Domain Theory and Multi-Variable
Calculus
Abbas Edalat
Imperial College London
www.doc.ic.ac.uk/~ae
Joint work with Andre Lieutier, Dirk Pattinson
Computational Model for Classical Spaces
• A research project since 1993:
Reconstruct basic mathematical analysis
• Embed classical spaces into the set of maximal
elements of suitable domains
x
{x}
X
Classical Space
DX
Domain
2
Computational Model for Classical Spaces
•
Other Applications:
Fractal Geometry
Measure & Integration Theory
Topological Representation of Spaces
Exact Real Arithmetic
Computational Geometry and Solid Modelling
Quantum Computation
3
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
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 ⊔i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 ⊑ ⊔i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.
• 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:
⊔i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
Topological embedding
{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
If : [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.
•
 : C0[0,1]n ↪ ( [0,1]n  IR), with f  If
is a topological embedding into a proper subset of maximal
elements of [0,1]n IR .
• We identify x and {x}, also f and If
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
⊔1in(ai ↘ bi)
are called step functions.
• 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
Interval Lipschitz constant in dimension one
• For f  ([0,1]  IR) we have:
iff for all x2x1
• x1, x2  ao, b(x1 – x2) ⊑ f(x1) – f(x2)
• b- (x1 – x2)  f(x1) – f(x2)  b+(x1 – x2) iff
• Graph(f) is within lines of
slope b- & b+ at each
point (x, f(x)), x  ao.
.
(x, f(x))
b+
Graph(f)
ba
12
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 for all i or
x=bottom
• Interval Lipschitz constants of real-valued functions in
Rn take values in (IR1× n)s
13
Interval Lipschitz constant in R
n
• 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-|)
14
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 .
15
Set of primitive maps
•  : ([0,1]n  IR)  (P([0,1]n  (IR1xn)s),  )
( P the power set constructor)
•  a↘b := (a,b)
•  ⊔i 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
16
The Derivative
• Definition. Given f : [0,1]n  IR the derivative of f
is:
df
n  (IR1xn)
:
[0,1]
s
dx
df
dx
= ⊔ {a↘b | f  (a,b) }
• Theorem. (Compare with the classical case.)
df
• is well–defined & Scott continuous.
dx
• If f 
C1[0,1],
then
• f  (a,b) iff a↘b ⊑
df
dx
df
f '
dx
17
f : x | x |: R  R
Examples
If : x  {| x |} : R  IR
d If
: R  IR
dx
{1} x  0

