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Topological Complexity and Degrees of Discontinuity

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					Level of Discontinuity                Tests in Computation Trees               Degrees of Discontinuity




                   Topological Complexity and Degrees of
                                Discontinuity

                                        Peter Hertling

                         Institut für Theoretische Informatik und Mathematik
                          Universität der Bundeswehr München, Germany


                              International Conference on
                           Infinity in Logic and Computation
                           Cape Town, 3–5 November 2007
Level of Discontinuity   Tests in Computation Trees   Degrees of Discontinuity



                          Introduction
Level of Discontinuity       Tests in Computation Trees    Degrees of Discontinuity



                              Introduction
       Goal
       Analyse the discontinuities appearing in computation problems
Level of Discontinuity       Tests in Computation Trees    Degrees of Discontinuity



                              Introduction
       Goal
       Analyse the discontinuities appearing in computation problems


       Motivation
       Discontinuities cause problems when computing real number
       functions.
Level of Discontinuity           Tests in Computation Trees     Degrees of Discontinuity



                                  Introduction
       Goal
       Analyse the discontinuities appearing in computation problems


       Motivation
       Discontinuities cause problems when computing real number
       functions.
            • Discontinuities in
                • numerical computation: instabilities.
                • computational geometry: degenerate configurations.
Level of Discontinuity             Tests in Computation Trees      Degrees of Discontinuity



                                    Introduction
       Goal
       Analyse the discontinuities appearing in computation problems


       Motivation
       Discontinuities cause problems when computing real number
       functions.
            • Discontinuities in
                • numerical computation: instabilities.
                • computational geometry: degenerate configurations.
            • In Computable Analysis (Turing machine model, computing
                with “finite” (rational, dyadic) approximations):
                           Computable functions are continuous.
Level of Discontinuity              Tests in Computation Trees      Degrees of Discontinuity



                                     Introduction
       Goal
       Analyse the discontinuities appearing in computation problems


       Motivation
       Discontinuities cause problems when computing real number
       functions.
            • Discontinuities in
                • numerical computation: instabilities.
                • computational geometry: degenerate configurations.
            • In Computable Analysis (Turing machine model, computing
                with “finite” (rational, dyadic) approximations):
                           Computable functions are continuous.
                Levels of discontinuity are topological levels of
                noncomputability.
Level of Discontinuity     Tests in Computation Trees   Degrees of Discontinuity



                              Overview

       I Level of Discontinuity
       = Number of Tests in Continuous Computation Trees
Level of Discontinuity           Tests in Computation Trees         Degrees of Discontinuity



                                    Overview

       I Level of Discontinuity
       = Number of Tests in Continuous Computation Trees
            • Level of Discontinuity
                • Hausdorff’s Reducible Sets/Difference Hierarchy
                • Schwarz Genus
Level of Discontinuity           Tests in Computation Trees       Degrees of Discontinuity



                                    Overview

       I Level of Discontinuity
       = Number of Tests in Continuous Computation Trees
            • Level of Discontinuity
                • Hausdorff’s Reducible Sets/Difference Hierarchy
                • Schwarz Genus
            • The Number of Tests in Computation Trees
                • Degenerate Configurations in Computational Geometry
                • Problems from Algebraic Topology and from Algebraic
                  Complexity Theory
                • The Topological Complexity of Zero Finding for Continuous
                  Functions in Various Settings
Level of Discontinuity           Tests in Computation Trees       Degrees of Discontinuity



                                    Overview

       I Level of Discontinuity
       = Number of Tests in Continuous Computation Trees
            • Level of Discontinuity
                • Hausdorff’s Reducible Sets/Difference Hierarchy
                • Schwarz Genus
            • The Number of Tests in Computation Trees
                • Degenerate Configurations in Computational Geometry
                • Problems from Algebraic Topology and from Algebraic
                  Complexity Theory
                • The Topological Complexity of Zero Finding for Continuous
                  Functions in Various Settings
       II Continuous Reducibility of Functions
       → Refinement of the Level of Discontinuity
            • Degrees of Discontinuity
Level of Discontinuity              Tests in Computation Trees   Degrees of Discontinuity



                         Level of Discontinuity of a Function
Level of Discontinuity              Tests in Computation Trees   Degrees of Discontinuity



                         Level of Discontinuity of a Function
       Let X , Y be topological spaces, f :⊆ X → Y a (possibly partial)
       function.
Level of Discontinuity              Tests in Computation Trees            Degrees of Discontinuity



                         Level of Discontinuity of a Function
       Let X , Y be topological spaces, f :⊆ X → Y a (possibly partial)
       function.
       Cα (f ) := dom(f ) ∩             {x ∈ Cβ (f ) | f |Cβ (f ) is discontinuous in x}
                                  β<α
Level of Discontinuity              Tests in Computation Trees            Degrees of Discontinuity



                         Level of Discontinuity of a Function
       Let X , Y be topological spaces, f :⊆ X → Y a (possibly partial)
       function.
       Cα (f ) := dom(f ) ∩             {x ∈ Cβ (f ) | f |Cβ (f ) is discontinuous in x}
                                  β<α
       Example
Level of Discontinuity              Tests in Computation Trees            Degrees of Discontinuity



                         Level of Discontinuity of a Function
       Let X , Y be topological spaces, f :⊆ X → Y a (possibly partial)
       function.
       Cα (f ) := dom(f ) ∩             {x ∈ Cβ (f ) | f |Cβ (f ) is discontinuous in x}
                                  β<α
       Example

                                                     C0 (f ) =
Level of Discontinuity              Tests in Computation Trees            Degrees of Discontinuity



                         Level of Discontinuity of a Function
       Let X , Y be topological spaces, f :⊆ X → Y a (possibly partial)
       function.
       Cα (f ) := dom(f ) ∩             {x ∈ Cβ (f ) | f |Cβ (f ) is discontinuous in x}
                                  β<α
       Example

                                                     C0 (f ) =   R2
Level of Discontinuity              Tests in Computation Trees            Degrees of Discontinuity



                         Level of Discontinuity of a Function
       Let X , Y be topological spaces, f :⊆ X → Y a (possibly partial)
       function.
       Cα (f ) := dom(f ) ∩             {x ∈ Cβ (f ) | f |Cβ (f ) is discontinuous in x}
                                  β<α
       Example

                                                     C0 (f ) =   R2
                                                     C1 (f ) =
Level of Discontinuity              Tests in Computation Trees            Degrees of Discontinuity



                         Level of Discontinuity of a Function
       Let X , Y be topological spaces, f :⊆ X → Y a (possibly partial)
       function.
       Cα (f ) := dom(f ) ∩             {x ∈ Cβ (f ) | f |Cβ (f ) is discontinuous in x}
                                  β<α
       Example

                                                     C0 (f ) =   R2
                                                     C1 (f ) =   {(x, y ) ∈ R2 |
                                                                  y = 0 ∨ y > 0 ∧ x = 0}
Level of Discontinuity              Tests in Computation Trees            Degrees of Discontinuity



                         Level of Discontinuity of a Function
       Let X , Y be topological spaces, f :⊆ X → Y a (possibly partial)
       function.
       Cα (f ) := dom(f ) ∩             {x ∈ Cβ (f ) | f |Cβ (f ) is discontinuous in x}
                                  β<α
       Example

                                                     C0 (f ) =   R2
                                                     C1 (f ) =   {(x, y ) ∈ R2 |
                                                                  y = 0 ∨ y > 0 ∧ x = 0}
                                                     C2 (f ) =
Level of Discontinuity              Tests in Computation Trees            Degrees of Discontinuity



                         Level of Discontinuity of a Function
       Let X , Y be topological spaces, f :⊆ X → Y a (possibly partial)
       function.
       Cα (f ) := dom(f ) ∩             {x ∈ Cβ (f ) | f |Cβ (f ) is discontinuous in x}
                                  β<α
       Example

                                                     C0 (f ) =   R2
                                                     C1 (f ) =   {(x, y ) ∈ R2 |
                                                                  y = 0 ∨ y > 0 ∧ x = 0}
                                                     C2 (f ) =   {(0, 0)}
Level of Discontinuity              Tests in Computation Trees            Degrees of Discontinuity



                         Level of Discontinuity of a Function
       Let X , Y be topological spaces, f :⊆ X → Y a (possibly partial)
       function.
       Cα (f ) := dom(f ) ∩             {x ∈ Cβ (f ) | f |Cβ (f ) is discontinuous in x}
                                  β<α
       Example

                                                     C0 (f ) =   R2
                                                     C1 (f ) =   {(x, y ) ∈ R2 |
                                                                  y = 0 ∨ y > 0 ∧ x = 0}
                                                     C2 (f ) =   {(0, 0)}
                                                     C3 (f ) =
Level of Discontinuity              Tests in Computation Trees            Degrees of Discontinuity



                         Level of Discontinuity of a Function
       Let X , Y be topological spaces, f :⊆ X → Y a (possibly partial)
       function.
       Cα (f ) := dom(f ) ∩             {x ∈ Cβ (f ) | f |Cβ (f ) is discontinuous in x}
                                  β<α
       Example

                                                     C0 (f ) =   R2
                                                     C1 (f ) =   {(x, y ) ∈ R2 |
                                                                   y = 0 ∨ y > 0 ∧ x = 0}
                                                     C2 (f ) =   {(0, 0)}
                                                     C3 (f ) =   ∅
Level of Discontinuity              Tests in Computation Trees            Degrees of Discontinuity



                         Level of Discontinuity of a Function
       Let X , Y be topological spaces, f :⊆ X → Y a (possibly partial)
       function.
       Cα (f ) := dom(f ) ∩             {x ∈ Cβ (f ) | f |Cβ (f ) is discontinuous in x}
                                  β<α
       Example

                                                     C0 (f ) =   R2
                                                     C1 (f ) =   {(x, y ) ∈ R2 |
                                                                   y = 0 ∨ y > 0 ∧ x = 0}
                                                     C2 (f ) =   {(0, 0)}
                                                     C3 (f ) =   ∅




       Lev(f ) := min{α | Cα (f ) = ∅},
Level of Discontinuity              Tests in Computation Trees            Degrees of Discontinuity



