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)

30

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