Transcript A New Foundation for Computational Geometry and Solid

Domain Theory, Computational Geometry
and Differential Calculus
Abbas Edalat
Imperial College London
www.doc.ic.ac.uk/~ae
With contributions from Andre Lieutier, Ali Khanban,
Marko Krznaric and Dirk Pattinson
First SQU Workshop on Topology and its Applications
December 2004
First rudimentary notion of a real number
• In his ``Commentaries on the Difficulties in the Postulates of
Euclid's Elements'', Omar Khayyam, the 11th century Persian
mathematician and poet, first showed the equivalence of Euclid's notion of
ratios with that of continued fractions.
• Then, in a stroke of genius, he defined two ratios as equal
``when they can be expressed by the ratio of integer numbers with as
great a degree of accuracy as we like.''
• Three centuries later, Ghiasseddin Jamshid Kashani, another Persian
mathematician, devised the first fixed point technique for computation in
analysis in the beginning of the 15th century:
He used a cubic polynomial in a recursive scheme to approximate thesine
of 1° correctly up to 17 decimal places
2
Computational Model for Classical Spaces
• Research project since 1993: Reconstruct mathematical
analysis in an order-theoretic setting
• Embed classical spaces into the set of maximal elements of
suitable partially ordered sets, called domains:
x
{x}
X
Classical Space
DX
Domain
3
Computational Model for Classical Spaces
• Applications:
Fractal Geometry
Measure & Integration Theory: Generalized Riemann Integration
Topological Representation of Spaces
Exact Real Arithmetic
Computational Geometry and Solid Modelling
Differential Calculus
Quantum Computation
4
Continuous Scott Domains
• A directed complete partial order (dcpo) is a poset (A, ⊑) , in
which every directed set {ai | iI }  A has a sup or lub supiI ai
• The way-below relation in a dcpo is defined by:
a ≪ b iff for all directed subsets {ai | iI }, the relation
b ⊑ supiI 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
• A dcpo is bounded complete if every bounded subset has a lub
• A continuous Scott Domain is an -continuous bounded complete
dcpo
5
Continuous functions
• The Scott topology of a dcpo has as closed subsets
downward closed subsets that are closed under the
lub of directed subsets, usually only T0.
• Proposition. If D is a dcpo then f:D D is Scott continuous iff
• (i) f is monotone, i.e. f(x) ⊑ f(y) if x ⊑ y, and
• (ii) f preserves sups of directed subsets, i.e. for any directed set
AD, we have: supaA f(a) = f(supA)
• Fact. The Scott topology on a continuous dcpo A
with basis B has basic open sets {a  A | b ≪ a } for
each b  B.
6
The Domain of nonempty compact
Intervals of R
• Let IR={ [a,b] | a, b  R}  {R}
• (IR, ) is a bounded complete dcpo with R as bottom:
supiI ai = iI 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  {x} : R  IR an embedding
• We identify {x} with x
{x}
R
IR
7
Continuous Functions
• Scott continuous f:[0,1]n  IR is given by lower and upper semicontinuous functions f -, f +:[0,1]n  R with f(x)=[f -(x),f +(x)]
• f : [0,1]n  R, f  C0[0,1]n, has continuous extension
i(f ): [0,1]n  IR
x
 {f (x)}
• Scott continuous maps [0,1]n  IR with:
f ⊑ g  x  R . f(x) ⊑ g(x)
is another continuous Scott domain.
• The function space [0,1]n  IR is a continuous Scott domain.
 : C0[0,1]n ↪ ( [0,1]n  IR), with f  i(f)
is a topological embedding into a proper subset of maximal
elements of [0,1]n IR .
• We identify x and {x}, also f and i(f)
8
Step Functions
• Single-step function:
a↘b : [0,1]n  IR, with a  I[0,1]n, b=[b-,b+]  IR:
x 
b
x  ao

