Domain-theoretic Models of Differential Calculus and Geometry Abbas Edalat Imperial College London ae BRICS, February 2003 Contributions

Domain-theoretic Models of Differential CalculusDomain-theoretic Models of Differential Calculus and Geometry and Geometry

Abbas Edalat

Imperial College London

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

BRICS, February 2003 Contributions from Andre Lieutier, Ali Khanban, Marko Krznaric

(I) Domain Theory and Differential Calculus(I) Domain Theory and Differential Calculus(II) Domain-theoretic Solution of Differential Equations(II) Domain-theoretic Solution of Differential Equations(III) Domain-theoretic Model of Geometry and Solid Modelling(III) Domain-theoretic Model of Geometry and Solid Modelling

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

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

• Topological Representation of Spaces

• Exact Real Arithmetic

Part (I)Part (I)Domain Theory and Differential Domain Theory and Differential

CalculusCalculus

Synthesize Differential Calculus developed by Newton and Leibnitz in the 17th century with Computer Science developed in the 20th century

5

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 for

interval-valued functions • Domain of C1 functions• Domain of Ck functions

7

Continuous Scott DomainsContinuous Scott Domains

• A directed complete partial order (dcpo) is a poset (A, ⊑) , in which every directed set {ai | iI } A has a sup or lub ⊔iI ai

• The way-below relation in a dcpo is defined by:

a ≪ b iff for all directed subsets {ai | iI }, the relation b ⊑⊔iI ai implies that there exists i I such that a ⊑ ai

• If a ≪ b then a gives a finitary approximation to b• B A is a basis if for each a A , {b B | b ≪ a } is directed

with lub a• A dcpo is (-)continuous if it has a (countable) basis• The Scott topology on a continuous dcpo A with basis B has basic

open sets {a A | b ≪ a } for each b B• A dcpo is bounded complete if every bounded subset has a lub • A continuous Scott Domain is an -continuous bounded complete

dcpo

8

• Let IR={ [a,b] | a, b R} {R}

• (IR, ) is a bounded complete dcpo with R as bottom: ⊔iI ai = iI ai

• a ≪ b ao b

• (IR, ⊑) is -continuous: countable basis {[p,q] | p < q & p, q Q}

• (IR, ⊑) is, thus, a continuous Scott domain.• Scott topology has basis:

↟a = {b | ao 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

9

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 .

10

Step FunctionsStep Functions

• Single-step function: a↘b : [0,1] IR, with a I[0,1], b IR:

b x ao x otherwise

• Lubs of finite and bounded collections of single- step functions

⊔1in(ai ↘ bi)

are called step function.

• Step functions with ai, bi rational intervals, give a basis for [0,1] IR

11

Step Functions-An ExampleStep Functions-An Example

0 1

R

b1

a3

a2

a1

b3

b2

12

Refining the Step FunctionsRefining the Step Functions

0 1

R

b1

a3

a2

a1

b3

b2

13

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]

14

• 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

15

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)}

16

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 (hence f continuous) 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)}

17

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 .

18

(a1,b1) (a2,b2) iff a2 ⊑ a1 & b1 ⊑ b2

ni=1 (ai,bi) iff {ai↘bi | 1 i n}

bounded.

iI (ai,bi) iff {ai↘bi | iI } bounded

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

19

The Indefinite IntegralThe Indefinite Integral

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

a↘b := (a,b)

⊔i I ai ↘ bi := iI (ai,bi)

is well-defined and Scott continuous.• But unlike the classical case, the

indefinite integral is not 1-1.

20

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}↘

21

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 ⊑

22

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

23

The Derivative OperatorThe Derivative Operator

• : ([0,1] IR) ([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|} : [-1,1] IR g : x {–|x|} : [-1,1] IR

(f+g) (0) (0) + (0) dx

d

dx

d

dx

df f

dx

d

dx

df

dx

dg

dx

d

24

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 it

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.

25

• 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

26

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

27

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≡

28

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

29

• 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 least and greatest functions h with the above properties in each connected component of dom(g) which intersects dom(f) .

30

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

• We will define L(f,g), G(f,g) in general and show that:1. (f,g) Cons iff L(f,g) G(f,g).