                         Level of Discontinuity of a Function
       Let X , Y be topological spaces, f :⊆ X → Y a (possibly partial)
       function.
       Cα (f ) := dom(f ) ∩             {x ∈ Cβ (f ) | f |Cβ (f ) is discontinuous in x}
                                  β<α
       Example

                                                     C0 (f ) =   R2
                                                     C1 (f ) =   {(x, y ) ∈ R2 |
                                                                   y = 0 ∨ y > 0 ∧ x = 0}
                                                     C2 (f ) =   {(0, 0)}
                                                     C3 (f ) =   ∅
                                                     Lev(f ) =



       Lev(f ) := min{α | Cα (f ) = ∅},
Level of Discontinuity              Tests in Computation Trees            Degrees of Discontinuity



                         Level of Discontinuity of a Function
       Let X , Y be topological spaces, f :⊆ X → Y a (possibly partial)
       function.
       Cα (f ) := dom(f ) ∩             {x ∈ Cβ (f ) | f |Cβ (f ) is discontinuous in x}
                                  β<α
       Example

                                                     C0 (f ) = R2
                                                     C1 (f ) = {(x, y ) ∈ R2 |
                                                                 y = 0 ∨ y > 0 ∧ x = 0}
                                                     C2 (f ) = {(0, 0)}
                                                     C3 (f ) = ∅
                                                     Lev(f ) = 3



       Lev(f ) := min{α | Cα (f ) = ∅},
Level of Discontinuity              Tests in Computation Trees               Degrees of Discontinuity



                         Level of Discontinuity of a Function
       Let X , Y be topological spaces, f :⊆ X → Y a (possibly partial)
       function.
       Cα (f ) := dom(f ) ∩             {x ∈ Cβ (f ) | f |Cβ (f ) is discontinuous in x}
                                  β<α
       Example

                                                     C0 (f ) = R2
                                                     C1 (f ) = {(x, y ) ∈ R2 |
                                                                 y = 0 ∨ y > 0 ∧ x = 0}
                                                     C2 (f ) = {(0, 0)}
                                                     C3 (f ) = ∅
                                                     Lev(f ) = 3



       Lev(f ) := min{α | Cα (f ) = ∅},                lev(f , x) := min{α | x ∈ Cα (f )}.
Level of Discontinuity   Tests in Computation Trees   Degrees of Discontinuity



                           Properties
Level of Discontinuity              Tests in Computation Trees      Degrees of Discontinuity



                                      Properties




            • If Lev(f ) ≥ α then Lev(f ) = α + Lev(f |Cα (f ) ).
Level of Discontinuity              Tests in Computation Trees      Degrees of Discontinuity



                                      Properties




            • If Lev(f ) ≥ α then Lev(f ) = α + Lev(f |Cα (f ) ).
            • Lev(f ◦ g) ≤ Lev(f ) · Lev(g).
Level of Discontinuity          Tests in Computation Trees   Degrees of Discontinuity



                    Level and Admissible Representations
Level of Discontinuity           Tests in Computation Trees     Degrees of Discontinuity



                    Level and Admissible Representations
            • The level is invariant under admissible representations
Level of Discontinuity           Tests in Computation Trees     Degrees of Discontinuity



                    Level and Admissible Representations
            • The level is invariant under admissible representations

       A function δ :⊆ Σω → X is an admissible representation
       iff δ is surjective, continuous, and for every cont. f :⊆ Σω → X
       there is some cont. h :⊆ Σω → Σω with f = δ ◦ h
       (use adm. repr. of X for computations over X via oracle Turing
       machines).
Level of Discontinuity           Tests in Computation Trees     Degrees of Discontinuity



                    Level and Admissible Representations
            • The level is invariant under admissible representations

       A function δ :⊆ Σω → X is an admissible representation
       iff δ is surjective, continuous, and for every cont. f :⊆ Σω → X
       there is some cont. h :⊆ Σω → Σω with f = δ ◦ h
       (use adm. repr. of X for computations over X via oracle Turing
       machines).
       Let X , Y be T0 -spaces with countable bases, δX :⊆ Σω → X
       and δY :⊆ Σω → Y admissible representations of X and Y .
Level of Discontinuity                 Tests in Computation Trees                Degrees of Discontinuity



                    Level and Admissible Representations
            • The level is invariant under admissible representations

       A function δ :⊆ Σω → X is an admissible representation
       iff δ is surjective, continuous, and for every cont. f :⊆ Σω → X
       there is some cont. h :⊆ Σω → Σω with f = δ ◦ h
       (use adm. repr. of X for computations over X via oracle Turing
       machines).
       Let X , Y be T0 -spaces with countable bases, δX :⊆ Σω → X
       and δY :⊆ Σω → Y admissible representations of X and Y .

                                                                         F
       A function F :⊆ Σω → Σω real-                            Σω      −→   Σω
       izes f if for all p ∈ dom(f δX )
                                                             δX ↓            ↓ δY
                     f δX (p) = δY F (p).
                                                                    X   −→   Y
                                                                         f
Level of Discontinuity           Tests in Computation Trees     Degrees of Discontinuity



                    Level and Admissible Representations


            • The level is invariant under admissible representations
Level of Discontinuity           Tests in Computation Trees     Degrees of Discontinuity



                    Level and Admissible Representations


            • The level is invariant under admissible representations


       Theorem
Level of Discontinuity            Tests in Computation Trees     Degrees of Discontinuity



                    Level and Admissible Representations


            • The level is invariant under admissible representations


       Theorem
            • (global)
                Lev(f ) ≤ α ⇐⇒ there exists F realizing f with Lev(F ) ≤ α.
Level of Discontinuity             Tests in Computation Trees    Degrees of Discontinuity



                    Level and Admissible Representations


            • The level is invariant under admissible representations


       Theorem
            • (global)
                Lev(f ) ≤ α ⇐⇒ there exists F realizing f with Lev(F ) ≤ α.
            • (local)
                For every function F realizing f and all α,
                Cα (f ) ⊆ δX (Cα (F )).
Level of Discontinuity              Tests in Computation Trees       Degrees of Discontinuity



                    Level and Admissible Representations


            • The level is invariant under admissible representations


       Theorem
            • (global)
                Lev(f ) ≤ α ⇐⇒ there exists F realizing f with Lev(F ) ≤ α.
            • (local)
                For every function F realizing f and all α,
                Cα (f ) ⊆ δX (Cα (F )).
                If Lev(f ) is defined then there exists a function F realizing f
                with Cα (f ) = δX (Cα (F )) for all α.
Level of Discontinuity   Tests in Computation Trees   Degrees of Discontinuity



       Hausdorff’s Reducible Sets/Difference Hierarchy
Level of Discontinuity              Tests in Computation Trees              Degrees of Discontinuity



       Hausdorff’s Reducible Sets/Difference Hierarchy
       Let X be a topological space, M ⊆ X . Hausdorff’s residues:
                         
                         M
                                                                if α = 0
          Rα (M) :=       R (M) ∩ Rβ (M) \ Rβ (M) if α = β + 1
                          β
                           β<α Rβ (M)             if α is a limit number.
                         
Level of Discontinuity              Tests in Computation Trees              Degrees of Discontinuity



       Hausdorff’s Reducible Sets/Difference Hierarchy
       Let X be a topological space, M ⊆ X . Hausdorff’s residues:
                         
                         M
                                                                if α = 0
          Rα (M) :=       R (M) ∩ Rβ (M) \ Rβ (M) if α = β + 1
                          β
                           β<α Rβ (M)             if α is a limit number.
                         


       Lemma
       Rα (M) = M ∩ C2·α (cfM ).
Level of Discontinuity              Tests in Computation Trees              Degrees of Discontinuity



       Hausdorff’s Reducible Sets/Difference Hierarchy
       Let X be a topological space, M ⊆ X . Hausdorff’s residues:
                         
                         M
                                                                if α = 0
          Rα (M) :=       R (M) ∩ Rβ (M) \ Rβ (M) if α = β + 1
                          β
                           β<α Rβ (M)             if α is a limit number.
                         


       Lemma
       Rα (M) = M ∩ C2·α (cfM ).

       Hausdorff called a set M reducible if there exists an α with
       Rα (M) = ∅.
Level of Discontinuity              Tests in Computation Trees              Degrees of Discontinuity



       Hausdorff’s Reducible Sets/Difference Hierarchy
       Let X be a topological space, M ⊆ X . Hausdorff’s residues:
                         
                         M
                                                                if α = 0
          Rα (M) :=       R (M) ∩ Rβ (M) \ Rβ (M) if α = β + 1
                          β
                           β<α Rβ (M)             if α is a limit number.
                         


       Lemma
       Rα (M) = M ∩ C2·α (cfM ).

       Hausdorff called a set M reducible if there exists an α with
       Rα (M) = ∅.
       Equivalent to: Lev(cfM ) is defined.
Level of Discontinuity              Tests in Computation Trees              Degrees of Discontinuity



       Hausdorff’s Reducible Sets/Difference Hierarchy
       Let X be a topological space, M ⊆ X . Hausdorff’s residues:
                         
                         M
                                                                if α = 0
          Rα (M) :=       R (M) ∩ Rβ (M) \ Rβ (M) if α = β + 1
                          β
                           β<α Rβ (M)             if α is a limit number.
                         


       Lemma
       Rα (M) = M ∩ C2·α (cfM ).

