let T[0,1] = Image ( ), i.e. T[0,1] iff

x is the nonempty intersection of a family of single ties:

= iI (ai,bi)

• Domain of ties: ( T[0,1] , )

• Theorem. ( T[0,1] , ) is a continuous Scott domain.

23

• Define : (T[0,1] , ) ([0,1] IR)

∆ ⊓ { | f ∆ }

dx

d

dx

df

The Fundamental Theorem of CalculusThe Fundamental Theorem of Calculus

• Theorem. : (T[0,1] , ) ([0,1] IR)

is upper adjoint to : ([0,1] IR) (T[0,1] , )

In fact, Id = ° and Id ⊑ ° dx

d

dx

d

dx

d

24

Fundamental Theorem of CalculusFundamental Theorem of Calculus

• For f, g C1[0,1], let f ~ g if f = g + r, for some r R.

• We have:

x.{f(x)}

f

R}c|cg(x)}.{{

g

x

~]1,0[1C ]1,0[0C

x

dx

d≡

IR]1,0[ T[0,1]

dx

d

25

F.T. of Calculus: Isomorphic versionF.T. of Calculus: Isomorphic version

• For f , g [0,1] IR, let f ≈ g if f = g a.e.

• We then have:

x.{f(x)}

f

R}c|cg(x)}.{{

g

x

~]1,0[1C ]1,0[0C

x

dx

d≡

IR)/]1,0([T[0,1]

dx

d≡

26

A Domain for A Domain for CC11 Functions Functions

• If h C1[0,1] , then ( Ih , Ih ) ([0,1] IR) ([0,1] IR)

• What pairs ( f, g) ([0,1] IR)2 approximate a differentiable function?

• We can approximate ( Ih, Ih ) in ([0,1] IR)2

i.e. ( f, g) ⊑ ( Ih ,Ih ) with f ⊑ Ih and g ⊑ Ih

27

• Proposition (f,g) Cons iff there is a continuous h: dom(g) R

with f Ih ⊑ and g ⊑ .

dx

Ih d

Function and Derivative ConsistencyFunction and Derivative Consistency

• Define the consistency relation:Cons ([0,1] IR) ([0,1] IR) with(f,g) Cons if (f) ( g)

• In fact, if (f,g) Cons, there are always a least and a greatest functions h with the above properties.

28

Approximating function: f = ⊔i ai↘bi

• (⊔i ai↘bi, ⊔j cj↘dj) Cons is a finitary property:

Consistency for basis elementsConsistency for basis elements

L(f,g) = least function

G(f,g)= greatest function

• (f,g) Cons iff L(f,g) G(f,g) . Cons is decidable on the basis.• Up(f,g) := (fg , g) where fg : t [ L(f,g)(t) , G(f,g)(t) ]

fg(t)

t

Approximating derivative: g = ⊔j cj↘dj

29

• Lemma. Cons ([0,1] IR)2 is Scott closed.

• Theorem.D1 [0,1]:= { (f,g) ([0,1]IR)2 | (f,g) Cons}is a continuous Scott domain, which can be given an effective structure.

The Domain of The Domain of CC11 FunctionsFunctions

• Define D1c := {(f0,f1) C1C0 | f0 = f1 }

• Theorem. : C1[0,1] C0[0,1] ([0,1] IR)2

restricts to give a topological embedding D1

c ↪ D1

(with C1 norm) (with Scott topology)

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Higher Interval DerivativeHigher Interval Derivative

• Proposition. For f C2[0,1], the following are equivalent: • If 2(a,b)x a0. b f (x) bx1,x2 a0. b (x1 – x2) ⊑ If (x1) – If (x2)

• a↘b ⊑ If

• Let 1(a,b) = (a,b)

• Definition. (the second tie) f 2(a,b) P([0,1] IR) if 1(a,b)

• Note the recursive definition, which can be extended to higher derivatives.

dx

df

31

Higher Derivative and Indefinite Higher Derivative and Indefinite Integral Integral

• For f : [0,1] IR we define:

: [0,1] IR by

• Then = ⊔f 2(a,b) a↘b

: ([0,1] IR) (P([0,1] IR), ) a↘b := (a,b)

⊔i I ai ↘ bi := iI (ai,bi)

is well-defined and Scott continuous.

2

2

dx

fd

dx

df

dx

d

dx

fd2

2

2

2

dx

fd

2(2)

(2)

(2)

(2)2

32

Domains of Domains of C C 22 functionsfunctions

• D2c := {(f0,f1,f2) C2C1C0 | f0 = f1, f1 = f2}

• Theorem. restricts to give a topological embedding D2

c ↪ D2

• Define Cons (f0,f1,f2) iff f0 f1 f2 (2)

Theorem. Cons (f0,f1,f2) is decidable on basis elements.

(The present algorithm to check is NP-hard.)

• D2 := { (f0,f1,f2) (I[0,1]IR)3 | Cons (f0,f1,f2) }

33

Domains of Domains of C C kk functionsfunctions

• Dk := { (fi)0ik (I[0,1]IR)k+1 | Cons (fi)0ik }

• D := { (fk)k0 ( I[0,1]IR)ω | k0. (fi)0ik Dk }∞

(i)• Let (fi)0ik (I[0,1]IR)k+1

Define Cons (fi)0ik iff 0ik fi

• The decidability of Cons on basis elements for k 3 is an

open question.