otherwise
• Lubs of finite and bounded collections of single- step
functions
f= sup1in(ai ↘ bi)
are called step functions with N(f):= n (number of pieces)
• Step functions with ai, bi rational intervals give a basis for
[0,1]n  IR. They are used to approximate C0 functions.
9
Step Functions-An Example in R
R
b3
a3
b1
b2
a1
a2
0
1
10
Refining the Step Functions
R
b3
a3
b1
a1
b2
a2
0
1
11
Kleene’s Fixed Point Theorem
• Theorem. If D is a dcpo with bottom (least element) d0
and if f : D  D is Scott continuous then f has a least
fixed point given by supiN f i(d0)
12
Initial Value Problems in Domain
•
Consider initial value problems given by
y  ( y1 ,...., y n )  v( y1 ,...., yn )
y(0)  0
where y=(y1,…,yn): [-a,a]Rn and v: [-K,K]n[-M,M]n are continuous..
•
ODE solving:
•
•
•
•
Mathematical Analysis: no direct computational content
Numerical Analysis: no correctness guarantee
Interval Analysis: no convergence guarantee
Computable Analysis: no data types
•
1.
2.
3.
Domain Theory provides a method which
is based on proper data types
has guaranteed speed of convergence
is directly implementable on a digital computer
13
Picard's Classical Theorem
• Standard assumption: v: [-K, K]n  [-M, M]n and aM ≤K.
• Theorem: Suppose v is Lipschitz: |v(x) - v(y)| ≤ L |x – y|
for some L > 0 and all x, y [-K, K]n. then, there exists a unique
y: [-a, a]  [-K, K]n with
y i  vi (y1 ,....,yn ) for i  1,.....,n
• Proof: For y: [-a, a]  [-K, K]n let Pv(y): [-a, a]  [-K, K]n with
x
(Pv (y))i : x   vi (y1 ,...,y n )dt
o
By Banach's fixed point theorem, the sequence yk defined by
y0 = any initial continuous function, and yk+1 = Pv(yk) converges to
the unique fixed point.
14
Translation to Domain Theory: Road map
•
•
•
•
•
Formulate the problem as fixed point equation over
[-a,a]I[-K,K]n
Apply Kleene's Theorem to obtain the least fixed point
Relate domain-theoretic fixed points to the classical problem
Show that the computations restrict to bases of the involved
domains.
Furthermore, we give:
(i) Estimates on the speed of convergence
(ii) Estimates on the (algebraic) complexity of the problem.
15
Translation to Domain Theory: Solutions
• Interval valued approximations to the solution:
• S = { y: [-a, a]  I[-K, K]n | y Scott continuous}
•
where
I[-K, K]n = {A | A [-K, K]n is a compact rectangle}
is the sub-domain of rectangles contained in [-K, K]n with reverse inclusion.
• The order on S is pointwise:
•
If f, g S then f ⊑ g iff f(x) ⊑ g(x) for all x  [-a, a].
16
Translation into Domain Theory: Vector Field
• Computing v(y) requires vector fields with interval input:
•
V= { u: I[-K, K]n  I[-M, M]n | u Scott continuous } with pointwise
ordering
• Relationship to Classical Vector Fields: Extensions
• u V extends a classical v: [-K, K]n  [-M, M]n if
•
u({x}) = { v(x) } for all x  [-K, K]n.
• The greatest or canonical extension of v is given by
Iv: I[-K, K]n  I[-M, M]n with ((Iv)(A))i =vi[A]
17
The Domain Theoretic Picard Operator
• Let u V.
• The domain-theoretic Picard operator Pu:SS is given by
x
(Pu (y))i : x   vi (y1 ,...,y n )dt
o
where for f = [f- -, f +]: [-a, a]  I[-K, K]
[ x f  (t)dt, x f  (t)dt] if 0  x
0
 0
x
if x  0
0 f(t)dt  0 x 
[ f (t)dt, x f  (t)dt] if x  0
0
 0
• The monotone convergence theorem shows that Pu is Scott continuous.
18
Relationship to the Classical Problem
• Assume u is an extension of v: [-K, K]n  [-M, M]n.
• Suppose y is the least fixed point of Pu
• Every solution f satisfies y ⊑ f
• If y = [y-, y+] and y- = y+, then y- is the unique solution.
• That is, we look for fixed points of width 0:
• For (A1, …..,An)  I[-K, K]n, we write
Ai = [Ai-, Ai+] and define the width of A by
• w(A)=max{Ai+-Ai- | 1≤i≤n}
• w(f)=sup{w(f(x))| x[-a,a]}
19
The Lipschitz Case
• Recall: The classical proof assumes that v is Lipschitz.
• Suppose u is interval Lipschitz, that is
•
w(u(x)) ≤L w(x) for all x I[-K, K]n.
• Then w(Pu(y)) ≤ a L w(y) for all y S.
• Assuming aL < 1 (which can be actually removed), we obtain
• Theorem:
y0 = [-K, K]n and yk+1 = Pu(yk). Then
y = sup kN yk satisfies Pu(y) = y and w(y) = 0.
• In particular, y - = y+ is the unique solution.
20
Approximations of the Vector Field
• Proposition: The map P: V→(S→S) with u→ Pu is Scott continuous.
• This allows us to use approximations of the vector field to obtain the
solution:
• Theorem: Suppose u = supk uk and y0 = [-K, K]n and
yk 1  Pu k (yk )
. Then y = sup k N yk satisfies Pu(y) =y and w(y) = 0.
Speed of Convergence:
• If aL < c < 1 and d(u, uk) < 2Mck (c - aL), then w(yk) ≤ck w(y0), where
d(u, u k )  sup{d(u(x)), u k (x))| x  I[-K,K]n } with theHausdorff distance:




d(A, B)  max{|A i  Bi |, | A i  Bi |: 1  i  n} for A, B  IR n
21
Restrinting to a Base
• Let a,K,MQ.
• Denote the class of rational step functions I[-K,K]n→I[-M,M]n by VQ.
• We also have the class SL Q of rational piecewise linear step functions
f:[-a,a]→I[-K,K]n
We again put N(f) = number of pieces in f
In the above example N(f)=3
22
Computing with SLQ and VQ
• Pu restricts to the base:
• Suppose u VQ and y  SLQ then:
•
The map u(y) is piecewise constant, hence Pu(y) SLQ Pu(y) SLQ .
• Pu(y) can be computed in time O(N(u)N(y)), i.e.
•
N(Pu(y))  O(N(u)N(y)).
23
Summary
• Suppose u = supk uk with uk VQ , y0= [-K, K]n and yk 1  Pu (yk )
then
k
•
yk SLQ for all k  N
• y = supk yk has width 0 and is the unique solution
•
w(yk) O(ck) if d(u, uk)  O(ck).
• Given an elementary function u, one can use continued fractions for to obtain
uk with the above properties..
• These properties are preserved in an implementation using rational arithmetic.
24
PART II: A Domain-Theoretic Model for
Differential Calculus
• Overall Aim:
Synthesize Computer Science with Differential Calculus
• Plan of the talk:
1. Primitives of continuous interval-valued function in Rn
2. Derivative of a continuous function in Rn
3. Fundamental Theorem of Calculus for interval-valued
functions in Rn
4. Domain of C1 functions in Rn
5. Inverse and implicit functions in domain theory
25
Operations 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+]
• Recall that for real x, we identify {x} with x.
26
Basic Construction n=1
• What is a primitive map of a single step function a↘b ?
• Classically,
f
 {F  a | a  R}
with
F' f
• We expect  a↘b  ([0,1]  IR)
• For what f  C1[0,1], should we have If   a↘b ?
• f should satisfy:

x  a . b  f ' (x)  b


27
Interval Lipschitz contant
• 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 Lipschitz constant
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)}
28
Definition of Interval Lipschitz constant
• f  ([0,1]  IR) has an interval Lipschitz constant
b  IR at a  I[0,1] if x1, x2  ao,
b(x1 – x2) ⊑ f(x1) – f(x2).
• The tie of a with b, is
(a,b) := { f | 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)
+
b
and Graph(f) is
(x, f(x))
Graph(f)
within lines of slope
+
b
b & b at each point
(x, f(x)), x  ao.
a
.
29
For Classical Functions
Let f  C1[0,1]; the following are equivalent:
• f  (a,b)
• x  ao . b-  f ′(x)  b+
• x1,x2 [0,1], x1,x2  ao.
b(x1 – x2) ⊑ f (x1) – f (x2)
• a↘b ⊑ f ′
Thus, (a,b) is our candidate for  a↘b .
30
Set of primitive maps
•  : ([0,1]  IR)  (P([0,1]  IR),  )
( P the power set constructor)
•  a↘b := (a,b)
•  ⊔i I ai ↘ bi := iI (ai,bi)
•  is well-defined and Scott continuous.
•  f is always a non-empty set.
31
The Derivative
• Definition. Given f : [0,1]  IR the derivative of f
is:
df
: [0,1]  IR
dx
df
dx
= ⊔ {a↘b | f  (a,b) }
• Theorem. (Compare with the classical case.)
df
• is well–defined & Scott continuous.
dx
• If f 
then
df
• f  (a,b) iff a↘b ⊑ dx
C1[0,1],
df
f '
dx
32
f : x | x |: R  R
Examples
x | x |
If : x  {| x |} : R  IR
d If
: R  IR
dx
{1} x  0

x [1,1] x  0
{1}
0 x

x
f : x  x 2 sin(x 1 ) : R  R
f : x  x sin(x 1 ) : R  R
d If
: R  IR
dx
x0
If 
x 
[ 1,1] x  0
d If
: R  IR
dx
If 
x 

x0
x0
33
Fundamental Theorem of Calculus
•
f g
iff g
⊑ df
(interval version)
dx
• If g
C0
df
then f g iff g =
dx
(classical version)
34
Idea of Domain for C1 Functions
• If h C1[0,1] , then
( h , h′ )  ([0,1]  IR)  ([0,1]  IR)
• We can approximate ( h, h′ ) in ([0,1]  IR)2
i.e. ( f, g) ⊑ ( h ,h′ ) with f ⊑ h and g ⊑h′
• What pairs ( f, g)  ([0,1]  IR)2 approximate a
differentiable function?
35
Function and Derivative Consistency
• Define the consistency relation:
Cons  ([0,1]  IR)  ([0,1] 
IR) with
(f,g)  Cons if (f)  ( g)  
• Proposition
(f,g)  Cons iff there is a continuous h: dom(g)  R
with f ⊑ h and g ⊑ dh .
dx
• 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) .
36
Consistency Test for (f,g)
• 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 +].
• Let O be a connected component of dom(g) with
O  dom(f)  . For x , y  O, let:
x
 

f (y)  g (u)du
y


S(f,g) (x, y)  
x


f (y)  g  (u)du

y


y  x




x  y


• Define the upper envelop: s(f,g)(x) := supyOdom(f) S(f,g)(x,y)
• Proposition. s(f,g) is the least function with
f-
ds(f, g)
 s(f,g) and g ⊑ dx
37
Similarly, let:
x
 


f (y)  g (u)du y  x 
y




T(f,g) (x, y)  

