Domain Theory and Multi-Variable Calculus Abbas Edalat Imperial College London Joint work with Andre Lieutier, Dirk Pattinson

2 Computational Model for Classical Spaces X Classical Space x DX Domain {x} A research project since 1993: Reconstruct basic mathematical analysis Embed classical spaces into the set of maximal elements of suitable domains

3 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

4 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 R n 2.Derivative of a continuous function in R n 3.Fundamental Theorem of Calculus for interval-valued functions in R n 4.Domain of C 1 functions in R n 5.Inverse and implicit functions in domain theory

5 Continuous Scott Domains A directed complete partial order (dcpo) is a poset (A, ⊑ ), in which every directed set {a i | i  I }  A has a sup or lub ⊔ i  I a i The way-below relation in a dcpo is defined by: a ≪ b iff for all directed subsets {a i | i  I }, the relation b ⊑ ⊔ i  I a i implies that there exists i  I such that a ⊑ a i 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

6 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 T 0. Fact. The Scott topology on a continuous dcpo A with basis B has basic open sets {a  A | b ≪ a } for each b  B.

7 Let IR={ [a,b] | a, b  R}  {R} (IR,  ) is a bounded complete dcpo with R as bottom: ⊔ i  I a i =  i  I a i a ≪ b  a o  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 | a o  b}  x {x} R I R x  {x} : R  IR Topological embedding The Domain of nonempty compact Intervals of R

8 Continuous Functions Scott continuous f:[0,1] n  IR is given by lower and upper semi- continuous functions f -, f + :[0,1] n  R with f(x)=[f - (x),f + (x)] f : [0,1] n  R, f  C 0 [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.  : C 0 [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

9 Step Functions Single-step function: a ↘ b : [0,1] n  IR, with a  I[0,1] n, b=[b -,b + ]  IR : b x  a o x   otherwise Lubs of finite and bounded collections of single- step functions ⊔ 1  i  n (a i ↘ b i ) are called step functions. Step functions with a i, b i rational intervals, give a basis for [0,1] n  IR. They are used to approximate C 0 functions.

10 Step Functions-An Example in R 01 R b1b1 a3a3 a2a2 a1a1 b3b3 b2b2

11 Refining the Step Functions 01 R b1b1 a3a3 a2a2 a1a1 b3b3 b2b2

12 Interval Lipschitz constant in dimension one For f  ([0,1]  IR) we have:  x 1, x 2  a o, b(x 1 – x 2 ) ⊑ f(x 1 ) – f(x 2 ) iff for all x 2  x 1 b - (x 1 – x 2 )  f(x 1 ) – f(x 2 )  b + (x 1 – x 2 ) iff Graph(f) is within lines of slope b - & b + at each point (x, f(x)), x  a o. (x, f(x)) b+b+ a Graph(f). b-b-

13 Functions of several varibales (IR) 1× n row n-vectors with entries in IR For dcpo A, let (A n ) s = smash product of n copies of A: x  (A n ) s if x=(x 1,…..x n ) with x i non-bottom for all i or x=bottom Interval Lipschitz constants of real-valued functions in R n take values in (IR 1× n ) s

14 Interval Lipschitz constant in R n f  ([0,1] n  IR) has an interval Lipschitz constant b  (IR 1xn ) s in a  I[0,1] n if  x, y  a o, b(x – y) ⊑ f(x) – f(y). Proposition. If f  (a,b), then f(x)  Maximal (IR) for x  a o and for all x,y  a o. |f(x)-f(y)|  k ||x-y|| with k=max i (|b i + |, |b i - |) The tie of a with b, is  (a,b) := { f |  x,y  a o. b(x – y) ⊑ f(x) – f(y)}

15 Let f  C 1 [0,1] n ; the following are equivalent: f   (a,b)  x  a o. b -  f ´(x)  b +  x,y  a o, b(x – y) ⊑ f (x) – f (y) a ↘ b ⊑ f ´ For Classical Functions Thus,  (a,b) is our candidate for  a ↘ b.

16 Set of primitive maps  : ([0,1] n  IR)  ( P ([0,1] n  (IR 1xn ) s ),  ) ( P the power set constructor)  a ↘ b :=  (a,b)  ⊔ i  I a i ↘ b i :=  i  I  (a i,b i )  is well-defined and Scott continuous.  g can be the empty set for 2  n Eg. g=(g 1,g 2 ), with g 1 (x, y)= y, g 2 (x, y)=0

17 The Derivative Definition. Given f : [0,1] n  IR the derivative of f is: : [0,1] n  (IR 1xn ) s = ⊔ {a ↘ b | f   (a,b) } Theorem. (Compare with the classical case.) is well–defined & Scott continuous. If f  C 1 [0,1], then f   (a,b) iff a ↘ b ⊑

18 Examples

19 Relation with Clarke’s gradient For a locally Lipschitz f : [0,1] n  R ∂ f (x) := convex-hull{ lim m f ´(x m ) | x m  x} It is a non-empty compact convex subset of R n 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.