2. 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

31

f

1

1

Function and Derivative Information Function and Derivative Information

g

1

2

32

f

1

1

UpdatingUpdating

g

1

2

33

• Let O be a connected component of dom(g) with O dom(f) . For x , y O define:

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

yxduug

xyduug

yxdx

y

x

y

)(

)(

),(

• Define: L(f,g)(x) := supyOdom(f)(f –(y) + d–+(x,y)) and G(f,g)(x) := infyOdom(f)(f +(y) + d+–(x,y))

• Theorem. (f, g) Con iff x O. L (f, g) (x) G (f, g) (x).

yxduug

xyduug

yxdx

y

x

y

)(

)(

),(

• For x dom(g), let g(x) = [g (x), g+(x)] where g , g+: dom(g) R are lower and upper semi-continuous.

Similarly we define f , f +: dom(f) R. Write f = [f –, f +].

34

Updating Linear step FunctionsUpdating Linear step Functions

• A linear single-step function: a [↘ b–, b +] : [0,1] IR, with b–, b +: ao R linear

[b–(x) , b +(x)] x ao x otherwise

We write this simply as a b ↘ with b=[b–, b +] . x

b

ba

• Hence L(f,g) is the max of k+2 linear maps.

• Similarly for G(f,g)(x).

myy

• Proposition. For x O, we have: L(f,g)(x) = max {f –(x) , limsup f –(y) + d–+(x , y) | ym O dom(f) }

• For (f, g) = (⊔1in ai↘bi , ⊔1jm cj↘dj) with f linear g standard,

the rational end–points of ai and cj induce a partition y0 < y1 < y2 < … < yk of the connected component O of dom(g).

35

f

1

1

Updating Algorithm Updating Algorithm

g

1

2

36

f

1

1

Updating Algorithm (left to right)Updating Algorithm (left to right)

g

1

2

37

f

1

1

Updating Algorithm (left to right)Updating Algorithm (left to right)

g

1

2

38

f

1

1

Updating Algorithm (right to left)Updating Algorithm (right to left)

g

1

2

39

f

1

1

Updating Algorithm (right to left)Updating Algorithm (right to left)

g

1

2

40

f

1

1

Updating Algorithm (similarly for Updating Algorithm (similarly for upper one)upper one)

g

1

2

41

f

1

1

Output of the Updating Algorithm Output of the Updating Algorithm

g

1

2

42

• 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

D1c ↪ D1

(with C1 norm) (with Scott topology)

43

Higher Interval DerivativeHigher Interval Derivative• 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

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

• a↘b ⊑ If

44

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

45

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 seems to be NP-hard.)

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

46

Domains of Domains of C C kk functionsfunctions

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

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

(i)• Let (fi)0ik ([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.

47

Part (II)Part (II)Domain-theoretic Solution of Differential EquationsDomain-theoretic Solution of Differential Equations

• Develop proper data types for ordinary differential equations.

• Solve initial value problem up to any given precision.

48

• Theorem. In a neighbourhood of t0, there is a unique solution, which is the unique fixed point of:

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

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

for some k>0 .

t0

t

Picard’s TheoremPicard’s Theorem

• = v(t,x) with v: R2 R continuous

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

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

dt

dx

49

• Up⃘�Apv: (f,g) (t . (x0 + t . v(t,f(t))) dt , t. v(u,f(u)))

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

t

t0

Picard’s Solution ReformulatedPicard’s Solution Reformulated

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

t0

• P: f t . (x0 + v(t , f(t)) dt)

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

t

t0

• Apv: (f,g) (f , t. v(t,f(t)))

50

• To obtain Picard’s theorem with domain theory, we have to make sure that derivative updating preserves consistency.

• (f , g) is strongly consistent, (f , g) S-Cons, if h ⊒ g we have: (f , h) Cons

• Q(f,g)(x) := supyODom(f) (f –(y) + d+–(x,y))

R(f,g)x) := infyODom(f) (f +(y) + d–+(x,y))

• Theorem. If f –, f +, g–, g+: [0,1] R are bounded and g–, g+ are continuous a.e. (e.g. for polynomial step functions f and g), then (f,g) is strongly consistent iff for any connected component O of dom(g) with O dom(f) , we have: x O. Q(f,g)(x), R(f,g)(x) [f –(x) , f +(x) ]

• Thus, on basis elements strong consistency is decidable.