x




f (y)  g (u)du x  y 


y
• The lower envelop: t(f,g)(x) := inf yOdom(f) (f +(y) + T (f,g) (x,y))
•
Proposition. t(f,g) is the greatest function with
t(f,g)  f+ and g ⊑ dt(f, g)
dx
• Theorem. (f, g)  Con iff for all component O of dom(g) with
O dom(f)   and all x  O. s (f, g) (x)  t (f, g) (x).
38
Consistency for basis elements
• (⊔i ai↘bi, ⊔j cj↘dj)  Cons is a finitary property:
t(f,g)= greatest function
Approximating function:
f=
⊔i ai↘bi
s(f,g) = least function
Approximating derivative: g = ⊔j cj↘dj
39
Function and Derivative Information
f
1
1
g
2
1
40
f
Least and greatest functions
1
1
g
2
1
41
Decidability of consistency
•
For (f, g) = (⊔1in ai↘bi , ⊔1jm cj↘dj),
the rational end–points of ai and cj induce a partition
y0 < y1 < y2 < … < yk of the connected component O of dom(g).
• Proposition. For x  O, we have:
s(f,g)(x) = max {f –(x) , limsup f –(y) + S(f,g) (x , y) | ym  O  dom(f) }
y  ym
• Hence s(f,g) is the max of k+2 rational linear maps.
•
Similarly for t(f,g)(x).
• Thus, s(f,g) t(f,g) is decidable.
• Theorem. Consistency is decidable.
42
The Domain of C1 Functions
• 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.
• Theorem.
 : C0[0,1]  D1 [0,1]
f
 (f , df ) is topological embedding, giving a
dx
computational model for continuous functions and their differential
properties. It restricts to give an embedding of C1[0,1].
43
Functions of several varibales
• (IR)1× n row n-vectors with entries in IR
• For dcpo A, let
(An)s = smash product of n copies of A:
x(An)s if x=(x1,…..xn) with xi non-bottom
or x=bottom
• We work with (IR1× n)s
44
Definition of Interval Lipschitz constant
• f  ([0,1]n  IR) has an interval Lipschitz constant
b  (IR1xn)s in a  I[0,1]n if x, y  ao,
b(x – y) ⊑ f(x) – f(y).
• The tie of a with b, is
(a,b) := { f | x,y  ao. b(x – y) ⊑ f(x) – f(y)}
• Proposition. If f(a,b), then f(x)  Maximal (IR) for x  ao and
for all x,y  ao. |f(x)-f(y)| k ||x-y|| with k=max i (|bi+|, |bi-|)
45
For Classical Functions
Let f  C1[0,1]n; the following are equivalent:
• f  (a,b)
• x  ao . b-  f ´(x)  b+
•  x,y  ao.
b(x – y) ⊑ f (x) – f (y)
• a↘b ⊑ f ′
Thus, (a,b) is our candidate for  a↘b .
46
Set of primitive maps
•  : ([0,1]n  IR)  (P([0,1]n  (IR1xn)s),  )
( P the power set constructor)
•  a↘b := (a,b)
•  supi I ai ↘ bi := iI (ai,bi)
•  is well-defined and Scott continuous.
•  g can be the empty set for 2  n
Eg. g=(g1,g2), with g1(x , y)= y , g2(x, ,y)=0
47
The Derivative
• Definition. Given f : [0,1]n  IR the derivative of f
is:
df
n  (IR1xn)
:
[0,1]
s
dx
df
dx
=sup {a↘b | f  (a,b) }
• Theorem. (Compare with the classical case.)
df
• is well–defined & Scott continuous.
dx
• If f 
then
df
• f  (a,b) iff a↘b ⊑ dx
C1[0,1],
d If
 If'
dx
48
Fundamental Theorem of Calculus
•
df
f g iff g ⊑
dx
• If g
C0
df
then f g iff g =
dx
(interval version)
(classical version)
49
Relation with Clarke’s gradient
For a locally Lipschitz f : [0,1]n  IR
∂ f (x)=convex{ lim mf ´(xm) | x mx}
It is a non-empty compact convex subset of Rn
Theorem:
For locally Lipschitz f : [0,1]n  IR, the following holds:
df
The domain-theoretic derivative at x , i.e. dx is the smallest
n-dimensional rectangle with sides parallel to the coordinate
planes that contains ∂ f (x)
• In dimension one the two coincide.
•
•
•
•
•
•
50
Idea of Domain for C1 Functions
• If h C1[0,1]n , then
( h , h′ )  ([0,1]n  IR)  ([0,1]n  IR)ns
• We can approximate ( h, h´ ) in
([0,1]n  IR)  ([0,1]n  IR)ns
i.e. ( f, g) ⊑ ( h ,h´ ) with f ⊑ h and g ⊑h´
• What pairs ( f, g)  ([0,1]n  IR)  ([0,1]n  IR)ns
approximate a differentiable function?
51
Function and Derivative Consistency
• Define the consistency relation:
Cons  ([0,1]n  IR)  ([0,1]n  IR)ns with
(f,g)  Cons if (f)  ( g)  
• Proposition
(f,g)  Cons iff there is a continuous h: dom(g)  R
with f ⊑ h and g ⊑ dh .
dx
• 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) .
52
Basis elements
• Definition. g:[0,1]n  (IRn)s the domain of g is
dom(g) = {x | g(x) non-bottom}
• Basis element: (f, g1,g2,….,gn)  ([0,1]n IR)  ([0,1]n IR)ns
• Each f, gi :[0,1]n  IR is a rational step function.
• Recall: the intersection of a closed and an open set is a crescent.
• dom(g) is partitioned by disjoint crescents in each of which g is a constant
rational interval. .
Eg. For n=2:
A step function gi with four single step
functions with two horizontal and two
vertical rectangles as their domains
and a hole inside, and with eight vertices.
[-2,2]
[-3,3]
[-1,1]
[-2,2]
53
Corners and their coaxial points
[-2,2]
[-3,3]
[-1,1]
[-2,2]
• Step function
• corners
• their coaxial points
• vertices=corners  coaxial points
54
Decidability of Consistency
• (f,g)  Cons if (f)  ( g)  
• First we check if g is integrable, i.e. if  g  
• In classical calculus, g:[0,1]n  Rn will be integrable
by Green’s theorem iff for any piecewise smooth closed
non-intersecting path
• p:[0,1] [0,1]n with p(0)=p(1)
1
 g(p(t)) p'(t)dt  0
0
• We generalize this to the type g:[0,1]n  (IRn)s
55
Interval-valued path integral
• For vIRn , uRn define the interval-valued scalar product
v  u : {w  u | w  v}  [(v  u) - , (v  u)  ] with
n
(v  u)   v i
-
i 1
σ(-u i )
n
u i and (v  u)   v i