x  [ 1,1] x  0
{1}
0 x

x | x |
x
f : x  x 2 sin( x 1 ) : R  R
d If
: R  IR
dx
x0
If 
x 
[ 1,1] x  0
18
Relation with Clarke’s gradient
For a locally Lipschitz f : [0,1]n  R
∂ f (x) := convex-hull{ lim mf ´(xm) | x mx}
It is a non-empty compact convex subset of Rn
Theorem:
For locally Lipschitz f : [0,1]n  R
The domain-theoretic derivative at x is the smallest
n-dimensional rectangle with sides parallel to the
coordinate planes that contains ∂ f (x)
• In dimension one, the two notions coincide.
•
•
•
•
•
•
19
In dimension two
• f: R2R with f(x1, x2) = max ( min (x1, x2) , x2-x1 )
x2=x1/2
r=([-1,0],[-1,1])
(-1,1)
([-1,1],[-1,1])
(0, -1)
t=([-1,1],[0,1])
∂ f (0)=
convex((-1,1),(-1,0),(01))
x2=2x1
s=([0,1],[-1,0])
(1,0)
x2=x1
20
Fundamental Theorem of Calculus
df
dx
•
f g
iff g ⊑
•
If g
df
then f g iff g =
dx
C0
(interval version)
(classical version)
21
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?
22
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) .
23
Consistency in dimension one
• (⊔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
24
Function and Derivative Information
f
1
1
g
2
1
25
f
Least and greatest functions
1
1
g
2
1
26
Solving Initial Value Problems
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,
27
Solution
f
1
.
1
g
1
1
28
Solution
f
1
.
1
g
1
1
29
Solution
f
1
.
1
g
1
1
30
Basis of ([0,1]n  IR)  ([0,1]n  IR)ns
• 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.
• dom(g) is partitioned by disjoint crescents (intersection of closed
and open sets) in each of which g is a constant rational interval.
Eg. For n=2:
A step function gi with four single step
[-1,1]
functions with two horizontal and two
vertical rectangles as their domains
and a hole inside, and with eight vertices.
[-2,2]
[-3,3]
[-2,2]
31
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 type g:[0,1]n  (IRn)s
32
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 v 0  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
33
Generalized Green’s Theorem
• Definition. g:[0,1]n  (IRn)s the domain of g is
dom(g) = {x | g(x) non-bottom}
• Theorem.  g   iff for any 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
• We can replace piecewise smooth with piecewise linear.
• For step functions, the lower and upper path integrals will
depend linearly on the nodes of the path, so their extreme
values will be reached when these nodes are at the corners of
dom(g).
• Since there are finitely many of these extreme paths, zero
containment can be decided in finite time for all paths.
34
Locally minimal paths
• Definition. Given x,ydom(g), a non-self-intersecting path
p:[0,1] closure(dom(g)) with p(0)=y and p(1)=x is
locally minimal if its length is minimal in its homotopic
class of paths from y to x.
.x
.
y
35
Minimal surface
• Step function g:[0,1]n  (IRn)s . Let O be a component of dom(g) . Let
x,yclosure(O).
• Consider the following supremum over all piecewise linear paths p in
closure(O) with p(0 )= y and p(1)= x.
1
Vg (x, y)  sup p{L  g(p(t))  p' (t)dt}
0
• Theorem. If g satisfies the zero-containment condition, then there is a non-selfintersecting locally minimal piecewise linear path p with
1
Vg (x, y)  L g(p(t))  p' (t)dt
0
• 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
36
Maximal surface
• Step function g:[0,1]n  (IRn)s . Let O be a component of dom(g) . Let
x,ycl(O).
• Consider the following infimum over all piecewise linear paths p in cl(O) with
p(0 )= y and p(1)= x.
1
Wg (x, y)  inf p {U  g(p(t))  p' (t)dt}
0
• Theorem. If g satisfies the zero-containment condition, then there is a non-selfintersecting locally minimal piecewise linear path q with
1
Wg (x, y)  U g(q(t))  q' (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
37
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)  f  (y) and
s(f, g)(x)  sup {S (f, g) (x, y) | y  O x }
with O x the component of x in dom(g)
• Proposition.
s(f, g)(x)  max {S (f, g) (x, y) | y  vertices( O x )}
• Theorem. s(f,g): dom(g) R is the least continuous function
with f -  s(f,g) and g ⊑ ds(f, g)
dx
38
Maximal surface for (f,g)
T(f, g) (x, y)  Wg (x, y)  f  (y) and
t(f, g)(x)  inf {T(f, g) (x, y) | y  O x }
t(f, g)(x)  inf {T(f, g) (x, y) | y  vertices( O x )}
Theorem. t(f,g): dom(g) R is the least continuous function with
dt(f, g)
+
t(f,g)  f and g ⊑ dx
39
Decidability of Consistency
• Theorem. Consistency is decidable.
Proof: In s(f,g)t(f,g) we compare two rational
piecewise-linear surfaces, which is decidable.
40
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
f
 (f , df ) is topological embedding into maximal
dx
elements of D1 , giving a computational model for continuous
functions and their differential properties.
41
Inverse and Implicit Function theorems
•
Definition. Given f:[-1,1]nRn the mean derivative at x0 is the
linear map represented by the matrix M with
1 df
df

Mij= [( (x 0 )) ij  ( (x 0 )) ij ]
2 dx
dx
•
Theorem. Let f:[-1,1]nRn such that the mean derivative M of
df
-1
(0) -I ||<1/n. Then:
f at 0 is invertible with || M
dx
1. The map f has a Lipschitz inverse in a neighbourhood of 0 .
2. Given an increasing sequence of step functions converging to f we
can effectively obtain an increasing sequence of step functions
converging to f-1
3. If f is C1 and given also an increasing sequence of step functions
converging to f ´we can also effectively obtain an increasing
sequence of step functions converging to (f-1)'
42
Further Work
•
•
•
•
A robust CAD
PDE’s
Differential Topology
Differential Geometry
43
THE END
http://www.doc.ic.ac.uk/~ae