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

51

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

• Let v : [0,1] IR IR be Scott continuous and

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

(f,g) ( f , t. v (t , f(t) ))

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, t. v (t , f(t) ) ) S-Cons

• Then the continuous map Up � Apv has a least fixed point above (f, t.v (t , f(t))) given by

(fs, gs) = ⊔n 0 (Up � Apv )n (f, t.v (t , f(t) ) )

52

• Then (f , [-a,a ] ↘ [-M ,M ] ) S-Cons, hence (f, t. v(t , f(t) ) ) S-Cons since ([-a,a ] ↘ [-M ,M ]) ⊑ t. v (t , f(t) )

• Theorem. The domain-theoretic solution

(fs, gs) = ⊔n 0 (Up � Apv )n (f, t. v (t , f(t) ))

gives the unique classical solution through (0,0).

The Classical Initial Value ProblemThe Classical Initial Value Problem

• Suppose v = Ih for a continuous h : [-1,1] R R which satisfies the Lipschitz property around (t0,x0) =(0,0).

• Then h is bounded by M say in a compact rectangle K around the origin. We can choose positive a 1 such that [-a,a] [-Ma,Ma] K.

• Put f = ⊔n 0 fn where fn = [-a/2n,a /2n] ↘ [-Ma/2n , Ma/ 2n ]

53

Computation of the solution for a given precision Computation of the solution for a given precision >0

• Let (un , wn) := (Pv )n (fn , t. vn (t , fn(t) ) ) with un = [un

- ,un+]n

• We express f and v as lubs of step functions:

f = ⊔n 0 fn v = ⊔n 0 vn

• Putting Pv := Up � Apv the solution is obtained as:

• For all n 0 we have: un- un+1

- un+1+ un

+ with un+ - un

- 0

• Compute the piecewise linear maps un- , un

+ until

the first n 0 with un+ - un

-

• (fs, gs) = ⊔n 0 (Pv )n (f , t. v (t , f(t) ) ) = ⊔n 0 (Pv )

n (fn , t. vn (t , fn(t) ) ) n

54

ExampleExample

1

f

g

1

1

1

v

v is approximated by a sequence of step functions, v0, v1, …

v = ⊔i vi

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

for t [0,1] with

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

dt

dx

a3

b3

a2

b2

a1

b1

v3

v2

v1

The initial condition is approximated by rectangles aibi:

{(1/2,9/8)} = ⊔i aibi,

t

t

.

55

SolutionSolution

1

f

g

1

1

1

At stage n we find un

- and un +

.

56

SolutionSolution

1

f

g

1

1

1 .


- and un +

57

SolutionSolution

1

f

g

1

1

1 un - and un

+ tend to

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

.


- and un +

58

Computing with polynomial step functionsComputing with polynomial step functions

Part III: Part III: A Domain-Theoretic Model of A Domain-Theoretic Model of

GeometryGeometry

• To develop a Computable model for Geometry and Solid Modelling, so that:

• the model is mathematically sound, realistic;

• the basic building blocks are computable;

• it bridges theory and practice.

60

Why do we need a data type for solids?Why do we need a data type for solids?

Answer: To develop robust algorithms!

Lack of a proper data type and use of real RAM in which comparison of real numbers is decidable give unreliable programs in practice!

61

The Intersection of two linesThe Intersection of two lines

With floating point arithmetic, find the point P of the intersection L1 L2. Then: min_dist(P, L1) > 0, min_dist(P, L2) > 0.

1L

2LP

62

The Convex Hull AlgorithmThe Convex Hull Algorithm

A, B & C nearly collinear A

B

C

With floating point we can get:

63

A, B & C nearly collinear


A

B

C


(i) AC, or

64



A

B

C


(i) AC, or(ii) just AB, or

65



A

B

C


(i) AC, or(ii) just AB, or(iii) just BC, or

66



(i) AC, or(ii) just AB, or(iii) just BC, or(iv) none of them.

The quest for robust algorithms is the most fundamental unresolved problem in solid modelling and computational geometry.


A

B

C

67

• Subset A X topological space.

Membership predicate A : X {tt, ff }

is continuous iff A is both open and closed.

Axff

Axttx

A Fundamental Problem in Topology and GeometryA Fundamental Problem in Topology and Geometry

• In particular, for A Rn, A , A Rn

A : Rn {tt, ff } is not continuous.

• Most engineering is done, however, in Rn.

68

• There is discontinuity at the boundary of the set. x

Non-computability of the Membership PredicateNon-computability of the Membership Predicate

• This predicate is not computable:

x If x is on the boundary, we cannot in general determine if it is in or out at any finite stage of computation.

FalseTrue

69

Non-computable Operations in Classical CG & SMNon-computable Operations in Classical CG & SM

• A: Rn {tt, ff} not continuous means it is not computable, even for simple objects like A=[0,1]n.