i 1
σ(u i )
ui
where σ(r)  sign(r) {-,0,}, v  [v , v  ] and v0  1.
1
 gdt :  g(p(t)) p'(t)dt,i.e.
0
p
 gdt  [L gdt , U  gdt ]
p
p
1
with
p
1
L  gdt   (g(p(t)) p'(t)) dt and U gdt   (g(p(t)) p'(t)) dt
-
0
p
0
p
56
Generalized Green’s Theorem
• Theorem.  g   iff for all piecewise smooth non-intersecting
path p:[0,1] dom(g) with p(0)=p(1), we have zero-containment:
1
0   g(p(t)) p'(t)dt
0
• For step functions, we can and will replace “smooth” with “linear”.
• Then, the lower and upper path integrals will depend piecewise
linearly on the coordinates of nodes the path, so their extreme
values will be reached when these nodes are at the corners of
dom(g) or their coaxial points.
• Since there are finitely many of these extreme paths, zero
containment can be decided in finite time for all paths.
57
Minimal surface for g
• Step function g:[0,1]n  (IRn)s with  g   . Let O be a component of dom(g)
. Let x,ycl(O).
• Consider the following supremum over all piecewise linear paths p in cl(O) with
p(0 )= y and p(1)= x.
1
Vg (x, y)  sup p {L  g(p(t)) p'(t)dt}
0
• The above supremum is attanied.
• For fixed y, the map Vg (. ,y): cl(O) R is a rational piecewise linear function.
• It is the least continuous function or surface with:
• Vg (y ,y)=0
and g ⊑
dVg (., y )
dx
58
Maximal surface for g
• Step function g:[0,1]n  (IRn)s . Let O be a component of dom(g) . Let
x,ycl(O).
• The following infimum is attained over all piecewise linear paths p in cl(O) with
p(0 )= y and p(1)= x.
1
Wg (x, y)  infp {U  g(p(t)) p'(t)dt}
0
• For fixed y, the map Wg (. ,y): cl(O) R is a rational piecewise linear function.
• It is the greatest continuous function or surface with:
• Wg (y ,y)=0
and g ⊑
dWg (., y )
dx
59
Minimal surface for (f,g)
• (f,g)([0,1]n  IR)  ([0,1]n  IR)ns rational step function
• Assume we have determined that  g  
• Put
S(f,g) (x,y)  Vg (x,y)  liminf f  (y) and
s(f,g)(x)  sup {S(f,g) (x, y) | y  O x }
with O x thecomponentof x in dom(g)
• Proposition.
s(f,g)(x)  max{S(f,g) (x,y) | y  vertices(Ox )}
Theorem. s(f,g): dom(g) R is the least continuous function with
ds(f, g)
dx
f -  s(f,g) and g ⊑
60
Maximal surface for (f,g)
T(f,g) (x,y)  Wg (x,y)  limsup f  (y)
and
t(f,g)(x)  inf {T(f,g) (x,y) | y Ox }
We have: t(f,g)(x)  min{T(f,g) (x,y) | y  vertices(Ox )}
Theorem. t(f,g): dom(g) R is the least continuous function with
t(f,g)  f + and g ⊑ dt(f, g)
dx
Theorem. Consistency is decidable.
Proof: In s(f,g)t(f,g) we compare two rational piecewise-linear
surfaces, which is decidable.
61
The Domain of C1 Functions
• Lemma. Cons  ([0,1]n  IR) ([0,1]n  IR)ns is Scott closed.
• Theorem.
D1 [0,1]n:= { (f,g) | (f,g)  Cons} is a continuous Scott domain that
can be given an effective structure.
• Theorem.
 : C0[0,1]n  D1 [0,1]n
df
f
 (f , ) is topological embedding, and so is its
dx
1
restriction to C [0,1]n , giving a computational model for
continuous functions and their differential properties.
62
Example
f
We solve:
dx
= v(t,x), x(t0) =x0
dt
for t [0,1] with
v(t,x) = t and t0=1/2, x0=9/8.
b1 b2 b3
1
v is approximated by a
sequence of step
functions, v0, v1, …
.
aa3
2
a1
1
g
1
t
The initial condition is
approximated by
rectangles aibi:
v1
v2
v3
v = ⊔i vi
{(1/2,9/8)} =
v
1
t
⊔i aibi,
63
Solution
f
At stage n we find
un - and un +
1
.
1
g
1
1
64
Solution
f
At stage n we find
un - and un +
1
.
1
g
1
1
65
Solution
f
At stage n we find
un - and un +
1
.
1
g
1
1
un - and un + tend to
the exact solution:
f: t  t2/2 + 1
66
Computing with polynomial step functions
67
Some Open Problems
• Topologically, what sort of a set is (a,b) ?
• Topologically, what sort of a set is C0 in the set of maximal
elements of D0=[0,1]IR? Similarly, for C1 as a subset of
maximal elements of D1.
• For a domain of twice differentiable functions, is consistency
decidable? (It is true for n=1.)
• Decidability of consistency for 3rd and higher derivatives (not
known even for n=1).
• Construct a domain for analytic functions.
• For solving PDE’s, obtain a domain-theoretic version of the
finite difference and finite element methods.
68
Part III:
A Domain-Theoretic Model of Geometry
• To develop a model for Computational Geometry
and Solid Modelling, so that:
•
the model is mathematically sound, realistic;
•
the basic building blocks are computable;
•
it bridges theory and practice.
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!
70
The Intersection of two lines
L1
L2
P
With floating point arithmetic, find the point P of the
intersection L1  L2. Then:
min_dist(P, L1) > 0,
min_dist(P, L2) > 0.
71
The Convex Hull Algorithm
A, B & C nearly collinear
A
With floating point we
can get:
B
C
72
The Convex Hull Algorithm
A, B & C nearly collinear
A
With floating point we
can get:
(i) AC, or
B
C
73
The Convex Hull Algorithm
A, B & C nearly collinear
A
With floating point we
can get:
(i) AC, or
(ii) just AB, or
B
C
74
The Convex Hull Algorithm
A, B & C nearly collinear
A
With floating point we
can get:
(i) AC, or
(ii) just AB, or
(iii) just BC, or
B
C
75
The Convex Hull Algorithm
A, B & C nearly collinear
A
With floating point we
can get:
(i) AC, or
(ii) just AB, or
(iii) just BC, or
(iv) none of them.
B
C
The quest for robust algorithms is the most fundamental
unresolved problem in solid modelling and computational
geometry.
76
A Fundamental Problem in Topology and
Geometry
• Subset A  X topological space.
Membership predicate A : X  {tt, ff }
 tt
x
 ff
x A
x A
is continuous iff A is both open and closed.
• In particular, for A  Rn, A  , A  Rn
A : Rn  {tt, ff } is not continuous.
• Most engineering is done, however, in Rn.
77
Non-computability of the
Membership Predicate
• There is discontinuity at the
boundary of the set.
True
False
x
• This predicate is not computable:
If x is on the boundary, we cannot
in general determine if it is in or
out at any finite stage of
computation.
x
78
Non-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.
79
Intersection of two 3D cubes
80
Intersection of two 3D cubes
81
Intersection of two 3D cubes
82
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.
83
Computable Topology and 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 nonobservable, like x = 0.
open
open
•
• Redefine: A : X {tt, ff}
tt

