Domain Theory and Multi-Variable Calculus

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Domain Theory and Multi-Variable Calculus

• Abbas Edalat
• Imperial College London
• www.doc.ic.ac.uk/ae
• Joint work with Andre Lieutier, Dirk Pattinson

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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

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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

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A Domain-Theoretic Model for Differential
Calculus

• Overall Aim Synthesize Computer Science with
  Differential Calculus
• Plan of the talk
• Primitives of continuous interval-valued function
  in Rn
• Derivative of a continuous function in Rn
• Fundamental Theorem of Calculus for
  interval-valued functions in Rn
• Domain of C1 functions in Rn
• Inverse and implicit functions in domain theory

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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 bya
  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

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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.

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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 ?-continuouscountable basis p,q
  p lt q p, q ? Q
• (IR, ?) is, thus, a continuous Scott domain.
• Scott topology has basis?a b ao ? b

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Continuous Functions

• Scott continuous f0,1n ? IR is given by lower
  and upper semi-continuous functions f -, f
  0,1n ? R with f(x)f -(x),f (x)
• f 0,1n ? R, f ? C00,1n, has continuous
  extension If 0,1n ? IR
  x ? f (x)

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

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Step Functions

• 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 0,1n ? IR. They
  are used to approximate C0 functions.

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Step Functions-An Example in R
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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Interval Lipschitz constant in dimension one

• For f ? (0,1 ? IR) we have
• ?x1, x2 ? ao, b(x1 x2) ? f(x1) f(x2) iff
  for all x2?x1
• b- (x1 x2) ? f(x1) f(x2) ? b(x1 x2)
  iff

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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 Ax?(An)s if x(x1,..xn) with xi
  non-bottom for all i or xbottom
• Interval Lipschitz constants of real-valued
  functions in Rn take values in (IR1 n)s

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Interval Lipschitz constant in R n

• f ? (0,1n ? IR) has an interval Lipschitz
  constantb ? (IR1xn)s in a ? I0,1n 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 kmax i (bi, bi-)

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For Classical Functions

• Let f ? C10,1n 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 .
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Set of primitive maps

• ? (0,1n ? IR) ? (P(0,1n ? (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

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The Derivative
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Examples
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Relation with Clarkes gradient

• For a locally Lipschitz f 0,1n ? 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 0,1n ? 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.

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In dimension two

• f R2?R with f(x1, x2) max ( min (x1, x2) ,
  x2-x1 )

? f (0) convex((-1,1),(-1,0),(01))
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Fundamental Theorem of Calculus
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Idea of Domain for C1 Functions

• If h ?C10,1n , then ( h , h ) ? (0,1n
  ? IR) ? (0,1n ? IR)ns

• We can approximate ( h, h ) in
• (0,1n ? IR) ? (0,1n ? IR)ns
• i.e. ( f, g) ? ( h ,h ) with f ? h and g
  ?h

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

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Function and Derivative Consistency

• Define the consistency relationCons ? (0,1n ?
  IR) ? (0,1n ? IR)ns 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) .

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Consistency in dimension one

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

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Function and Derivative Information
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Least and greatest functions
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Solving Initial Value Problems
v is approximated by a sequence of step
functions, v0, v1, v ?i vi
.
t
The initial condition is approximated by
rectangles ai?bi (1/2,9/8) ?i ai?bi,
v
t
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Solution
.
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Solution
.
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Solution
.
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Basis of (0,1n ? IR) ? (0,1n ? IR)ns

• Definition. g0,1n ? (IRn)s the domain of g
  is dom(g) x g(x)
  non-bottom
• Basis element (f, g1,g2,.,gn) ? (0,1n? IR) ?
  (0,1n? IR)ns
• Each f, gi 0,1n ? 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 n2A step function gi with four single
  stepfunctions with two horizontal and
  twovertical rectangles as their domainsand a
  hole inside, and with eight vertices.

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Decidability of Consistency

• (f,g) ? Cons if (?f) ? (? g) ? ?
• First we check if g is integrable, i.e. if ? g ?
  ?
• In classical calculus, g0,1n ? Rn will be
  integrable by Greens theorem iff for any
  piecewise smooth closed non-intersecting path
• p0,1? 0,1n with p(0)p(1)

• We generalize this to type g0,1n ? (IRn)s

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Interval-valued path integral

• For v?IRn , u?Rn define the interval-valued
  scalar product

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Generalized Greens Theorem

• Definition. g0,1n ? (IRn)s the domain of g is
  dom(g) x g(x)
  non-bottom
• Theorem. ? g ? ? iff for any piecewise smooth
  non-intersecting path p0,1? dom(g) with
  p(0)p(1), we have zero-containment

• 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.

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Locally minimal paths

• Definition. Given x,y?dom(g), a
  non-self-intersecting pathp0,1?
  closure(dom(g)) with p(0)y and p(1)x
  islocally minimal if its length is minimal in
  its homotopicclass of paths from y to x.

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Minimal surface

• Step function g0,1n ? (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.

• 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

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Maximal surface

• Step function g0,1n ? (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.

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Minimal surface for (f,g)

• (f,g)?(0,1n ? IR) ? (0,1n ? IR)ns rational
  step function
• Assume we have determined that ? g ? ?

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Maximal surface for (f,g)
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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.

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The Domain of C1 Functions

• Lemma. Cons ? (0,1n ? IR)? (0,1n ? IR)ns is
  Scott closed.
• Theorem.D1 0,1n (f,g) (f,g) ? Cons is a
  continuous Scott domain that can be given an
  effective structure.

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Inverse and Implicit Function theorems

• Definition. Given f-1,1n?Rn the mean
  derivative at x0 is the linear map represented by
  the matrix M with
• 
  Mij

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Further Work

• A robust CAD
• PDEs
• Differential Topology
• Differential Geometry

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THE END

• http//www.doc.ic.ac.uk/ae