• x A is not decidable even for simple objects: for A = [0,) R, we just have the undecidability of x 0.

• The Boolean operation is not continuous, hence noncomputable, wrt the natural notion of topology on subsets: : C(Rn) C(Rn) C(Rn), where C(Rn) is compact subsets with the Hausdorff metric.

70

Intersection of two 3D cubesIntersection of two 3D cubes

71


72


73

This is Really Ironical!This is Really Ironical!

• Topology and geometry have been developed to study continuous functions and transformations on spaces.

• The membership predicate and the binary operation for are the fundamental building blocks of topology and geometry.

• Yet, these fundamental functions are not continuous in classical topology and geometry.

74

Elements of a Computable Topology/GeometryElements of a Computable Topology/Geometry

• The membership predicate A : X {tt, ff} fails to be continuous on A, the boundary of A.

• For any open or closed set A, the predicate x A is non-observable, like x = 0.

• Redefine: A : X {tt, ff}

with the Scott topology on {tt, ff}.

Ax

IntAxff

IntAxtt

x c

tt ff

open

open

observable observable

Non-observable

A is now a continuous function.

75

Elements of a Computable Topology/GeometryElements of a Computable Topology/Geometry

• Note that A=B iff int A=int B & int Ac=int Bc,

i.e. sets with the same interior and exterior have the same membership predicate.

• We now change our view: In analogy with classical set theory where every set is completely determined by its membership predicate, we define a (partial) solid object to be given by any continuous map:

f : X { tt, ff }

• Then:f –1{tt} is open; it’s called the interior of the object. f –1{ff} is open; it’s called the exterior of the object.

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Partial Solid ObjectsPartial Solid Objects

• We have now introduced partial solid objects, since X \ (f –1{tt} f –1{ff}) may have non-empty interior.

• We partially order the continuous functions:f, g : X {tt, ff } f ⊑ g x X . f(x) ⊑ g(x)

• f ⊑ g f –1{tt} g –1{tt} & f –1{ff} g –1{ff}Therefore, f ⊑ g means g has more information about an idealized real solid object.

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The Geometric (Solid) Domain of The Geometric (Solid) Domain of XX

• The geometric (solid) domain S (X) of X is the poset (X {tt, ff }, ⊑ )

• S(X) is isomorphic to the poset SO(X) of pairs of disjoint

open sets (O1,O2) ordered componentwise by inclusion:

21

212

1

11

),(

otherwise

}){},{(

)( )(

OO

OOOxff

Oxtt

x

fffttff

XSXS O

78

• Theorem For a second countable locally compact Hausdorff space X (e.g. Rn), S(X) is bounded complete and –continuous with (U1, U2) <<

(V1, V2) iff the closures of U1 and U2 are compact subsets of V1 and V2

respectively.

• Theorem If X is Hausdorff, then: (O1, O2) Maximal (S X, ⊑) iff

O1= int O2c & O2= int O1

c.

This is a regular solid object: O1=int O1 & O2=int O2

Properties of the Geometric (Solid) DomainProperties of the Geometric (Solid) Domain

• Definition (O1, O2) , O1 ≠ ∅ ≠ O2 , is a classical object if O1 ∪ O2 = X.

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ExamplesExamples

• A = {xR2 |x| ≤ 1} ⃒� [1, 2]represented in the model byArep = (int A, int Ac) =

( {x | ⃒� x| < 1}, R2 \ A )is a classical (but non-regular) solid object.

• B = {xR2 |x| ≤ 1} ⃒�represented by Brep=

({x | ⃒� x| < 1} , {x | ⃒� x| > 1}) Maximal (SR2, ⊑)is a regular solid object with Arep ⊑ Brep.

80

Boolean operations and predicatesBoolean operations and predicates

Theorem All these operations are Scott continuous and preserve classical solid objects.

),()),( , ),((

:

22112121 BABABBAA

SXSXSX

),(),(

:

1221 OOOO

SXSX

),()),( , ),((

:

22112121 BABABBAA

SXSXSX

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Subset InclusionSubset Inclusion

otherwise.

)),(),,((

},{:

21

12

2121 BAff

XBAtt

BBAA

ffttSXXSb

• Subset inclusion is Scott continuous.

.)},{( compact} |),{( 221 cb ASXAAXS• Let

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General Minkowski operatorGeneral Minkowski operator

• For smoothing out sharp corners of objects.

• SbRn = { (A, B) SRn | Bc is bounded} {( , )}.∪ ∅ ∅

All real solids are represented in SbRn.