x   ff


x  IntA
tt
ff
observable
observable
x  IntAc
x  A
with the Scott topology on {tt, ff}.
• A is now a continuous function.

Non-observable
84
Elements 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.
85
Partial 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.
86
The Geometric (Solid) Domain of X
• 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:
S(X )

f

( f 1{tt}, f 1{ ff })

(O1 , O2 )
O1  O2  
tt

x   ff


x  O1
x  O2
otherwise
SO ( X )
87
Properties of the Geometric (Solid) Domain
• Theorem For a second countable locally compact Hausdorff space X
(e.g. Rn), S(X) is bounded complete and –continuous (i.e. a
continuous Scott domain) 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 O1c.
This is a regular solid object: O1=int O1 & O2=int O2
• Definition
(O1, O2) , O1 ≠ ∅ ≠ O2 , is a classical object if O1 ∪ O2 = X.
88
Examples
• A = {xR2 ⃒ |x| ≤ 1}  [1, 2]
represented in the model by
Arep = (int A, int Ac) =
( {x ⃒ |x| < 1}, R2 \ A )
is a classical (but non-regular) solid object.
• B = {xR2 ⃒ |x| ≤ 1}
represented by Brep=
({x ⃒ |x| < 1} , {x ⃒ |x| > 1})  Maximal (SR2, ⊑)
is a regular solid object with Arep ⊑ Brep.
89
Boolean operations and predicates
 : SX  SX
(O1 , O2 )  (O2 , O1 )
 : SX  SX  SX
(( A1 , A2 ) , ( B1 , B2 ))  ( A1  B1 , A2  B2 )
 : SX  SX  SX
(( A1 , A2 ) , ( B1 , B2 ))  ( A1  B1 , A2  B2 )
Theorem All these operations are Scott continuous and preserve
classical solid objects.
90
Subset Inclusion
• Let Sb X  {( A1, A2 )  SX | A2 compact}{(, )}.
c
 : Sb X  SX  {tt , ff }
tt