34

• Theorem. There exists a neighbourhood of t0 where there is a unique solution, the fixed point of:

P: C0 [t0-k , t0+k] C0 [t0-k , t0+k]

f t . (x0 + F(t , f(t) ) dt)

for some k>0 .

t0

t

Picard’s TheoremPicard’s Theorem

• = F(t,x) with F: R2 R

x(t0) = x0 with (t0,x0) R2

where F is Lipschitz in x uniformly in t for some neighbourhood of (t0,x0).

dt

dx

35

• Up⃘�ApF: (f,g) (t . (x0 + g dt , t . F(t,f(t)))

has a fixed point (f,g) with f = g = t . F(t,f(t))

t

t0

Picard’s Solution ReformulatedPicard’s Solution Reformulated

• Up: (f,g) ( t . (x0 + g(t) dt) , g )t

t0

• P: f t . (x0 + F(t , f(t)) dt)

can be considered as upgrading the information about the function f and the information about its derivative g.

t

t0

• ApF: (f,g) (f , t. F(t,f(t)))

36

• We now have the basic framework to obtain Picard’s theorem with domain theory.

• However, we have to make sure that derivative updating preserves consistency.

• Say (f , g) is strongly consistent, (f , g) S-Cons, if h ⊒ g. (f , h) Cons

• On basis elements, strong consistency is decidable.

A domain-theoretic Picard’s theoremA domain-theoretic Picard’s theorem

37

A domain-theoretic Picard’s theoremA domain-theoretic Picard’s theorem

• Let F : [0,1] IR IR and

ApF : ([0,1] IR)2 ([0,1] IR)2

(f,g) ( f , F (. , f ) )

Up : ([0,1] IR)2 ([0,1] IR)2 Up(f,g) = (fg , g) where fg (t) = [ L (f,g) (t) , G (f,g) (t) ]

• Consider any initial value f [0,1] IR with

(f, F (. , f ) ) S-Cons

• Then the continuous map P = Up � ApF has a least fixed point above (f, F (. , f ))

• Theorem. If F = Ih for a map h : [0,1] R R which satisfies the Lipschitz property of Picard’s theorem, then the domain-theoretic solution coincides with the classical solution.

38

ExampleExample

1

f

g

1

1

1

F

F is approximated by a sequence of step functions, F1, F2, …

F = ⊔i Fi

We solve: = F(t,x), x(t0) =x0

for t [0,1] with

F(t,x) = t and t0=1/2, x0=9/8.

dt

dx

a3

b3

a2

b2

a1

b1

F3

F2

F1

The initial condition is approximated by rectangles aibi:

{(1/2,9/8)} = ⊔i aibi,

t

t

.

39

SolutionSolution

1

f

g

1

1

1

At each stage we find Li and Gi

.

40

SolutionSolution

1

f

g

1

1

1 .

At each stage we find Li and Gi

41

SolutionSolution

1

f

g

1

1

1 Li and Gi tend to

the exact solution:f: t t2/2 + 1

.

At each stage we find Li and Gi

42

Further WorkFurther Work

• Solving Differential Equations with Domains

• Differential Calculus with Several Variables

• Implicit and Inverse Function Theorems

• Reconstruct Geometry and Smooth Mathematics with Domain Theory

• Continuous processes, robotics,…

43

THE ENDTHE END

http://www.doc.ic.ac.uk/~aehttp://www.doc.ic.ac.uk/~ae

44

45

Higher Interval DerivativeHigher Interval Derivative

• Proposition. For f C2[0,1], the following are equivalent: • If 2(a,b)x a0. b f (x) bx1,x2 ≫ a. b (x1 – x2) ⊑ If (x1) – If (x2)

• a↘b ⊑ If

• Let 1(a,b) = (a,b)

• Definition. (the second tie) f 2(a,b) P(I[0,1] IR) if 1(a,b)

• Note the recursive definition, which can be extended to higher derivatives.

dx

df

46

Higher Interval DerivativeHigher Interval Derivative

• For f : I[0,1] IR we define:

: I[0,1] IR by

• Then = ⊔f 2(a,b) a↘b

: (I[0,1] IR) (P(I[0,1] IR), ) a↘b := (a,b)

2

2

dx

fd

dx

df

dx

d

dx

fd2

2

2

2

dx

fd

2(2)

(2)

47

Domains of Domains of C C 22and and C C kk functionsfunctions

• D2c := {(f0,f1,f2) C2C1C0 | f0 = f1, f1 = f2}

• Theorem. restricts to give a topological embedding D2

c ↪ D2

• Dk := { (fi)0ik (I[0,1]IR)k+1 | 0ik fi }(i)

• D := { (fk)k0 ( I[0,1]IR) | k0. fk Dk }∞

• D2 := { (f0,f1,f2) (I[0,1]IR)3 | f0 f1 f2 }(2)

48

Consistency Test for Consistency Test for (f,g)(f,g)

yxduug

xyduug

yxdx

y

x

y

)(

)(

),(

yxduug

xyduug

yxdx

y

x

y

)(

)(

),(

• Also define: L(x) := supyODom(f)(f –(y) + d–+(x,y)) and G(x) := infyODom(f)(f +(y) + d+–(x,y))

• For x Dom(g), let g({x})=[g (x),g+(x)] where g ,g+: Dom(g) R are semi-continuous functions.

Similarly we define f , f+: Dom(f) R. • Let O be a connected component of Dom(g) with

O Dom(f) . For x , y O define:

49

• Theorem. (f, g) Con iff x O. L(x) G(x). For (f, g) = (⊔1in ai↘bi, ⊔1jm cj↘dj)

the rational end–points of ai and cj induce a partition X = {x0 < x1 < x2 < … < xk} of O.

• Proposition. For arbitrary x O, there isp, where 0 p k, with: L(x) = f –(xp) + d–+(x,xp).

• Similarly for G(x).

Consistency TestConsistency Test