• Define: __ : SRn SbRn SRn

((A,B) , (C,D)) (↦ A ⊕ C , (Bc ⊕ Dc)c) where A ⊕ C = { a+c | a A, c C }

• Theorem __ is Scott continuous.

83

• (A, B) is a computable partial solid object if there exists a total recursive function ß:NN such that : ( K ß(n) ) n 0 is an increasing chain with:

An effectively given solid domainAn effectively given solid domain

• The geometric domain SX can be given effective structure for any locally compact second countable Hausdorff space, e.g. Rn, Sn, Tn, [0,1]n.

• Consider X=Rn. The set of pairs of disjoint open rational bounded polyhedra of the form K = (L1 , L2) , with L1 L2 = , gives a basis for SX.

• Let Kn= (π1 ( K n ) , π2 ( K n) ) be an enumeration of this basis.

(A , B) = ( ∪n π1 ( K ß(n) ) , ∪n π2 ( K ß(n) ) )

84

Computing a Solid ObjectComputing a Solid Object

• In this model, a solid object is represented by its interior and exterior.

• The interior and the exterior

are approximated by two

nested sequence of rational polyhedra.

85

Computable Operations on the Solid DomainComputable Operations on the Solid Domain

• F: (SX)n SX or F: (SX)n { tt, ff }

is computable if it takes computable sequences of partial solid objects to computable sequences.

• Theorem All the basic Boolean operations and predicates are computable wrt any effective enumeration of either the partial rational polyhedra or the partial dyadic voxel sets.

86

Quantative Measure of ConvergenceQuantative Measure of Convergence

• In our present model for computable solids, there is no quantitative measure for the convergence of the basis elements to a computable solid.

• We will enrich the notion of domain-theoretic computability to include a quantitative measure of convergence.

87

• We strengthen the notion of a computable solid by using the Hausdorff distance d between compact sets in Rn.

• d(C,D) = min{ r>0 | C Dr & D Cr } where Dr = { x | y D. |x-y| r }

• (A , B) S [–k, k]n is Hausdorff computable

if there exists an effective chain Kß(n) of basis elements with ß :NN a total recursive function such that:

(A , B) = ( ∪n π1 ( K ß(n) ) , ∪n π2 ( K ß(n) ) )

with

d (π1 ( K ß(n) ) , A ) < 1/2n , d (π2 ( K ß(n) ) , B) < 1/2n

Hausdorff ComputabilityHausdorff Computability

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Hausdorff computabilityHausdorff computability

• Two solid objects which have a small Hausdorff distance from each other are visually close.

• The Hausdorff distance gives a natural quantitative measure for approximation of solid objects.

• However, the intersection or union of two Hausdorff computable solid objects may fail to be Hausdorff computable.

• Examples of such failure are nontrivial to construct.

89

Boolean Intersection is not Hausdorff computableBoolean Intersection is not Hausdorff computable

nQ

0r 1r 1nr nr r

21

121n

n21

0

.computableleft

rational,

nn

nn

QQ

rrr

However:

Q([0,1] {0})= [r,1] {0} R2

is not Hausdorffcomputable.

is Hausdorff computable.

)0,0( )0,1(

)1,1()1,0(

90

Lebesgue ComputabilityLebesgue Computability

• (A , B) S [–k, k]d is Lebesgue computable iff there

exists an effective chain Kß(n) of basis elements with ß

:NN a total recursive function such that:

(A , B) = ( ∪n π1 ( K ß(n) ) , ∪n π2 ( K ß(n) ) )

µ(A) - µ(π1 ( K ß(n) ) ) < 1/2 n & µ(B) - µ(π2 ( K ß(n) ) ) < 1/2

n

• A computable function is Lebesgue computable if it

preserves Lebesgue computable sequences.

• Theorem Boolean operations are Lebesgue computable.

91

• Hausdorff computable ⇏ Lebesgue computableComplement of a Cantor set with Lebesgue measure 1– r with r =lim rn: left computable but non-computable real.

, ,0m

n

0m1 rsrsrrs mnmnnn

Hausdorff and Lebesgue computabilityHausdorff and Lebesgue computability

0 1

s0

• At stage n remove 2n open mid-intervals of length sn/2n.

• start with

• stage 1

• stage 2

21s

21s

92

Hausdorff and Lebesgue computabilityHausdorff and Lebesgue computability

• Lebesgue computable ⇏ Hausdorff computable

Let 0 < rn Q with rn ↗ r, left computable, non-computable 0 < r < 1.