(( A1 , A2 ), ( B1 , B2 ))   ff


A2  B1  X
A1  B2  
otherwise.
• Subset inclusion is Scott continuous.
91
General 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.
92
An 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) is a computable partial solid object if there exists a total
recursive function ß:NN such that : ( K ß(n) ) n 0 is an
increasing chain with:
(A , B) = supnK ß(n) =( ∪n π1 ( K ß(n) ) , ∪n π2 ( K ß(n) ) )
93
Computing 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.
94
Computable 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.
95
Quantative Measure of Convergence
• In our present model for computable solids,
we require a quantitative measure for the
convergence of the basis elements to a
computable solid.
• We will enrich the notion of domain-theoretic
computability in two different ways to include a
quantitative measure of convergence.
96
Hausdorff Computability
• 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 :NN 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
97
Hausdorff 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.
98
Boolean Intersection is not Hausdorff
computable
(1,1)
(0,1)
rn rat ional,rn  r
left comput able.
Q   Qn
Qn
n0
is Hausdorff
computable.
1
2
1
2 n1
1
2n
(0,0)
r0 r1
rn 1 rn r
(1,0)
However:
Q([0,1]  {0})
= [r,1]  {0}  R2
is not Hausdorff
computable.
99
Lebesgue Computability
• (A , B)  S [–k, k]d is Lebesgue computable iff there
exists an effective chain Kß(n) of basis elements with
ß :NN a total recursive function such that:
(A , B) = ( ∪n π1 ( K ß(n) ) ,
µ(A) - µ(π1 ( K ß(n) ) ) < 1/2 n
&
∪n π2 ( K ß(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.
100
• Hausdorff computable ⇏ Lebesgue computable
Complement of a Cantor set with Lebesgue measure 1– r
with r =lim rn: left computable but non-computable real.
sn  rn  rn1 ,
• start with
n
s
m 0
m
 rn ,