       Hausdorff called a set M reducible if there exists an α with
       Rα (M) = ∅.
       Equivalent to: Lev(cfM ) is defined.
       For subsets M of a Polish space:
       M is in Fσ ∩ Gδ ⇐⇒ M is reducible ⇐⇒ Lev(cfM ) is defined.
Level of Discontinuity         Tests in Computation Trees      Degrees of Discontinuity



                          The Schwarz Genus
       Let p : X → Y be continuous and surjective. A function
       s :⊆ Y → X is a section if p(s(y )) = y for all y ∈ dom(s).
Level of Discontinuity             Tests in Computation Trees    Degrees of Discontinuity



                              The Schwarz Genus
       Let p : X → Y be continuous and surjective. A function
       s :⊆ Y → X is a section if p(s(y )) = y for all y ∈ dom(s).

       minLev(p) := min{Lev(s) | s : Y → X is a total section},
                  g(p) := min{k | (∃ open V1 , . . . , Vk ⊆ Y ) ( k Vi = Y ∧
                                                                  i=1
                                  (∀i ≤ k ) (∃ cont. section si : Vi → X ))}.
       g(p) is called Schwarz genus.
Level of Discontinuity             Tests in Computation Trees     Degrees of Discontinuity



                              The Schwarz Genus
       Let p : X → Y be continuous and surjective. A function
       s :⊆ Y → X is a section if p(s(y )) = y for all y ∈ dom(s).

       minLev(p) := min{Lev(s) | s : Y → X is a total section},
                  g(p) := min{k | (∃ open V1 , . . . , Vk ⊆ Y ) ( k Vi = Y ∧
                                                                  i=1
                                  (∀i ≤ k ) (∃ cont. section si : Vi → X ))}.
       g(p) is called Schwarz genus.

       Proposition
            • Always minLev(p) ≤ g(p).
            • Let Y be a locally connected metric space and p : X → Y
                a covering map. If g(p) or minLev(p) is finite, then

                                     g(p) = minLev(p).
Level of Discontinuity      Tests in Computation Trees   Degrees of Discontinuity



              The Number of Tests in Computation Trees
Level of Discontinuity       Tests in Computation Trees     Degrees of Discontinuity



              The Number of Tests in Computation Trees
       A Computation Tree over the real numbers is a tree containing
       unary and binary nodes where
Level of Discontinuity                     Tests in Computation Trees        Degrees of Discontinuity



              The Number of Tests in Computation Trees
       A Computation Tree over the real numbers is a tree containing
       unary and binary nodes where
         • unary nodes contain operations of the form
           “xi := f (xi1 , . . . , xik )” where
                    • xi1 , . . . , xik : values computed earlier,
                    • xi : value computed in the current node,
                    • f : some arithmetic operation, often only constants,
                         id, +, −, ∗, /, also exp, log, | · |, and others,
            • binary nodes contain tests of the form “xi ◦ 0” with
                ◦ ∈ {>, <, ≥, ≤},
            • the last node (leaf) on each path is unary.
Level of Discontinuity                     Tests in Computation Trees        Degrees of Discontinuity



              The Number of Tests in Computation Trees
       A Computation Tree over the real numbers is a tree containing
       unary and binary nodes where
         • unary nodes contain operations of the form
           “xi := f (xi1 , . . . , xik )” where
                    • xi1 , . . . , xik : values computed earlier,
                    • xi : value computed in the current node,
                    • f : some arithmetic operation, often only constants,
                         id, +, −, ∗, /, also exp, log, | · |, and others,
            • binary nodes contain tests of the form “xi ◦ 0” with
                ◦ ∈ {>, <, ≥, ≤},
            • the last node (leaf) on each path is unary.
       The computation
Level of Discontinuity                     Tests in Computation Trees        Degrees of Discontinuity



              The Number of Tests in Computation Trees
       A Computation Tree over the real numbers is a tree containing
       unary and binary nodes where
         • unary nodes contain operations of the form
           “xi := f (xi1 , . . . , xik )” where
                    • xi1 , . . . , xik : values computed earlier,
                    • xi : value computed in the current node,
                    • f : some arithmetic operation, often only constants,
                         id, +, −, ∗, /, also exp, log, | · |, and others,
            • binary nodes contain tests of the form “xi ◦ 0” with
                ◦ ∈ {>, <, ≥, ≤},
            • the last node (leaf) on each path is unary.
       The computation
            • starts in the root with the input in some registers x1 , . . . , xn .
            • ends if a leaf is reached. Then the value in that leaf is the
                result of the computation.
Level of Discontinuity      Tests in Computation Trees   Degrees of Discontinuity




       Computation trees over the real numbers are the common
       computation model in Computational Geometry.
Level of Discontinuity        Tests in Computation Trees    Degrees of Discontinuity




       Computation trees over the real numbers are the common
       computation model in Computational Geometry.

       A complexity theory has been built on this model by Blum,
       Shub and Smale (1989).
Level of Discontinuity        Tests in Computation Trees        Degrees of Discontinuity




       Computation trees over the real numbers are the common
       computation model in Computational Geometry.

       A complexity theory has been built on this model by Blum,
       Shub and Smale (1989).

       In practice problematic: the comparisons “xi ≥ 0”, etc
       (unstable).
Level of Discontinuity           Tests in Computation Trees   Degrees of Discontinuity



                         Continuous Computation Trees
Level of Discontinuity           Tests in Computation Trees   Degrees of Discontinuity



                         Continuous Computation Trees
       Similar to computation trees as above but
Level of Discontinuity            Tests in Computation Trees   Degrees of Discontinuity



                         Continuous Computation Trees
       Similar to computation trees as above but
            • over arbitrary topological spaces X , Y ,
            • with arbitrary continuous operations,
            • tests of the form “x ∈ O” for open sets O.
Level of Discontinuity            Tests in Computation Trees   Degrees of Discontinuity



                         Continuous Computation Trees
       Similar to computation trees as above but
            • over arbitrary topological spaces X , Y ,
            • with arbitrary continuous operations,
            • tests of the form “x ∈ O” for open sets O.

       Let T be a continuous computation tree.
Level of Discontinuity             Tests in Computation Trees   Degrees of Discontinuity



                         Continuous Computation Trees
       Similar to computation trees as above but
            • over arbitrary topological spaces X , Y ,
            • with arbitrary continuous operations,
            • tests of the form “x ∈ O” for open sets O.

       Let T be a continuous computation tree.

              Size(T ) := the number of its leaves
                          =   1 + the number of comparison nodes
Level of Discontinuity              Tests in Computation Trees      Degrees of Discontinuity



                         Continuous Computation Trees
       Similar to computation trees as above but
            • over arbitrary topological spaces X , Y ,
            • with arbitrary continuous operations,
            • tests of the form “x ∈ O” for open sets O.

       Let T be a continuous computation tree.

              Size(T ) := the number of its leaves
                          =   1 + the number of comparison nodes
          size(T , x) := the number of leaves that can be reached
                              if x may be disturbed slightly at the beginning
Level of Discontinuity              Tests in Computation Trees      Degrees of Discontinuity



                         Continuous Computation Trees
       Similar to computation trees as above but
            • over arbitrary topological spaces X , Y ,
            • with arbitrary continuous operations,
            • tests of the form “x ∈ O” for open sets O.

       Let T be a continuous computation tree.

              Size(T ) := the number of its leaves
                          =   1 + the number of comparison nodes
          size(T , x) := the number of leaves that can be reached
                              if x may be disturbed slightly at the beginning
         size (T , x) := the number of leaves that can be reached
                              if x may be disturbed slightly at any time
Level of Discontinuity              Tests in Computation Trees      Degrees of Discontinuity



                         Continuous Computation Trees
       Similar to computation trees as above but
            • over arbitrary topological spaces X , Y ,
            • with arbitrary continuous operations,
            • tests of the form “x ∈ O” for open sets O.

       Let T be a continuous computation tree.

              Size(T ) := the number of its leaves
                          =   1 + the number of comparison nodes
          size(T , x) := the number of leaves that can be reached
                              if x may be disturbed slightly at the beginning
         size (T , x) := the number of leaves that can be reached
                              if x may be disturbed slightly at any time

       Clear: size(T , x) ≤ size (T , x) ≤ Size(T )
Level of Discontinuity        Tests in Computation Trees     Degrees of Discontinuity


       For a continuous computation tree T let fT be the function
       computed by T .
Level of Discontinuity             Tests in Computation Trees       Degrees of Discontinuity


       For a continuous computation tree T let fT be the function
       computed by T .

       Theorem
       For any continuous computation tree T

                         lev(fT , x) ≤ size(T , x) ≤ size (T , x)

       for all x ∈ X and
                                  Lev(fT ) ≤ Size(T ).
Level of Discontinuity             Tests in Computation Trees       Degrees of Discontinuity


       For a continuous computation tree T let fT be the function
       computed by T .

       Theorem
       For any continuous computation tree T

                         lev(fT , x) ≤ size(T , x) ≤ size (T , x)

       for all x ∈ X and
                                  Lev(fT ) ≤ Size(T ).

       Theorem
       Let f be a function with Lev(f ) < ω. Then there exists a
       continuous computation tree T with

                         lev(fT , x) = size(T , x) = size (T , x)

       for all x ∈ X and
                                  Lev(fT ) = Size(T ).
Level of Discontinuity    Tests in Computation Trees   Degrees of Discontinuity



            Degenerate Configurations in Computational
                           Geometry
Level of Discontinuity        Tests in Computation Trees   Degrees of Discontinuity



            Degenerate Configurations in Computational
                           Geometry

       Let f be a geometric function to be computed.
Level of Discontinuity        Tests in Computation Trees   Degrees of Discontinuity



            Degenerate Configurations in Computational
                           Geometry

       Let f be a geometric function to be computed.
       Yap (1990) distinguishes between
Level of Discontinuity         Tests in Computation Trees   Degrees of Discontinuity



            Degenerate Configurations in Computational
                           Geometry

       Let f be a geometric function to be computed.
       Yap (1990) distinguishes between
        inherent                     algorithm-induced
                         and
        degeneracies x               degeneracies x
Level of Discontinuity         Tests in Computation Trees   Degrees of Discontinuity



            Degenerate Configurations in Computational
                           Geometry

       Let f be a geometric function to be computed.
       Yap (1990) distinguishes between
        inherent                     algorithm-induced
                         and
        degeneracies x               degeneracies x

       Can be modeled by:
Level of Discontinuity         Tests in Computation Trees          Degrees of Discontinuity



            Degenerate Configurations in Computational
                           Geometry

       Let f be a geometric function to be computed.
       Yap (1990) distinguishes between
        inherent                     algorithm-induced
                         and
        degeneracies x               degeneracies x

       Can be modeled by:

        lev(f , x) > 1   and         size(T , x) > 1 or size (T , x) > 1
Level of Discontinuity         Tests in Computation Trees          Degrees of Discontinuity



            Degenerate Configurations in Computational
                           Geometry

       Let f be a geometric function to be computed.
       Yap (1990) distinguishes between
        inherent                     algorithm-induced
                         and
        degeneracies x               degeneracies x

       Can be modeled by:

        lev(f , x) > 1   and         size(T , x) > 1 or size (T , x) > 1


       Yap: “It seems that induced degeneracies subsume inherent
       degeneracies”.
Level of Discontinuity          Tests in Computation Trees           Degrees of Discontinuity



            Degenerate Configurations in Computational
                           Geometry

       Let f be a geometric function to be computed.
       Yap (1990) distinguishes between
        inherent                      algorithm-induced
                          and
        degeneracies x                degeneracies x

       Can be modeled by:

        lev(f , x) > 1    and         size(T , x) > 1 or size (T , x) > 1


       Yap: “It seems that induced degeneracies subsume inherent
       degeneracies”.