0r nr r0 1

• Then [r,1] {0} R2 is Lebesgue computable but not Hausdorff computable.

93

• Hausdorff computable ⇏ Lebesgue computable

• Lebesgue computable ⇏ Hausdorff computable

• Theorem: A regular solid object is computable iff it is Hausdorff computable.

• However: A computable regular solid object may not be Lebesgue computable.

Hausdorff and Lebesgue Computable ObjectsHausdorff and Lebesgue Computable Objects

94

Conclusion Conclusion

Our model satisfies: A well-defined notion of computability Reflects the observable properties of geometric objects Is closed under basic operations Captures regular and non-regular sets Supports a methodology for designing robust

algorithms

95

Data-types for Computational Geometry and Systems of Data-types for Computational Geometry and Systems of Linear EquationsLinear Equations

• The Convex Hull

• Voronoi Diagram or the Post Office problem

• Delaunay Triangulation

• The Partial Circle through three partial points

96

The Outer Convex Hull AlgorithmThe Outer Convex Hull Algorithm

97

The Inner Convex Hull AlgorithmThe Inner Convex Hull Algorithm

Top right cornersTop left corners

bottom right cornersbottom left corners

98


99


100

• Let Hm: (R2)m C(R2) be the classical convex Hull map, with C(R2) the set of compact subsets of R2, with the Hausdorff metric.

• Let (IR2, ) be the domain of rectangles in R2.

The Convex Hull mapThe Convex Hull map

• For x=(T1,T2,…,Tm)(IR2)m, define:

Cm : (IR2)m SR2,

Cm(x) = (Im(x),Em(x)) with:

Em(x):={(Hm (y))c | y(R2)m, yiTi, 1 i m}

Im(x):= {(Hm (y))0 | y(R2)m, yiTi, 1 i m}

T1

T2T3

T4

• Each rectangle TIR2 has vertices T1,T2,T3,T4, going anti-clockwise from the right bottom corner.

101

The Convex Hull is Computable!The Convex Hull is Computable!

• Proposition: Em(x)=(H4m((Ti1,Ti

2,Ti3,Ti

4))1im)c

Im(x)=Int({Hm((Tin))1im) | n=1,2,3,4}).

• Theorem: The map Cm : (IR2)m SR2 is Scott continuous, Hausdorff and Lebesgue computable.

• Complexity:

1. Em(x) is O(m log m).

2. Im(x) is also O(m log m).

We have precisely the complexity of the classical convex hull algorithm in R2 and R3.

102

Voronoi DiagramsVoronoi Diagrams

• We are given a finite number of points in the plane.

• Divide the plane into components closest to these points.

• The problem is equivalent to the Delaunay triangulation of the points:

(1) Triangulate the set of given points so that the interior of the circumference circles do not contain any of the given points. (2) Draw the perpendicular bisectors of

the edges of the triangles.

103

• Recall that the Voronoi diagram is dual to the Delaunay triangulation: Given a finite number of points of the plane find the triples of points so that the interior of the circle through

any triple does not contain any other points. . .

.. .

.

Voronoi Diagram & Partial CirclesVoronoi Diagram & Partial Circles

• The centre of the circle through the three vertices of a triangle is the intersection of the perpendicular bisectors of the three edges of the triangle.

• The partial circle of three partial points in the plane is obtained by considering the Partial Perpendicular Bisector of two partial points in the plane.

104

Partial Perpendicular Bisector of Two Partial PointsPartial Perpendicular Bisector of Two Partial Points

105

PPBs for Three Partial PointsPPBs for Three Partial Points

Partial Centre

106

Partial CirclesPartial Circles

The Interior is the intersection of the interiors of the three blue circles.

The exterior is the union of the exteriors of the three red circles.

Each partial circle is defined by its interior and exterior. The exterior (interior) consists of all those points of the plane which are outside (inside) all circles passing through any three points in the three rectangles.

107


With more exact partial points, the boundaries of the interior and exterior of the partial circle get closer to each other.

108


• The limit of the area between the interior and exterior of the partial circle, and the Hausdorff distance between their boundaries, is zero.

• We get a Scott continuous map C : (IR2)3SR2

• We obtain a robust Voronoi algorithm.

109

Current and Further WorkCurrent and Further 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,…

110

THE ENDTHE END

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

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

Documents

Domain-theoretic Models of Differential Calculus and Geometry Abbas Edalat Imperial College London ae BRICS, February 2003 Contributions