Domain Theory and Differential Calculus

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Domain Theory and Differential Calculus
Abbas Edalat Imperial College http//www.doc.i
c.ac.uk/ae
Joint work with Andre Lieutier
Dassault Systemes
Oxford 17/2/2003
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Computational Model for Classical Spaces

• A research project since 1993
• Reconstruct some basic mathematics
• Embed classical spaces into the set of maximal
  elements of suitable domains

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Computational Model for Classical Spaces

• Previous Applications
• Fractal Geometry
• Measure Integration Theory
• Exact Real Arithmetic
• Computational Geometry/Solid Modelling

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Non-smooth Mathematics
Smooth Mathematics

• Geometry
• Differential Topology
• Manifolds
• Dynamical Systems
• Mathematical Physics
• .
• .
• All based on differential
  calculus

• Set Theory
• Logic
• Algebra
• Point-set Topology
• Graph Theory
• Model Theory
• .
• .

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A Domain-Theoretic Model for Differential
Calculus

• Indefinite integral of a Scott continuous
  function
• Derivative of a Scott continuous function
• Fundamental Theorem of Calculus
• Domain of C1 functions
• (Domain of Ck functions)
• Picards Theorem
  A data-type for differential
  equations

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The Domain of nonempty compact Intervals of R

• (IR, ?) is a bounded complete dcpo ?i?I ai
  ?i?I ai
• a b ? ao ? b
• (IR, ?) is ?-continuous
• Basis p,q p lt q p, q ? Q
• (IR, ?) is, thus, a continuous Scott domain.
• Scott topology has basis ?a b a b

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Continuous Functions

• Scott continuous maps 0,1 ? IR with
  f ? g ? ?x ? R . f(x) ? g(x)is another
  continuous Scott domain.
• ? C00,1 ? ( 0,1 ? IR), with f ?
  Ifis a topological embedding into a proper
  subset of maximal elements of 0,1 ? IR .

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Step Functions

• Finite lubs of consistent single step functions
  
  
• ?1?i?n(ai ? bi)
• with ai, bi rational intervals, give a basis
  for
• 0,1 ? IR

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Step Functions-An Example
R
b3
a3
b1
b2
a1
a2
0
1
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Refining the Step Functions
R
b3
a3
b1
a1
b2
a2
0
1
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Operations in Interval Arithmetic

• For a a, a ? IR, b b, b ? IR,and ?
  , , ? we have a b xy x ?
  a, y ? b
• For example
• a b a b, a b

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The Basic Construction

• What is the indefinite integral of a single step
  function a?b ?

• We expect ? a?b ? (0,1 ? IR)

• For what f ? C10,1, should we have If ? ? a?b
  ?

• Intuitively, we expect f to satisfy

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Interval Derivative
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Definition of Interval Derivative

• f ? (0,1 ? IR) has an interval derivativeb ?
  IR at a ? I0,1 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)

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For Classical Functions
Thus, ?(a,b) is our candidate for ? a?b .
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Properties of Ties

• ?(a1,b1) ? ?(a2,b2) iff a2 ? a1 b1 ? b2
• ?ni1 ?(ai,bi) ? ? iff ai?bi 1? i ? n
  consistent.
• ?i?I ?(ai,bi) ? ? iff ai?bi i?I
  consistent iff ?J ?finite I ?i?J
  ?(ai,bi) ? ?
• In fact, ?(a,b) behaves like a?b
  we call ?(a,b) a single-step tie.

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The Indefinite Integral

• ? (0,1 ? IR) ? (P(0,1 ? IR), ? )
• (
  P the power set)

• ? a?b ?(a,b)
• ? ?i ?I ai ? bi ?i?I ?(ai,bi)
• ? is well-defined and Scott continuous.
• But unlike the classical case, ? is not 1-1.

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Example

• (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

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The Derivative Operator

• (I0,1 ? IR) ? (I0,1 ? 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 I0,1
  ? IR g x ?
  x I0,1 ? IR
• (fg) ? ?x . (1 1) ?x
  . 0

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The Derivative
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Examples
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Domain of Ties, or Indefinite Integrals

• Recall ? (0,1 ? IR) ? (P(0,1 ? IR), ? )

• Domain of ties ( T0,1 , ? )
• Theorem. ( T0,1 , ? ) is a continuous Scott
  domain.

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The Fundamental Theorem of Calculus

• Define (T0,1 , ?) ? (0,1 ? IR)
• ? ? ? f ?
  ?