       Our Theorem:      lev(fT , x) ≤ size(T , x) ≤ size (T , x).
Level of Discontinuity       Tests in Computation Trees   Degrees of Discontinuity



                         Topological Complexity
       of a problem P over the real numbers:
Level of Discontinuity         Tests in Computation Trees   Degrees of Discontinuity



                         Topological Complexity
       of a problem P over the real numbers:
       comptotal (P)
           ARI
       := minARI−trees T Size(T ) − 1
Level of Discontinuity       Tests in Computation Trees    Degrees of Discontinuity



                         Topological Complexity
       of a problem P over the real numbers:
       comptotal (P)
             ARI
       := minARI−trees T Size(T ) − 1
       = the minimum of the total number of comparisons in the
       tree,
       where the minimum is taken over all computation trees that
       solve the problem using a certain set ARI of operations.
Level of Discontinuity        Tests in Computation Trees   Degrees of Discontinuity



                          Topological Complexity
       of a problem P over the real numbers:
       comptotal (P)
             ARI
       := minARI−trees T Size(T ) − 1
       = the minimum of the total number of comparisons in the
       tree,
       where the minimum is taken over all computation trees that
       solve the problem using a certain set ARI of operations.


       Another variant:
Level of Discontinuity        Tests in Computation Trees   Degrees of Discontinuity



                          Topological Complexity
       of a problem P over the real numbers:
       comptotal (P)
             ARI
       := minARI−trees T Size(T ) − 1
       = the minimum of the total number of comparisons in the
       tree,
       where the minimum is taken over all computation trees that
       solve the problem using a certain set ARI of operations.


       Another variant:
                 path
       compARI (P)
       := the minimum of the maximum number of comparisons on
       a computation path in the tree,
       where the minimum is taken over all computation trees that
       solve the problem using a certain set ARI of operations.
Level of Discontinuity               Tests in Computation Trees      Degrees of Discontinuity




       For all computation trees:
                              path
                         compARI (P) ≥ log2 (comptotal (P)) + 1) .
                                                 ARI
Level of Discontinuity               Tests in Computation Trees      Degrees of Discontinuity




       For all computation trees:
                              path
                         compARI (P) ≥ log2 (comptotal (P)) + 1) .
                                                 ARI




       In fact, trees with ARI = {cont. op.} or ARI = {+, −, ∗, /, | · |}
       can be balanced leading to
                              path
                         compARI (P) = log2 (comptotal (P)) + 1)
                                                 ARI
Level of Discontinuity               Tests in Computation Trees      Degrees of Discontinuity




       For all computation trees:
                              path
                         compARI (P) ≥ log2 (comptotal (P)) + 1) .
                                                 ARI




       In fact, trees with ARI = {cont. op.} or ARI = {+, −, ∗, /, | · |}
       can be balanced leading to
                              path
                         compARI (P) = log2 (comptotal (P)) + 1)
                                                 ARI



       Hence, in these cases, comptotal (P) is finer and therefore more
                                  ARI
       interesting!
                                        X ! Computation Trees
        and their disjunction t1 _ t2 : Tests in fTRUE FALSEg are also total topological tests.
Level of Discontinuity                                                          Degrees of Discontinuity
        Lemma 2.10 Let t1 t2 : X ! fTRUE FALSEg be total topological tests and T1, T2, T3
        CCTs. Then the two CCTs in Figure 1 are equivalent to each other and the two CCTs in
        Figure 2 are via:
                      equivalent to
       Balancing at each point. each other. Furthermore, in both cases the branching number
        is the same


                          ?
                          H                                               ?         H
                     ; H t1 HH +                      ()                     ; H t1 ^ tHH +
                 ?       HH         ?
                                    H                               ? HH               2
                                                                                                  ?
                T1             ; H t2 HH +                     ; H tHHH +                       T3
                           ?      H H             ?           ? HH1       ?
                          T2                     T3          T1                     T2

                                     Figure 1: Equivalent CCTs, Part 1
                         Figure: Equivalent computation trees, Part 1


                               ?
                               H          ()
       and similarly in the other direction.                           ? H
                               ; H t1 HH +                        ; H t1 _ tHH +
                          ?
                          H      HH               ?           ?      HH     2
                                                                              ?
                     ; H t2 HH +                 T3          T1          ; H tHHH +
                 ?       HH             ?                              ? HH1        ?
                T1                   T2                                 T2                      T3
                                        X ! Computation Trees
        and their disjunction t1 _ t2 : Tests in fTRUE FALSEg are also total topological tests.
Level of Discontinuity                                                          Degrees of Discontinuity
        Lemma 2.10 Let t1 t2 : X ! fTRUE FALSEg be total topological tests and T1, T2, T3
        CCTs. Then the two CCTs in Figure 1 are equivalent to each other and the two CCTs in
        Figure 2 are via:
                      equivalent to
       Balancing at each point. each other. Furthermore, in both cases the branching number
        is the same


                          ?
                          H                                               ?          H
                     ; H t1 HH +                      ()                      ; H t1 ^ tHH +
                 ?       HH         ?
                                    H                               ? HH                2
                                                                                                  ?
                T1             ; H t2 HH +                     ; H tHHH +                       T3
                           ?      H H             ?           ? HH1       ?
                          T2                     T3          T1                     T2

                                     Figure 1: Equivalent CCTs, Part 1
                         Figure: Equivalent computation trees, Part 1


                               ?
                               H          ()
       and similarly in the other direction.                H             ?
                           ; H t1 HH +               ; H t1 _ tHH +
                         ? HH
       Possible for computation trees ?
                         HHH +                     ? HH 2
                                           using the operations     ?
                                                                    H
       {+, −, ∗, /, | · |} because with |T3| one can compute;min and H +
              ; H t2
                 ?       HH             ?
                                         ·        T1             H t1 H max.
                                                                          ?      HH               ?
                T1                   T2                                 T2                      T3
Level of Discontinuity   Tests in Computation Trees   Degrees of Discontinuity



   Topological Complexity of Problems over the Reals:
                      Examples
Level of Discontinuity       Tests in Computation Trees   Degrees of Discontinuity



   Topological Complexity of Problems over the Reals:
                      Examples
       Sorting real numbers. Three versions:
Level of Discontinuity               Tests in Computation Trees   Degrees of Discontinuity



   Topological Complexity of Problems over the Reals:
                      Examples
       Sorting real numbers. Three versions:
          1. Input: a vector (x1 , . . . , xn ) ∈ Rn .
Level of Discontinuity             Tests in Computation Trees          Degrees of Discontinuity



   Topological Complexity of Problems over the Reals:
                      Examples
       Sorting real numbers. Three versions:
          1. Input: a vector (x1 , . . . , xn ) ∈ Rn .
             Output: a vector (xπ(1) , . . . , xπ(n) ) ∈ Rn where π is a
             permutation of {1, . . . , n} such that xπ(1) ≤ . . . ≤ xπ(n) .
Level of Discontinuity             Tests in Computation Trees          Degrees of Discontinuity



   Topological Complexity of Problems over the Reals:
                      Examples
       Sorting real numbers. Three versions:
          1. Input: a vector (x1 , . . . , xn ) ∈ Rn .
             Output: a vector (xπ(1) , . . . , xπ(n) ) ∈ Rn where π is a
             permutation of {1, . . . , n} such that xπ(1) ≤ . . . ≤ xπ(n) .
             Top. Compl.: 0, if arbitrary cont. op.’s are allowed.
Level of Discontinuity               Tests in Computation Trees        Degrees of Discontinuity



   Topological Complexity of Problems over the Reals:
                      Examples
       Sorting real numbers. Three versions:
          1. Input: a vector (x1 , . . . , xn ) ∈ Rn .
             Output: a vector (xπ(1) , . . . , xπ(n) ) ∈ Rn where π is a
             permutation of {1, . . . , n} such that xπ(1) ≤ . . . ≤ xπ(n) .
             Top. Compl.: 0, if arbitrary cont. op.’s are allowed.
          2. Input: a vector (x1 , . . . , xn ) ∈ Rn .
Level of Discontinuity             Tests in Computation Trees          Degrees of Discontinuity



   Topological Complexity of Problems over the Reals:
                      Examples
       Sorting real numbers. Three versions:
          1. Input: a vector (x1 , . . . , xn ) ∈ Rn .
             Output: a vector (xπ(1) , . . . , xπ(n) ) ∈ Rn where π is a
             permutation of {1, . . . , n} such that xπ(1) ≤ . . . ≤ xπ(n) .
             Top. Compl.: 0, if arbitrary cont. op.’s are allowed.
          2. Input: a vector (x1 , . . . , xn ) ∈ Rn .
             Output: a permutation π of {1, . . . , n} such that
             xπ(1) ≤ . . . ≤ xπ(n) .
Level of Discontinuity             Tests in Computation Trees          Degrees of Discontinuity



   Topological Complexity of Problems over the Reals:
                      Examples
       Sorting real numbers. Three versions:
          1. Input: a vector (x1 , . . . , xn ) ∈ Rn .
             Output: a vector (xπ(1) , . . . , xπ(n) ) ∈ Rn where π is a
             permutation of {1, . . . , n} such that xπ(1) ≤ . . . ≤ xπ(n) .
             Top. Compl.: 0, if arbitrary cont. op.’s are allowed.
          2. Input: a vector (x1 , . . . , xn ) ∈ Rn .
             Output: a permutation π of {1, . . . , n} such that
             xπ(1) ≤ . . . ≤ xπ(n) .
             Top. Compl.: n − 1, even if arbitrary cont. op.’s are allowed.
Level of Discontinuity              Tests in Computation Trees          Degrees of Discontinuity