s
m 0
m
r
0
1
• stage 1
s0
• stage 2
s1
2
s1
2
• At stage n remove 2n open mid-intervals of length sn/2n.
101
Hausdorff and Lebesgue computability
• Lebesgue computable ⇏ Hausdorff computable
Let 0 < rn  Q with rn ↗ r, left computable, noncomputable 0 < r < 1.
• Then [r,1]  {0}  R2 is Lebesgue computable but not
Hausdorff computable.
0
r0
rn
r
1
102
Hausdorff and Lebesgue Computable
Objects
• 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.
103
Summary
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
104
Data-types for Computational Geometry
and Systems of Linear Equations
• The Convex Hull
• Voronoi Diagram or the Post Office problem
• Delaunay Triangulation
• The Partial Circle through three partial points
105
The Outer Convex Hull Algorithm
106
The Inner Convex Hull Algorithm
Top left corners
bottom left corners
Top right corners
bottom right corners
107
The Convex Hull Algorithm
108
The Convex Hull Algorithm
109
The Convex Hull map
• 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.
3
2
T
T
• Each rectangle TIR2 has vertices T1,T2,T3,T4,
going anti-clockwise from the right bottom corner.
• For
x=(T1,T2,…,Tm)(IR2)m,
define:
T4
T1
Cm : (IR2)m  SR2,
Cm(x) = (Im(x),Em(x)) with:
Em(x):={(Hm (y))c | y(R2)m, yiTi, 1  i  m}
Im(x):= {(Hm (y))0 | y(R2)m, yiTi, 1  i  m}
110
The Convex Hull is Computable!
• Proposition: Em(x)=(H4m((Ti1,Ti2,Ti3,Ti4))1im)c
Im(x)=Int({Hm((Tin))1im) | 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.
111
Voronoi 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.
112
Voronoi Diagram & Partial Circles
• 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.
.
.
. .
. .
• 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.
113
Partial Perpendicular Bisector of Two
Partial Points
114
PPBs for Three Partial Points
Partial Centre
115
Partial 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.
The exterior is the union of the exteriors of the three red circles.
The Interior is the intersection of the interiors of the three blue circles.
116
Partial Circles
With more exact partial points, the boundaries of the interior and exterior of
the partial circle get closer to each other.
117
Partial Circles
• In the limit, the area between the interior and exterior of the
partial circle, and the Hausdorff distance between their boundaries,
tends to zero.
• We get a Scott continuous map C : (IR2)3SR2
• We obtain a robust Voronoi algorithm.
118
Part IV
Inverse and Implicit Function Theorems
Domain of Manifolds
• The mains tool in multi-variable calculus.
• In CAD curves and surfaces are built from the implicit
function theorem, f(x,y,z) = 0 for a C1 map f: R3 R
• A domain-theoretic framework can be
the basis of a robust CAD.
• Implicit function theorem follows from
the inverse function theorem, both
classically and domain-theoretically.
119
Calculus of operations in D1
• For a vector function f = (fi)i := (fi)1≤i≤n we write
fD1 ([0,1]n  Rn) if fiD1 ([0,1]n  R) for 1 ≤i ≤ n
• The following operations are well-defined and Scott continuous:
•
-  - : D1 ([0,1]n  IRn)× D1 ([0,1]n  IRn)  D1 ([0,1]n  IRn)
(f, g)
 fg
with
(f  g)i = ( fi0  gi0 , fi1  gi1 ,…….., fin  gin )
•
- ◦ - : D1 (Rn  IRn)× D1 ([0,1]n  IRn)  D1 ([0,1]n  IRn)
(f, g)
 f◦g
(f  g) i  (Ifi0 (g j0 ) j , k 1 Ifik (g j0 ) j  g k1 ,.....,k 1 Ifik (g j0 ) j  g kn )
n
n
• This extends the higher dimensional chain rule.
120
Inverse Constructor (function part)
• Idea: First find the inverse of a map which differs from the
identity map by a contraction
• Let A=[-a,a]n and B=[-ca,ca]n for a> 0 and 0< c ≤1/2
• V: (A→IB)×(B→IB)→(B→IB)
•
(f,g)
→ - If ◦ (id+g) , where id is the identity map.
• Theorem. Suppose f:[-a,a]n→Rn has Lipschitz constant nc<1 wrt
the max norm and f(0)=0. Then
• V(f , .): (B→IB)→(B→IB) has a unique fixed point g satisfying
g - = g+ and g(0)=0
• id+f : [-a,a]n→Rn has inverse id+g
In fact: g = - f ◦ (id+g) implies (id +f)◦(id+g) = id+ g – g = id
121
Inverse Constructor (derivative part)
• Recall A=[-a,a]n and B=[-ca,ca]n for a> 0 and 0< c ≤1/2
• The classical functional (f, g)→ - f ◦ (id+g): C1×C1→C1
is extended to:
• T: D1(A → IB) × D1( B → IB ) → D1( B→IB )
(f , g ) → ( V((fi0)i , (gi0)i ) , S ((fij)ij ,(gi0)i, (gij)ij) )
with
S: (A → IB) × (B → IB) × (B→ IRn) → (B → IRn)
(h , u ,w) → - ( Ih ◦ (id +u) )  (x.id + w)
122
Invesre Constructor (derivative part)
• Suppose f:[-a,a]n→Rn has Lipschitz constant nc < 1 wrt the max
norm and f(0)=0. Then the functional
df
T((f,
), - ) : D1 (B  IB)  D1 (B  IB)
dx
has a unique fixed point ((gi0)i,(gij)ij) where (gi0)i is the unique
fixed point of V(f,-) and (gij)ij is the unique fixed point of
df
S( , (g i0 )i , - ) : (B  IB)  (B  IB)
dx
• If f '(x) exists for some x[-a,a]n, so does (gi0)i '(x) and we have:
(gi0)i ' (x) = (gij)ij (x)
123
Mean Differential
• Lemma u: [-a,a]n → Rn has Lipschitz constant nc<1 wrt the max
norm iff uiδ([-a,a]n ,[-c,c]) for 1 ≤i ≤ n
•
Definition. Given u:[-1,1]nRn , the mean differential at
x0 :[-1,1]n is the linear map represented by the matrix M with
1 du
du


M ij  [( (x 0 )) ij  ( (x 0 )) ij ]
2 dx
dx
• Proposition. Suppose the mean differential M of u:[-1,1]n  Rn
du
at 0 is invertible with k:=|| M-1 (0) - id || < 1/n. Then, for any c
dx
with k< c <1, there exists a>0 such that the map
f=M-1 u- id: [-a,a]n → Rn satisfies fiδ([-a,a]n ,[-c,c]) for 1 ≤i ≤ n;
thus the map f has Lipschitz constant nc<1 wrt the max norm
124
Inverse Function theorem
•
Theorem. Let u:[-1,1]n  Rn and suppose the mean differential
M of u at 0 is invertible with || M-1
du
(0) - id || < 1/n. Then:
dx
•
The map u has a Lipschitz inverse in a neighbourhood of 0 .
•
Given an increasing sequence of linear step functions converging
to u, we can effectively obtain an increasing sequence of linear
step functions converging to u-1
•
If furthermore u is C1 and given also an increasing sequence of
linear step functions converging to u' we can also effectively
obtain an increasing sequence of polynomial step functions
converging to (u-1)'
125
Current and Further Work
• A domain (w-continuous dcpo)
for Lipschitz manifolds with basis
given by rational piecewise linear
manifold (i.e., polyhedra).
• Construct the kernel of a robust CAD, based on
domain-theoretic approximations to curves and surfaces
obtained by the implicit function theorem.
126
َ‫شکرا‬
http://www.doc.ic.ac.uk/~ae