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Fundamental Theorem of Calculus

• For f, g ? C10,1, let f g if f g r, for
  some r ? R.
• We have

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F.T. of Calculus Isomorphic version

• For f , g ? 0,1 ? IR, let f g if f
  g a.e.
• We then have

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A Domain for C1 Functions

• What pairs ( f, g) ? (0,1 ? IR)2 approximate a
  differentiable function?

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Function and Derivative Consistency

• Define the consistency relationCons ? (0,1 ?
  IR) ? (0,1 ? IR) with(f,g) ? Cons if
  (?f) ? (? g) ? ?

• In fact, if (f,g) ? Cons, there are always a
  least and a greatest functions h with the above
  properties.

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Consistency for basis elements

• (?i ai?bi, ?j cj?dj) ? Cons is a finitary
  property

• (f,g) ? Cons iff L(f,g) ?G(f,g) . Cons is
  decidable on the basis.

• Up(f,g) (fg , g) where fg t ?
  L(f,g)(t) , G(f,g)(t)

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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)
  ? Consis a continuous Scott domain, which can be
  given an effective structure.

• Theorem.??? C10,1 ? C00,1 ? (0,1 ? IR)2
• restricts to give a topological embedding
  
• D1c ? D1
  (with C1 norm) (with Scott topology)

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Higher Interval Derivative
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Higher Derivative and Indefinite 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 ?i?I ? (ai,bi)
• ? is well-defined and Scott continuous.

(2)
(2)
2
(2)
2
(2)
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Domains of C 2 functions

• Theorem. Cons (f0,f1,f2) is decidable on basis
  elements.
• (The present algorithm to check is NP-hard.)
• D2 (f0,f1,f2) ? (I0,1?IR)3 Cons
  (f0,f1,f2)

• Theorem. ????? restricts to give a topological
  embedding D2c ? D2

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Domains of C k functions

• The decidability of Cons on basis elements for k
  ? 3 is an open question.

• Dk (fi)0?i?k ? (I0,1?IR)k1 Cons
  (fi)0?i?k

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Picards Theorem
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Picards Solution Reformulated

• ApF (f,g) ? (f , ?t. F(t,f(t)))

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A domain-theoretic Picards theorem

• We now have the basic framework to obtain
  Picards theorem with domain theory.
• However, we have to make sure that derivative
  updating preserves consistency.
• Say (f , g) is strongly consistent, (f , g)
  ?S-Cons, if
• ? h ? g. (f , h) ? Cons
• On basis elements, strong consistency is
  decidable.

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A domain-theoretic Picards theorem

• Consider any initial value f ? 0,1 ? IR with
• (f, F (. , f ) ) ?
  S-Cons
• Then the continuous map P Up ? ApF has a
  least fixed point above (f, F (. , f ))
• Theorem. If F Ih for a map h 0,1 ? R ? R
  which satisfies the Lipschitz property of
  Picards theorem, then the domain-theoretic
  solution coincides with the classical solution.

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Example
F is approximated by a sequence of step
functions, F1, F2, F ?i Fi
.
t
The initial condition is approximated by
rectangles ai?bi (1/2,9/8) ?i ai?bi,
F
t
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Solution
At each stage we find Li and Gi
.
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Solution
At each stage we find Li and Gi
.
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Solution
At each stage we find Li and Gi
.
Li and Gi tend to the exact solutionf t ?
t2/2 1
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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,

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THE ENDhttp//www.doc.ic.ac.uk/ae
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Higher Interval Derivative
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Higher Interval Derivative

• For f I0,1 ? IR we define I0,1 ?
  IR by
• Then ?f ? ?2(a,b) a?b
• ? (I0,1 ? IR) ? (P(I0,1 ? IR), ? )
• ? a?b ? (a,b)

(2)
(2)
2
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Domains of C 2and C k functions

• Theorem. ????? restricts to give a topological
  embedding D2c ? D2

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Consistency Test for (f,g)

• Also define L(x) supy?O?Dom(f)(f (y)
  d(x,y)) and G(x)
  infy?O?Dom(f)(f (y) d(x,y))

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Consistency Test

• Theorem. (f, g) ? Con iff ?x ? O. L(x) ? G(x).
  For (f, g) (?1?i?n ai?bi, ?1?j?m cj?dj)
• the rational endpoints of ai and cj induce a
  partition X x0 lt x1 lt x2 lt lt xk of O.
• Proposition. For arbitrary x ? O, there isp,
  where 0 ? p ? k, with L(x)
  f (xp) d(x,xp).
• Similarly for G(x).