   Topological Complexity of Problems over the Reals:
                      Examples
       Sorting real numbers. Three versions:
          1. Input: a vector (x1 , . . . , xn ) ∈ Rn .
             Output: a vector (xπ(1) , . . . , xπ(n) ) ∈ Rn where π is a
             permutation of {1, . . . , n} such that xπ(1) ≤ . . . ≤ xπ(n) .
             Top. Compl.: 0, if arbitrary cont. op.’s are allowed.
          2. Input: a vector (x1 , . . . , xn ) ∈ Rn .
             Output: a permutation π of {1, . . . , n} such that
             xπ(1) ≤ . . . ≤ xπ(n) .
             Top. Compl.: n − 1, even if arbitrary cont. op.’s are allowed.
          3. Input: a vector (x1 , . . . , xn ) ∈ Rn with xi = xj for i = j.
Level of Discontinuity              Tests in Computation Trees          Degrees of Discontinuity



   Topological Complexity of Problems over the Reals:
                      Examples
       Sorting real numbers. Three versions:
          1. Input: a vector (x1 , . . . , xn ) ∈ Rn .
             Output: a vector (xπ(1) , . . . , xπ(n) ) ∈ Rn where π is a
             permutation of {1, . . . , n} such that xπ(1) ≤ . . . ≤ xπ(n) .
             Top. Compl.: 0, if arbitrary cont. op.’s are allowed.
          2. Input: a vector (x1 , . . . , xn ) ∈ Rn .
             Output: a permutation π of {1, . . . , n} such that
             xπ(1) ≤ . . . ≤ xπ(n) .
             Top. Compl.: n − 1, even if arbitrary cont. op.’s are allowed.
          3. Input: a vector (x1 , . . . , xn ) ∈ Rn with xi = xj for i = j.
             Output: a permutation π of {1, . . . , n} such that
             xπ(1) ≤ . . . ≤ xπ(n) .
Level of Discontinuity              Tests in Computation Trees          Degrees of Discontinuity



   Topological Complexity of Problems over the Reals:
                      Examples
       Sorting real numbers. Three versions:
          1. Input: a vector (x1 , . . . , xn ) ∈ Rn .
             Output: a vector (xπ(1) , . . . , xπ(n) ) ∈ Rn where π is a
             permutation of {1, . . . , n} such that xπ(1) ≤ . . . ≤ xπ(n) .
             Top. Compl.: 0, if arbitrary cont. op.’s are allowed.
          2. Input: a vector (x1 , . . . , xn ) ∈ Rn .
             Output: a permutation π of {1, . . . , n} such that
             xπ(1) ≤ . . . ≤ xπ(n) .
             Top. Compl.: n − 1, even if arbitrary cont. op.’s are allowed.
          3. Input: a vector (x1 , . . . , xn ) ∈ Rn with xi = xj for i = j.
             Output: a permutation π of {1, . . . , n} such that
             xπ(1) ≤ . . . ≤ xπ(n) .
             Top. Compl.: 0, if arbitrary cont. op.’s are allowed.
Level of Discontinuity      Tests in Computation Trees   Degrees of Discontinuity



                         Algebraic Topology
Level of Discontinuity        Tests in Computation Trees      Degrees of Discontinuity



                          Algebraic Topology

       Smale (1987) invented the notion topological complexity and
       asked for the topological complexity of the following problem:
Level of Discontinuity          Tests in Computation Trees        Degrees of Discontinuity



                            Algebraic Topology

       Smale (1987) invented the notion topological complexity and
       asked for the topological complexity of the following problem:
       Input: a number δ > 0 and a vector (an−1 , . . . , a0 ) ∈ Cn such
       that the polynomial p(z) := z n + an−1 · z n−1 + . . . + a1 · z + a0
       has n pairwise different complex zeros.
Level of Discontinuity          Tests in Computation Trees        Degrees of Discontinuity



                            Algebraic Topology

       Smale (1987) invented the notion topological complexity and
       asked for the topological complexity of the following problem:
       Input: a number δ > 0 and a vector (an−1 , . . . , a0 ) ∈ Cn such
       that the polynomial p(z) := z n + an−1 · z n−1 + . . . + a1 · z + a0
       has n pairwise different complex zeros.
       Output: a vector (ζ1 , . . . , ζn ) ∈ Cn approximating the zeros
       z1 , . . . , zn of p in the sense |ζi − zi | ≤ δ for all i.
Level of Discontinuity          Tests in Computation Trees        Degrees of Discontinuity



                            Algebraic Topology

       Smale (1987) invented the notion topological complexity and
       asked for the topological complexity of the following problem:
       Input: a number δ > 0 and a vector (an−1 , . . . , a0 ) ∈ Cn such
       that the polynomial p(z) := z n + an−1 · z n−1 + . . . + a1 · z + a0
       has n pairwise different complex zeros.
       Output: a vector (ζ1 , . . . , ζn ) ∈ Cn approximating the zeros
       z1 , . . . , zn of p in the sense |ζi − zi | ≤ δ for all i.

       Comment:       The top.
       compl. of Smale’s prob-
       lem for sufficiently small
       δ is equal to the top.
       compl. of the problem to
       sort n pairwise different
       complex numbers.
Level of Discontinuity          Tests in Computation Trees                 Degrees of Discontinuity



                            Algebraic Topology

       Smale (1987) invented the notion topological complexity and
       asked for the topological complexity of the following problem:
       Input: a number δ > 0 and a vector (an−1 , . . . , a0 ) ∈ Cn such
       that the polynomial p(z) := z n + an−1 · z n−1 + . . . + a1 · z + a0
       has n pairwise different complex zeros.
       Output: a vector (ζ1 , . . . , ζn ) ∈ Cn approximating the zeros
       z1 , . . . , zn of p in the sense |ζi − zi | ≤ δ for all i.

       Comment:       The top.
       compl. of Smale’s prob-
                                                             Cn
       lem for sufficiently small
                                                  πn              d   pn
       δ is equal to the top.                                      d
       compl. of the problem to                 ©
                                                             ∼
                                                             =
                                                                    ‚
                                                                    d
       sort n pairwise different              Pn '                    Cn /Sn
                                                             ρn
       complex numbers.
Level of Discontinuity     Tests in Computation Trees   Degrees of Discontinuity




       Theorem (Vassiliev 1988)
Level of Discontinuity                 Tests in Computation Trees       Degrees of Discontinuity




       Theorem (Vassiliev 1988)
       Let

                         Dp (n) := the sum of the digits of n, written
                                     in base p, for a prime number p,
                         U(n) := n + 1 − min{Dp (n) | primes p}.

       Then for all sufficiently small δ > 0:

                    U(n) − 1 ≤ comptotal op. (Smale’s problem) ≤ n − 1
                                   cont.
Level of Discontinuity                 Tests in Computation Trees       Degrees of Discontinuity




       Theorem (Vassiliev 1988)
       Let

                         Dp (n) := the sum of the digits of n, written
                                     in base p, for a prime number p,
                         U(n) := n + 1 − min{Dp (n) | primes p}.

       Then for all sufficiently small δ > 0:

                    U(n) − 1 ≤ comptotal op. (Smale’s problem) ≤ n − 1
                                   cont.



       Remark: If n is a prime power then U(n) = n, hence, the top.
       compl. of Smale’s problem is n − 1.
Level of Discontinuity                 Tests in Computation Trees       Degrees of Discontinuity




       Theorem (Vassiliev 1988)
       Let

                         Dp (n) := the sum of the digits of n, written
                                     in base p, for a prime number p,
                         U(n) := n + 1 − min{Dp (n) | primes p}.

       Then for all sufficiently small δ > 0:

                    U(n) − 1 ≤ comptotal op. (Smale’s problem) ≤ n − 1
                                   cont.



       Remark: If n is a prime power then U(n) = n, hence, the top.
       compl. of Smale’s problem is n − 1.
       Question: What is top. complexity of Smale’s problem if n is not
       a prime power?
Level of Discontinuity        Tests in Computation Trees     Degrees of Discontinuity




       Question: What is the top. complexity of Smale’s problem if n is
       not a prime power? Is it then n − 1 as well?
Level of Discontinuity            Tests in Computation Trees        Degrees of Discontinuity




       Question: What is the top. complexity of Smale’s problem if n is
       not a prime power? Is it then n − 1 as well?

       Theorem (De Concini et al 2003)
            • For n = 6 it is not 5 but 4,
            • If n = 3 · 2m , for some positive integer m, then it is also
                < n − 1.
Level of Discontinuity      Tests in Computation Trees   Degrees of Discontinuity



               Another Example from Algebraic Topology
Level of Discontinuity         Tests in Computation Trees   Degrees of Discontinuity



               Another Example from Algebraic Topology
       S n := {x ∈ Rn+1 | ||x|| = 1}.
Level of Discontinuity         Tests in Computation Trees   Degrees of Discontinuity



               Another Example from Algebraic Topology
       S n := {x ∈ Rn+1 | ||x|| = 1}.

       Borsuk’s Antipodal Theorem
Level of Discontinuity         Tests in Computation Trees     Degrees of Discontinuity



               Another Example from Algebraic Topology
       S n := {x ∈ Rn+1 | ||x|| = 1}.

       Borsuk’s Antipodal Theorem
       For n ≥ 1, for every continuous function f : S n → Rn there
       exists at least one point x ∈ S n with f (−x) = f (x).
Level of Discontinuity          Tests in Computation Trees       Degrees of Discontinuity



               Another Example from Algebraic Topology
       S n := {x ∈ Rn+1 | ||x|| = 1}.

       Borsuk’s Antipodal Theorem
       For n ≥ 1, for every continuous function f : S n → Rn there
       exists at least one point x ∈ S n with f (−x) = f (x).


       A function f : S n → Rm is called odd if f (−x) = −f (x) for all
       x ∈ Sn.
Level of Discontinuity          Tests in Computation Trees       Degrees of Discontinuity



               Another Example from Algebraic Topology
       S n := {x ∈ Rn+1 | ||x|| = 1}.

