Abbas Edalat Imperial College London ae Interval Derivative of Functions

Abbas Edalat

Imperial College London

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

Interval Derivative of FunctionsInterval Derivative of Functions

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


The Classical DerivativeThe Classical Derivative

• Let f: [a,b] R be a real-valued function.

The derivative of f at x is defined as

yx

f(y)f(x)lim(x)' f xy

when the limit exists (Cauchy 1821).

• If the derivative exists at x then f is continuous at x.

• However, a continuous function may not be differentiable at a point x and there are indeed continuous functions which are nowhere differentiable, the first constructed by Weierstrass:

) x π(a cosb (x) f n

0n

n

with 0 <b< 1 and a an odd positive integer.

3

Non Continuity of the DerivativeNon Continuity of the Derivative

• The derivative of f may exist in a neighbourhood O of x but the function

R O :' f

may be discontinuous at x, e.g.

12 xsin x x: f with f(0)=0

we have:

(0)' f (x) ' f lim

0)(x xcossin x x 2(x)' f

0(0)' f

0x

11

4

A Continuous Derivative for Functions?A Continuous Derivative for Functions?

• A computable function needs to be continuous with respect to the topology used for approximation.

• Can we define a notion of a derivative for real valued functions which is continuous with respect to a reasonable topology for these functions?

5

Dini’s Four Derivates of a Function (1890)Dini’s Four Derivates of a Function (1890)

yx

f(y)f(x)lim: (f)(x)D

yx

f(y)f(x)lim : (f)(x)D

yx


yx


xy

xy

xyu

xyu

l

l

• Upper right derivate at x

• Upper left derivate at x

• Lower right derivate at x

• Lower left derivate at x

• Clearly, f is differentiable at x iff its four derivates are equal, the common value will then be the derivative of f at x.

6

ExampleExample

2 )0(f)(D

1 (f)(0)D

2 (f)(0)D

1 (f)(0)Du

u

l

l

x0 sin xx

0 x 0

0 xsin x2x

x:f1

1

y0 ysin

0y ysin 2

y

f(y)

y0

f(y)f(0)1

1

7

Interval DerivativeInterval Derivative

• Put

) (f)(x)D , (f)(x)D (min yx


) (f)(x)D , (f)(x)D (max yx

f(y)f(x)lim : (f)(x)D

xy

uuxy

u

lll

• The interval derivative of f: [c,d] R is defined as

] (f)(x)Dlim , (f)(x)Dlim [

dx

df uxyxy

l

• Let IR={ [a,b] | a, b R} {R} and consider (IR, ) with R as bottom.

IR d][c, :dx

df

if both limits are finiteotherwise

8

ExampleExample12 xsin x x: f with f(0)=0

0)(x xcossin x x 2(x)' f

0(0)' f11

• We have already seen that

• We have 0)(x (x)' f (f)(x)D (f)(x)Du l

1,1][ ] (x)' flim , (x)' flim [

] (f)(x)Dlim , (f)(x)Dlim [ )0(dx

df

0x0x

u0x0x

l

• Thus

9

Interval-Limit of FunctionsInterval-Limit of Functions

• Let and be any extended real-valued function.RA:f RA

• ThenRA:f lim RA:f lim

• The interval limit of f is defined as

otherwise f(x)] lim , f(x) lim [

at x finitenot is f limor f lim if

IRA:lim(f)-int

x

10

ExamplesExamples

0 xif 1

0 xif 1 x

R(0,1]1,0)[:f

0 xif 1,1][

0 xif {1}

0 xif 1}{

x

IR1,1][:lim(f)-int

1sin x x

R)(0,,0)(:g

0 xif 1,1][

0 xif }{sin x x

IRR :lim(g)-int1

11

Interval-limit of Interval-valued FunctionsInterval-limit of Interval-valued Functions

• Let and be extended real-valued functions

with .

• Consider the interval-valued function

RA

otherwise (x)]f lim , (x)f lim [

at x finitenot are f limor f lim

IRA:lim(f)-int

x

IRA:]f,[ff

otherwise

finite are (x)f , (x)f if (x)]f(x),[f f(x)

• The interval-limit of f is now defined as

RA:f,f

ff

12

Continuity of the Interval DerivativeContinuity of the Interval Derivative

• Corollary. The interval derivative of f: [c,d] R

]) (f)D , (f)D [ (lim-intdx

df ul

is Scott continuous.

IR d][c, :dx

df

IRA:lim(f)-int

• Theorem. Given any

RA:]f,[ffor RA:f -

the interval limit is Scott continuous.

13

Computational Content of the Interval DerivativeComputational Content of the Interval Derivative

• Definition. (AE/AL in LICS’02) We say f: [c,d] R has interval Lipschitz constant in an open interval if

The set of all functions with interval Lipschitz constant b at a is called the tie of a with b and is denoted by .

]b , b[b d][c,a

y)(xb f(y)f(x) y)(xb x.y,ayx,

b)δ(a,

• Theorem. For f: [c,d] R we have:

b)}δ(a,f& ay|{b y)(dx

df thus, b)}δ(a,f|bsup{a

dx

df

• Recall the definition of single-step function

otherwise

a xif b x with IRd][c,:ba

14

Fundamental Theorems of CalculusFundamental Theorems of Calculus

• Continuous function versus continuously differentiable function

x

c

x

c

F(c)F(x) (t)dt F'

f(x) f(t)dt)'( for continuous f

for continuously differentiable F

• Lebesgue integrable function versus absolutely continuous function

x

c

x

c

F(c)F(x) (t)dt F'

a.e. f(x) f(t)dt)'(

for any Lebesgue integrable f

iff F is absolutely continuous

15

Locally Lipschitz functions Locally Lipschitz functions

• A map f: (c,d) R is locally Lipschitz if it is Lipschitz in a neighbourhood of each .d)(c,y

(y)dx

df d).(c,y withIRd)(c,:

dx

df

• The interval derivative induces a duality between locally Lipschitz maps versus bounded integral functions and their interval limits.