20 In dimension two f: R 2  R with f(x 1, x 2 ) = max ( min (x 1, x 2 ), x 2 -x 1 ) x 2 =x 1 /2 x 2 =x 1 x 2 =2x 1 (1,0) (0, -1) (-1,1) ([-1,1],[-1,1]) r=([-1,0],[-1,1]) s=([0,1],[-1,0]) t=([-1,1],[0,1]) ∂ f (0)= convex((-1,1),(-1,0),(01))

21 Fundamental Theorem of Calculus f  g iff g ⊑ (interval version) If g  C 0 then f  g iff g = (classical version)

22 If h  C 1 [0,1] n, then ( h, h ´ )  ([0,1] n  IR)  ([0,1] n  IR) n s Idea of Domain for C 1 Functions What pairs ( f, g)  ([0,1] n  IR)  ([0,1] n  IR) n s approximate a differentiable function? We can approximate ( h, h´ ) in ([0,1] n  IR)  ([0,1] n  IR) n s i.e. ( f, g) ⊑ ( h,h´ ) with f ⊑ h and g ⊑ h´

23 Function and Derivative Consistency Define the consistency relation: Cons  ([0,1] n  IR)  ([0,1] n  IR) n s 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). Proposition (f,g)  Cons iff there is a continuous h: dom(g)  R with f ⊑ h and g ⊑.

24 Approximating function: f = ⊔ i a i ↘ b i ( ⊔ i a i ↘ b i, ⊔ j c j ↘ d j )  Cons is a finitary property: Consistency in dimension one s(f,g) = least function t(f,g) = greatest function Approximating derivative: g = ⊔ j c j ↘ d j

25 f 1 1 Function and Derivative Information g 1 2

26 f 1 1 Least and greatest functions g 1 2

27 Solving Initial Value Problems 1 f g v v is approximated by a sequence of step functions, v 0, v 1, … v = ⊔ i v i We solve: = v(t,x), x(t 0 ) =x 0 for t  [0,1] with v(t,x) = t and t 0 =1/2, x 0 =9/8. a3a3 b3b3 a2a2 b2b2 a1a1 b1b1 v3v3 v2v2 v1v1 The initial condition is approximated by rectangles a i  b i : {(1/2,9/8)} = ⊔ i a i  b i, t t.

28 Solution 1 f g

29 Solution 1 f g

30 Solution 1 f g

31 Definition. g:[0,1] n  (IR n ) s the domain of g is dom(g) = {x | g(x) non-bottom} Basis element: (f, g 1,g 2,….,g n )  ([0,1] n  IR)  ([0,1] n  IR) n s Each f, g i :[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 g i with four single step functions with two horizontal and two vertical rectangles as their domains and a hole inside, and with eight vertices. Basis of ([0,1] n  IR)  ([0,1] n  IR) n s [-2,2] [-3,3][-1,1]

32 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  R n 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) We generalize this to type g:[0,1] n  (IR n ) s

33 Interval-valued path integral For v  IR n, u  R n define the interval-valued scalar product

34 Generalized Green’s Theorem Definition. g:[0,1] n  (IR n ) 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: 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.

35 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. y.y..x.x

36 Minimal surface Step function g:[0,1] n  (IR n ) 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. Theorem. If g satisfies the zero-containment condition, then there is a non-self- intersecting locally minimal piecewise linear path p with For fixed y, the map V g (.,y): cl(O)  R is a rational piecewise linear function. It is the least continuous function or surface with: V g (y,y)=0 and g ⊑

37 Maximal surface Step function g:[0,1] n  (IR n ) 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. Theorem. If g satisfies the zero-containment condition, then there is a non-self- intersecting locally minimal piecewise linear path q with For fixed y, the map W g (.,y): cl(O)  R is a rational piecewise linear function. It is the greatest continuous function or surface with: W g (y,y)=0 and g ⊑

38 Minimal surface for (f,g) (f,g)  ([0,1] n  IR)  ([0,1] n  IR) n s rational step function Assume we have determined that  g   Theorem. s(f,g): dom(g)  R is the least continuous function with f -  s(f,g) and g ⊑ Proposition. Put

39 Maximal surface for (f,g) Theorem. t(f,g): dom(g)  R is the least continuous function with t(f,g)  f + and g ⊑

40 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.

41 The Domain of C 1 Functions Lemma. Cons  ([0,1] n  IR)  ([0,1] n  IR) n s is Scott closed. Theorem. D 1 [0,1] n := { (f,g) | (f,g)  Cons} is a continuous Scott domain that can be given an effective structure. Theorem.  : C 0 [0,1] n  D 1 [0,1] n f  (f, ) is topological embedding into maximal elements of D 1, giving a computational model for continuous functions and their differential properties.

42 Inverse and Implicit Function theorems Definition. Given f:[-1,1] n  R n the mean derivative at x 0 is the linear map represented by the matrix M with M ij = Theorem. Let f:[-1,1] n  R n such that the mean derivative M of f at 0 is invertible with || M -1 -I ||<1/n. Then: 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 C 1 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 )'

43 Further Work A robust CAD PDE’s Differential Topology Differential Geometry

THE END