       Borsuk’s Antipodal Theorem
       For n ≥ 1, for every continuous function f : S n → Rn there
       exists at least one point x ∈ S n with f (−x) = f (x).


       A function f : S n → Rm is called odd if f (−x) = −f (x) for all
       x ∈ Sn.

       Easily shown to be
       Equivalent to Borsuk’s Antipodal Theorem:
Level of Discontinuity                Tests in Computation Trees         Degrees of Discontinuity



               Another Example from Algebraic Topology
       S n := {x ∈ Rn+1 | ||x|| = 1}.

       Borsuk’s Antipodal Theorem
       For n ≥ 1, for every continuous function f : S n → Rn there
       exists at least one point x ∈ S n with f (−x) = f (x).


       A function f : S n → Rm is called odd if f (−x) = −f (x) for all
       x ∈ Sn.

       Easily shown to be
       Equivalent to Borsuk’s Antipodal Theorem:
       For n ≥ 1,

                         min{Lev(f ) | f : S n → {−1, 1} odd} = n + 1.
Level of Discontinuity    Tests in Computation Trees   Degrees of Discontinuity




       Another Example:
Level of Discontinuity      Tests in Computation Trees   Degrees of Discontinuity




       Another Example: Problem: compute f := cfRn .
                                                 +
Level of Discontinuity            Tests in Computation Trees     Degrees of Discontinuity




       Another Example: Problem: compute f := cfRn .
                                                 +

       Proposition
                           path
       comptotal (f ) = compARI (f ) = 1
            ARI
       for {+, −, ∗, /, | · |} ⊆ ARI ⊆ {continuous functions}.
Level of Discontinuity               Tests in Computation Trees   Degrees of Discontinuity




       Another Example: Problem: compute f := cfRn .
                                                 +

       Proposition
                              path
       comptotal (f ) = compARI (f ) = 1
            ARI
       for {+, −, ∗, /, | · |} ⊆ ARI ⊆ {continuous functions}.
            • Every computation tree using arbitrary continuous
                functions needs at least 1 test.
Level of Discontinuity                 Tests in Computation Trees   Degrees of Discontinuity




       Another Example: Problem: compute f := cfRn .
                                                 +

       Proposition
                                path
       comptotal (f ) = compARI (f ) = 1
            ARI
       for {+, −, ∗, /, | · |} ⊆ ARI ⊆ {continuous functions}.
            • Every computation tree using arbitrary continuous
                functions needs at least 1 test.
            • There exists a computation tree using only 1 test and the
                arithmetic operations +, −, ∗, /, | · |:
                test whether min{x1 , . . . , xn } > 0.
Level of Discontinuity                 Tests in Computation Trees   Degrees of Discontinuity




       Another Example: Problem: compute f := cfRn .
                                                 +

       Proposition
                                path
       comptotal (f ) = compARI (f ) = 1
            ARI
       for {+, −, ∗, /, | · |} ⊆ ARI ⊆ {continuous functions}.
            • Every computation tree using arbitrary continuous
                functions needs at least 1 test.
            • There exists a computation tree using only 1 test and the
                arithmetic operations +, −, ∗, /, | · |:
                test whether min{x1 , . . . , xn } > 0.

       Now with analytic functions .....
Level of Discontinuity       Tests in Computation Trees   Degrees of Discontinuity


       Problem: compute f := cfRn .
                                +
Level of Discontinuity            Tests in Computation Trees     Degrees of Discontinuity


       Problem: compute f := cfRn .
                                +

       Theorem
                           path
       comptotal (f ) = compARI (f ) = n for ARI ⊆ {analytic functions}.
           ARI
Level of Discontinuity                 Tests in Computation Trees   Degrees of Discontinuity


       Problem: compute f := cfRn .
                                +

       Theorem
                                path
       comptotal (f ) = compARI (f ) = n for ARI ⊆ {analytic functions}.
           ARI
            • There exists a computation tree using only n tests and the
                comparisons xi > 0 for i = 1, . . . , n.
Level of Discontinuity                 Tests in Computation Trees   Degrees of Discontinuity


       Problem: compute f := cfRn .
                                +

       Theorem
                                path
       comptotal (f ) = compARI (f ) = n for ARI ⊆ {analytic functions}.
           ARI
            • There exists a computation tree using only n tests and the
                comparisons xi > 0 for i = 1, . . . , n.
Level of Discontinuity                 Tests in Computation Trees   Degrees of Discontinuity




       Problem: compute f := cfRn .
                                +

       Theorem
                                path
       comptotal (f ) = compARI (f ) = n for ARI ⊆ {analytic functions}.
           ARI
            • There exists a computation tree using only n tests and the
                comparisons xi > 0 for i = 1, . . . , n.
            • Rabin’s Theorem (1972) (proof corrected by Montaña et
                al(1994)): Every computation tree using only analytic
                functions as arithmetic operations needs at least n
                comparisons on the longest path in the tree!
Level of Discontinuity                 Tests in Computation Trees   Degrees of Discontinuity




       Problem: compute f := cfRn .
                                +

       Theorem
                                path
       comptotal (f ) = compARI (f ) = n for ARI ⊆ {analytic functions}.
           ARI
            • There exists a computation tree using only n tests and the
                comparisons xi > 0 for i = 1, . . . , n.
            • Rabin’s Theorem (1972) (proof corrected by Montaña et
                al(1994)): Every computation tree using only analytic
                functions as arithmetic operations needs at least n
                comparisons on the longest path in the tree!

       Note: A short proof for Rabin’s theorem was given by Vassiliev
       in 1997 using the Newton polyhedron of a power series.
Level of Discontinuity   Tests in Computation Trees   Degrees of Discontinuity



         The Topological Complexity of Zero Finding for
           Continuous Functions in Various Settings
Level of Discontinuity             Tests in Computation Trees         Degrees of Discontinuity



         The Topological Complexity of Zero Finding for
           Continuous Functions in Various Settings


                         F   := {f ∈ C[0, 1] | f (0) · f (1) < 0} ,
Level of Discontinuity              Tests in Computation Trees         Degrees of Discontinuity



         The Topological Complexity of Zero Finding for
           Continuous Functions in Various Settings


                          F   := {f ∈ C[0, 1] | f (0) · f (1) < 0} ,
                         Fnd := {f ∈ F | f is nondecreasing} ,
Level of Discontinuity              Tests in Computation Trees         Degrees of Discontinuity



         The Topological Complexity of Zero Finding for
           Continuous Functions in Various Settings


                          F   := {f ∈ C[0, 1] | f (0) · f (1) < 0} ,
                         Fnd := {f ∈ F | f is nondecreasing} ,
                         Finc := {f ∈ F | f is increasing} .
Level of Discontinuity              Tests in Computation Trees         Degrees of Discontinuity



         The Topological Complexity of Zero Finding for
           Continuous Functions in Various Settings


                          F   := {f ∈ C[0, 1] | f (0) · f (1) < 0} ,
                         Fnd := {f ∈ F | f is nondecreasing} ,
                         Finc := {f ∈ F | f is increasing} .


       A real number x ∈ [0, 1] is a zero of a function f : [0, 1] → R if
       f (x) = 0.
Level of Discontinuity              Tests in Computation Trees         Degrees of Discontinuity



         The Topological Complexity of Zero Finding for
           Continuous Functions in Various Settings


                          F   := {f ∈ C[0, 1] | f (0) · f (1) < 0} ,
                         Fnd := {f ∈ F | f is nondecreasing} ,
                         Finc := {f ∈ F | f is increasing} .


       A real number x ∈ [0, 1] is a zero of a function f : [0, 1] → R if
       f (x) = 0. For ε > 0, an ε–approximation of a number x ∗ is a
       number x with |x − x ∗ | ≤ ε.
Level of Discontinuity              Tests in Computation Trees         Degrees of Discontinuity



         The Topological Complexity of Zero Finding for
           Continuous Functions in Various Settings


                          F   := {f ∈ C[0, 1] | f (0) · f (1) < 0} ,
                         Fnd := {f ∈ F | f is nondecreasing} ,
                         Finc := {f ∈ F | f is increasing} .


       A real number x ∈ [0, 1] is a zero of a function f : [0, 1] → R if
       f (x) = 0. For ε > 0, an ε–approximation of a number x ∗ is a
       number x with |x − x ∗ | ≤ ε.

       Problem
       Fix some class G ∈ {F , Fnd , Finc } and some ε > 0.
Level of Discontinuity              Tests in Computation Trees         Degrees of Discontinuity



         The Topological Complexity of Zero Finding for
           Continuous Functions in Various Settings


                          F   := {f ∈ C[0, 1] | f (0) · f (1) < 0} ,
                         Fnd := {f ∈ F | f is nondecreasing} ,
                         Finc := {f ∈ F | f is increasing} .


       A real number x ∈ [0, 1] is a zero of a function f : [0, 1] → R if
       f (x) = 0. For ε > 0, an ε–approximation of a number x ∗ is a
       number x with |x − x ∗ | ≤ ε.

       Problem
       Fix some class G ∈ {F , Fnd , Finc } and some ε > 0.
       Input: A function f ∈ G, given: an oracle for function values.
Level of Discontinuity              Tests in Computation Trees         Degrees of Discontinuity



         The Topological Complexity of Zero Finding for
           Continuous Functions in Various Settings


                          F   := {f ∈ C[0, 1] | f (0) · f (1) < 0} ,
                         Fnd := {f ∈ F | f is nondecreasing} ,
                         Finc := {f ∈ F | f is increasing} .


       A real number x ∈ [0, 1] is a zero of a function f : [0, 1] → R if
       f (x) = 0. For ε > 0, an ε–approximation of a number x ∗ is a
       number x with |x − x ∗ | ≤ ε.