• The interval derivative of a locally Lipschitz map is never bottom:

• A locally Lipschitz map f is differentiable a.e. and

x

c f(c)f(x) (t)dt ' f

16

Primitive of a Scott Continuous MapPrimitive of a Scott Continuous Map

• Given Scott continuous is there

IR[0,1]:]g,[gg

R[0,1]:f

with

gdx

df

• In other words, does every Scott continuous function has a primitive with respect to the interval derivative?

• For example, is there a function f with ?]1,0[]1,0[dx

df

17

Total Splittings of IntervalsTotal Splittings of Intervals

• A total splitting of [0,1] is given by disjoint measurable subsets A, B with such that for any interval

we have:

BA[0,1]

q)(p [0,1]q][p,

0 B)q]λ([p, & 0 A)q]λ([p,

where is the Lebesgue measure.λ

• It follows that A and B are both dense with empty interior.

• Non-example: d][c, A'B & d][c, QA

18

Construction of a Total SplittingConstruction of a Total Splitting

• Construct a fat Cantor set in [0,1] with

0 1

• In the open intervals in the complement of construct

countably many Cantor sets with

00C

00C

8/1)Cλ(0i 0i

1/4)λ(C00

0)(i C0i

• In the open intervals in the complement of construct

Cantor sets with .

0i 0iC

0)(i C1i 16/1)Cλ(0i 1i

• Continue to construct with 0i jiC

j2

0i ji 1/2)Cλ(

• Put 1/2(B)(A) with [0,1] A'B & CA

0ji, ji

19

Primitive of a Scott Continuous Function Primitive of a Scott Continuous Function

• To construct with for a given

IR[0,1]:]g,[gg

• Take any total splitting (A,B) of [0,1] and put

gdx

df R[0,1]:f

B xif (x)g

A xif (x)g f(x)

• Theorem. g

dx

df

20

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

22

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

23

• 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

24

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 .

25

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

26

Step Functions-An ExampleStep Functions-An Example

0 1

R

b1

a3

a2

a1

b3

b2

27

Refining the Step FunctionsRefining the Step Functions

0 1

R

b1

a3

a2

a1

b3

b2

28

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]

29

• 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

30

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)}

31

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)}

32

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 .

33

(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

34

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.

35

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}↘

36

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 ⊑

37

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

38

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|} : [0,1] IR g : x {–|x|} : [0,1] IR

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

d

dx

d

dx

df f

dx

d

dx

df

dx

dg

dx

d

39

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.

40

• 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

41

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

42

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≡

43

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

44

• 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) .

45

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

46

f

1

1

Function and Derivative Information Function and Derivative Information

g

1

2

47

f

1

1

UpdatingUpdating

g

1

2

48

• 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 +].

49

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).

50

f

1

1

Updating Algorithm Updating Algorithm

g

1

2

51

f

1

1

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

g

1

2

52

f

1

1

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

g

1

2

53

f

1

1

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

g

1

2

54

f

1

1

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

g

1

2

55

f

1

1

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

g

1

2

56

f

1

1

Output of the Updating Algorithm Output of the Updating Algorithm

g

1

2

57

• 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)

58

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

59

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

60

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) }

61

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.

62

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.

63

• 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

64

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

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(t) dt) , 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)))

65

• 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 +(y) ]

• Thus, on basis elements strong consistency is decidable.

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

66

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) ) )

67

• 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 ]

68

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

69

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

.

70

SolutionSolution

1

f

g

1

1

1

At stage n we find un

- and un +

.

71

SolutionSolution

1

f

g

1

1

1 .


- and un +

72

SolutionSolution

1

f

g

1

1

1 un - and un

+ tend to

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

.


- and un +

73

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.

75

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!

76

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

77

The Convex Hull AlgorithmThe Convex Hull Algorithm

A, B & C nearly collinear A

B

C

With floating point we can get:

78

A, B & C nearly collinear


A

B

C


(i) AC, or

79



A

B

C


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

80



A

B

C


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

81



(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

82

• 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.

83

• 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

84

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.

85

Intersection of two 3D cubesIntersection of two 3D cubes

86


87


88

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.

89

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.

90

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.

91

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.

92

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

93

• 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.

94

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.

95

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

96

Subset InclusionSubset Inclusion

otherwise.

)),(),,((

},{:

21

12

2121 BAff

XBAtt

BBAA

ffttSXXSb

• Subset inclusion is Scott continuous.

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

97

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.

98

• (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 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) ) )

99

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.

100

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.

101

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.

102

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

• d(C,D) = min{ r | 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

103

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.

104

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(

105

Lebesgue ComputabilityLebesgue Computability

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

exists an effective chain K ß(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.

106

• 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

107

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.

108

• 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

109

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

110

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

111

The Outer Convex Hull AlgorithmThe Outer Convex Hull Algorithm

112

The Inner Convex Hull AlgorithmThe Inner Convex Hull Algorithm

Top right cornersTop left corners

bottom right cornersbottom left corners

113


114


115

• 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) | y(R2)m, yiTi, 1 i m}

Im(x):= {Hm (y) | 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.

116

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.

117

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.

118

• 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.

119

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

120

PPBs for Three Partial PointsPPBs for Three Partial Points

Partial Centre

121

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.

122


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

123


• 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 which is m log m on average.

124

Current and Further WorkCurrent and Further Work

• Let IR={ [a,b] | a, b R} {R}

• (IR, ) equipped with the Scott topology is a continuous Scott domain with R as bottom.

125

THE ENDTHE END

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


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