       Problem
       Fix some class G ∈ {F , Fnd , Finc } and some ε > 0.
       Input: A function f ∈ G, given: an oracle for function values.
       Output: An ε-approximation of a zero of f .
Level of Discontinuity            Tests in Computation Trees      Degrees of Discontinuity



         The Topological Complexity of Zero Finding for
           Continuous Functions in Various Settings


       Results for the three function classes F , Fnd , and Finc and for
       different classes ARI of allowed arithmetic operations:
            • ARI = {+, −, ∗, /, exp, log} and
                ARI = {all operations satisfying a Hölder condition on each
                bounded subset of their domain}:
                           z
                [Novak, Wo´ niakowski 1996].
            • {+, −, ∗, /, | · |} ⊆ ARI ⊆ {arbitrary continuous operations}:
                [H 1996].
            • ARI = {+, −, ∗, /}: [H 2002]
Level of Discontinuity   Tests in Computation Trees   Degrees of Discontinuity



                             Bisection
Level of Discontinuity              Tests in Computation Trees   Degrees of Discontinuity



                                        Bisection




            • Maximum number of tests on a path:
                  (log2 (1/(2ε)) ∼ log2 (1/(2ε)).
Level of Discontinuity         Tests in Computation Trees   Degrees of Discontinuity



                                   Bisection




            • Maximum number of tests on a path:
               (log2 (1/(2ε)) ∼ log2 (1/(2ε)).
            • Total number of tests: 1/(2ε) − 1
Level of Discontinuity        Tests in Computation Trees    Degrees of Discontinuity



                                 Trisection
       An algorithm for computing an ε-approximation of the zero of
       f ∈ Finc , using {+, −, ∗, /, | · |} and no tests:
Level of Discontinuity                 Tests in Computation Trees           Degrees of Discontinuity



                                          Trisection
       An algorithm for computing an ε-approximation of the zero of
       f ∈ Finc , using {+, −, ∗, /, | · |} and no tests:

       Start with [a, b] := [0, 1] and repeat the following loop
       max{0, − log3/2 (2ε) } times.
       Output: midpoint of the last interval [a, b].


                     begin {loop}
                       c := (b − a)/3;
                       for j = 0, . . . , 3 do begin xj := a + j · c;
                                                       zj := f (xj ) end;
                       for j = 1, 2 do rj := max{0, −zj+1 · zj−1 };
                       x := (x1 r1 + x2 r2 )/(r1 + r2 );
                       a := x − c; b := x + c
                     end {loop}
Level of Discontinuity           Tests in Computation Trees   Degrees of Discontinuity



                     Zero Finding for Increasing Functions


       Theorem
       For G = Finc and 0 < ε < 1/2.
Level of Discontinuity             Tests in Computation Trees       Degrees of Discontinuity



                     Zero Finding for Increasing Functions


       Theorem
       For G = Finc and 0 < ε < 1/2.
            • For
                {+, −, ∗, /, | · |} ⊆ ARI ⊆ {arbitrary continuous operations}
                                   comptotal (Finc , ε) = 0 .
                                       ARI
Level of Discontinuity             Tests in Computation Trees       Degrees of Discontinuity



                     Zero Finding for Increasing Functions


       Theorem
       For G = Finc and 0 < ε < 1/2.
            • For
                {+, −, ∗, /, | · |} ⊆ ARI ⊆ {arbitrary continuous operations}
                                    comptotal (Finc , ε) = 0 .
                                        ARI
            • [Novak, Wo´ niakowski 1996] For
                        z
                ARI = {+, −, ∗, /, exp, log}
                                    comptotal (Finc , ε) = 0 .
                                        ARI
Level of Discontinuity             Tests in Computation Trees       Degrees of Discontinuity



                     Zero Finding for Increasing Functions


       Theorem
       For G = Finc and 0 < ε < 1/2.
            • For
                {+, −, ∗, /, | · |} ⊆ ARI ⊆ {arbitrary continuous operations}
                                    comptotal (Finc , ε) = 0 .
                                        ARI
            • [Novak, Wo´ niakowski 1996] For
                        z
                ARI = {+, −, ∗, /, exp, log}
                                    comptotal (Finc , ε) = 0 .
                                        ARI
            • For ARI = {+, −, ∗, /}
                                    comptotal (Finc , ε) = 1
                                        ARI
Level of Discontinuity         Tests in Computation Trees   Degrees of Discontinuity



                         Results for comptotal (G, ε)
                                         ARI
Level of Discontinuity                 Tests in Computation Trees       Degrees of Discontinuity



                              Results for comptotal (G, ε)
                                              ARI



                         {+, −, ∗, /, | · |} ⊆ ARI ⊆ {arb. cont. op.}
          G = Finc
          G = Fnd
          G=F
Level of Discontinuity                 Tests in Computation Trees       Degrees of Discontinuity



                              Results for comptotal (G, ε)
                                              ARI



                         {+, −, ∗, /, | · |} ⊆ ARI ⊆ {arb. cont. op.}
          G = Finc                              0
          G = Fnd
          G=F
Level of Discontinuity                 Tests in Computation Trees       Degrees of Discontinuity



                              Results for comptotal (G, ε)
                                              ARI



                         {+, −, ∗, /, | · |} ⊆ ARI ⊆ {arb. cont. op.}
          G = Finc                              0
          G = Fnd                      log2 (ε−1 + 2) − 2
          G=F
Level of Discontinuity                 Tests in Computation Trees       Degrees of Discontinuity



                              Results for comptotal (G, ε)
                                              ARI



                         {+, −, ∗, /, | · |} ⊆ ARI ⊆ {arb. cont. op.}
          G = Finc                              0
          G = Fnd                      log2 (ε−1 + 2) − 2
          G=F                          log2 (ε−1 + 2) − 2
Level of Discontinuity                 Tests in Computation Trees         Degrees of Discontinuity



                              Results for comptotal (G, ε)
                                              ARI



                         {+, −, ∗, /, | · |} ⊆ ARI ⊆ {arb. cont. op.}   ARI = {+, −, ∗, /}
          G = Finc                              0
          G = Fnd                      log2 (ε−1 + 2) − 2
          G=F                          log2 (ε−1 + 2) − 2
Level of Discontinuity                 Tests in Computation Trees         Degrees of Discontinuity



                              Results for comptotal (G, ε)
                                              ARI



                         {+, −, ∗, /, | · |} ⊆ ARI ⊆ {arb. cont. op.}   ARI = {+, −, ∗, /}
          G = Finc                              0                              1
          G = Fnd                      log2 (ε−1 + 2) − 2
          G=F                          log2 (ε−1 + 2) − 2
Level of Discontinuity                 Tests in Computation Trees         Degrees of Discontinuity



                              Results for comptotal (G, ε)
                                              ARI



                         {+, −, ∗, /, | · |} ⊆ ARI ⊆ {arb. cont. op.}   ARI = {+, −, ∗, /}
          G = Finc                              0                              1
          G = Fnd                      log2 (ε−1 + 2) − 2                 ∼ log2 (1/ε)
          G=F                          log2 (ε−1 + 2) − 2
Level of Discontinuity                 Tests in Computation Trees         Degrees of Discontinuity



                              Results for comptotal (G, ε)
                                              ARI



                         {+, −, ∗, /, | · |} ⊆ ARI ⊆ {arb. cont. op.}   ARI = {+, −, ∗, /}
          G = Finc                              0                              1
          G = Fnd                      log2 (ε−1 + 2) − 2                 ∼ log2 (1/ε)
          G=F                          log2 (ε−1 + 2) − 2                  1/(2ε) − 1
Level of Discontinuity                 Tests in Computation Trees         Degrees of Discontinuity



                              Results for comptotal (G, ε)
                                              ARI



                         {+, −, ∗, /, | · |} ⊆ ARI ⊆ {arb. cont. op.}   ARI = {+, −, ∗, /}
          G = Finc                              0                              1
          G = Fnd                      log2 (ε−1 + 2) − 2                 ∼ log2 (1/ε)
          G=F                          log2 (ε−1 + 2) − 2                  1/(2ε) − 1


       Results for comppath (G, ε):
                       ARI
Level of Discontinuity                 Tests in Computation Trees         Degrees of Discontinuity



                              Results for comptotal (G, ε)
                                              ARI



                         {+, −, ∗, /, | · |} ⊆ ARI ⊆ {arb. cont. op.}   ARI = {+, −, ∗, /}
          G = Finc                              0                              1
          G = Fnd                      log2 (ε−1 + 2) − 2                 ∼ log2 (1/ε)
          G=F                          log2 (ε−1 + 2) − 2                  1/(2ε) − 1


       Results for comppath (G, ε): ∼ log2 (comptotal (G, ε)).
                       ARI                      ARI
Level of Discontinuity                 Tests in Computation Trees         Degrees of Discontinuity



                              Results for comptotal (G, ε)
                                              ARI



                         {+, −, ∗, /, | · |} ⊆ ARI ⊆ {arb. cont. op.}   ARI = {+, −, ∗, /}
          G = Finc                              0                              1
          G = Fnd                      log2 (ε−1 + 2) − 2                 ∼ log2 (1/ε)
          G=F                          log2 (ε−1 + 2) − 2                  1/(2ε) − 1


       Results for comppath (G, ε): ∼ log2 (comptotal (G, ε)).
                       ARI                      ARI



       Bisection algorithm:
Level of Discontinuity                 Tests in Computation Trees         Degrees of Discontinuity



                              Results for comptotal (G, ε)
                                              ARI



                         {+, −, ∗, /, | · |} ⊆ ARI ⊆ {arb. cont. op.}   ARI = {+, −, ∗, /}
          G = Finc                              0                              1
          G = Fnd                      log2 (ε−1 + 2) − 2                 ∼ log2 (1/ε)
          G=F                          log2 (ε−1 + 2) − 2                  1/(2ε) − 1


       Results for comppath (G, ε): ∼ log2 (comptotal (G, ε)).
                       ARI                      ARI



       Bisection algorithm: number of tests: 1/(2ε) − 1 ≈ 1/(2ε).
Level of Discontinuity                 Tests in Computation Trees         Degrees of Discontinuity



                              Results for comptotal (G, ε)
                                              ARI



                         {+, −, ∗, /, | · |} ⊆ ARI ⊆ {arb. cont. op.}   ARI = {+, −, ∗, /}
          G = Finc                              0                              1
          G = Fnd                      log2 (ε−1 + 2) − 2                 ∼ log2 (1/ε)
          G=F                          log2 (ε−1 + 2) − 2                  1/(2ε) − 1


       Results for comppath (G, ε): ∼ log2 (comptotal (G, ε)).
                       ARI                      ARI



       Bisection algorithm: number of tests: 1/(2ε) − 1 ≈ 1/(2ε).

       Surprise: There is an exponentially better algorithm with
       number of tests only log2 (ε−1 + 2) − 2 ≈ log2 (1/(2ε)).
Level of Discontinuity            Tests in Computation Trees                Degrees of Discontinuity


       Test-optimal algorithm
       for continuous f : [0, 1] → R with f (0) < 0 < f (1), and
       {+, −, ∗, /, | · |} ⊆ ARI ⊆ {arbitrary continuous operations}:




                                                      1         x       0     if x < 0
        sig : R \ {0} → {0, 1},      sig(x) :=          · (1 +     )=
                                                      2        |x|      1     if x > 0
Level of Discontinuity         Tests in Computation Trees       Degrees of Discontinuity


       Test-optimal algorithm
       for continuous f : [0, 1] → R with f (0) < 0 < f (1), and
       {+, −, ∗, /, | · |} ⊆ ARI ⊆ {arbitrary continuous operations}:
Level of Discontinuity         Tests in Computation Trees       Degrees of Discontinuity


       Test-optimal algorithm
       for continuous f : [0, 1] → R with f (0) < 0 < f (1), and
       {+, −, ∗, /, | · |} ⊆ ARI ⊆ {arbitrary continuous operations}:
Level of Discontinuity        Tests in Computation Trees   Degrees of Discontinuity



                         Degrees of Discontinuity
       Let V , W , X , Y be topological spaces,
       f :⊆ V → W and g :⊆ X → Y be functions.
Level of Discontinuity                 Tests in Computation Trees   Degrees of Discontinuity



                               Degrees of Discontinuity
       Let V , W , X , Y be topological spaces,
       f :⊆ V → W and g :⊆ X → Y be functions.
       Definition
                         f ≤0 g : ⇐⇒    (∃ cont. B) (∀x ∈ dom f )
                                        f (x) = gB(x)
                                        (Wadge reducibility)
Level of Discontinuity                 Tests in Computation Trees      Degrees of Discontinuity



                               Degrees of Discontinuity
       Let V , W , X , Y be topological spaces,
       f :⊆ V → W and g :⊆ X → Y be functions.
       Definition
                         f ≤0 g : ⇐⇒    (∃ cont. B) (∀x ∈ dom f )
                                        f (x) = gB(x)
                                        (Wadge reducibility)
                         f ≤1 g : ⇐⇒    (∃ cont. A, B) (∀x ∈ dom f )
                                        f (x) = AgB(x)
                                        (Weihrauch 1992)
Level of Discontinuity                 Tests in Computation Trees      Degrees of Discontinuity



                               Degrees of Discontinuity
       Let V , W , X , Y be topological spaces,
       f :⊆ V → W and g :⊆ X → Y be functions.
       Definition
                         f ≤0 g : ⇐⇒    (∃ cont. B) (∀x ∈ dom f )
                                        f (x) = gB(x)
                                        (Wadge reducibility)
                         f ≤1 g : ⇐⇒    (∃ cont. A, B) (∀x ∈ dom f )
                                        f (x) = AgB(x)
                                        (Weihrauch 1992)
                         f ≤2 g : ⇐⇒    (∃ cont. A, B) (∀x ∈ dom f )
                                        f (x) = A(x, gB(x))
                                        (Hirsch 1993, Weihrauch 1992)
Level of Discontinuity           Tests in Computation Trees   Degrees of Discontinuity




       Lemma
            • The relations ≤i are reflexive and transitive.
Level of Discontinuity           Tests in Computation Trees   Degrees of Discontinuity




       Lemma
            • The relations ≤i are reflexive and transitive.
            • f ≤0 g ⇒ f ≤1 g and f ≤1 g ⇒ f ≤2 g.
Level of Discontinuity            Tests in Computation Trees   Degrees of Discontinuity




       Lemma
            • The relations ≤i are reflexive and transitive.
            • f ≤0 g ⇒ f ≤1 g and f ≤1 g ⇒ f ≤2 g.
            • If f ≤i g then Lev(f ) ≤ Lev (g).
Level of Discontinuity         Tests in Computation Trees     Degrees of Discontinuity



                   Continuous Reductions Directly over R




       Theorem
       For i = 0, 1 the ≤i -relation is not well-founded on
       {F : R → {0, 1} | Lev(f ) = 2}.
Level of Discontinuity               Tests in Computation Trees            Degrees of Discontinuity



                         Continuous Reductions over Σω

       Theorem (H 1993, Selivanov 2007)
            • With any function f :⊆ Σω → {0, . . . , k − 1} one can
                associate a forest (= set of trees) F(f ) describing the
                discontinuities of f and define a relation ≤0 on trees and
                forests so that

                                  f ≤0 g ⇐⇒ F(f ) ≤0 G(g)

                for any f , g :⊆ Σω → {0, . . . , k − 1} of level < ω1 .
Level of Discontinuity               Tests in Computation Trees            Degrees of Discontinuity



                         Continuous Reductions over Σω

       Theorem (H 1993, Selivanov 2007)
            • With any function f :⊆ Σω → {0, . . . , k − 1} one can
                associate a forest (= set of trees) F(f ) describing the
                discontinuities of f and define a relation ≤0 on trees and
                forests so that

                                  f ≤0 g ⇐⇒ F(f ) ≤0 G(g)

                for any f , g :⊆ Σω → {0, . . . , k − 1} of level < ω1 .
            • Similarly for ≤1 and ≤2 for functions of finite level
                (works probably also for functions of level < ω1 : work in
                progress).
Level of Discontinuity              Tests in Computation Trees                  Degrees of Discontinuity



       Trees Measuring the Discontinuities of Functions
       Example
       For f : {0, 1}ω → {2, 3, 7} defined by
                                
                                7 if p = 0ω
                                
                       f (p) := 3 if (∃i ∈ N) p = 0i 1ω
                                
                                  2 otherwise.
                                



                  F(f ) =   {   2, 3, 7, 3,                  7,            7           }.
                                                           t                  d
                                                            t                  d
                                             2          2         3   2    3    3


                                                                                2
Level of Discontinuity           Tests in Computation Trees   Degrees of Discontinuity



                    The ≤0 Relation on Trees and Forests



       For trees T1 and T2 :

               T1 ≤0 T2 : ⇐⇒ (∃f : T1 → T2 ) f preserves ancestors
                                              and node values.
Level of Discontinuity              Tests in Computation Trees    Degrees of Discontinuity



                    The ≤0 Relation on Trees and Forests



       For trees T1 and T2 :

               T1 ≤0 T2 : ⇐⇒ (∃f : T1 → T2 ) f preserves ancestors
                                              and node values.

       For forests F1 and F2 :

                     F1 ≤0 F2 : ⇐⇒ (∀T1 ∈ F1 ) (∃T2 ∈ F2 ) T1 ≤0 T2 .
Level of Discontinuity           Tests in Computation Trees      Degrees of Discontinuity



                    The ≤2 Relation on Trees and Forests


       For trees T1 and T2 :

       T1 ≤2 T2 : ⇐⇒       (∃f : T1 → T2 ) f preserves ancestors and
                            if two nodes on a path in T1 have different values
                            then there images have different values as well.
Level of Discontinuity              Tests in Computation Trees      Degrees of Discontinuity



                    The ≤2 Relation on Trees and Forests


       For trees T1 and T2 :

       T1 ≤2 T2 : ⇐⇒          (∃f : T1 → T2 ) f preserves ancestors and
                               if two nodes on a path in T1 have different values
                               then there images have different values as well.

       For forests F1 and F2 :

                     F1 ≤2 F2 : ⇐⇒ (∀T1 ∈ F1 ) (∃T2 ∈ F2 ) T1 ≤2 T2 .
Level of Discontinuity          Tests in Computation Trees       Degrees of Discontinuity



   Continuous Degrees: Finite Refinement of the Level

       Corollary
       For any n ∈ N≥2 and k ∈ N, the number of ≡i -classes of
       functions f :⊆ Σω → {0, . . . , k − 1} of level ≤ n is finite.
Level of Discontinuity          Tests in Computation Trees       Degrees of Discontinuity



   Continuous Degrees: Finite Refinement of the Level

       Corollary
       For any n ∈ N≥2 and k ∈ N, the number of ≡i -classes of
       functions f :⊆ Σω → {0, . . . , k − 1} of level ≤ n is finite.

       But for ≤0 and ≤1 it grows quickly for growing k .
Level of Discontinuity          Tests in Computation Trees       Degrees of Discontinuity



   Continuous Degrees: Finite Refinement of the Level

       Corollary
       For any n ∈ N≥2 and k ∈ N, the number of ≡i -classes of
       functions f :⊆ Σω → {0, . . . , k − 1} of level ≤ n is finite.

       But for ≤0 and ≤1 it grows quickly for growing k .

       Not so for ≤2 once k ≥ n!
Level of Discontinuity          Tests in Computation Trees       Degrees of Discontinuity



   Continuous Degrees: Finite Refinement of the Level

       Corollary
       For any n ∈ N≥2 and k ∈ N, the number of ≡i -classes of
       functions f :⊆ Σω → {0, . . . , k − 1} of level ≤ n is finite.

       But for ≤0 and ≤1 it grows quickly for growing k .

       Not so for ≤2 once k ≥ n!

       Theorem
       For any n ∈ N, the number of ≡2 -classes of functions f defined
       on a subset of Σω and with discrete range of level ≤ n is finite.
Level of Discontinuity          Tests in Computation Trees       Degrees of Discontinuity



   Continuous Degrees: Finite Refinement of the Level

       Corollary
       For any n ∈ N≥2 and k ∈ N, the number of ≡i -classes of
       functions f :⊆ Σω → {0, . . . , k − 1} of level ≤ n is finite.

       But for ≤0 and ≤1 it grows quickly for growing k .

       Not so for ≤2 once k ≥ n!

       Theorem
       For any n ∈ N, the number of ≡2 -classes of functions f defined
       on a subset of Σω and with discrete range of level ≤ n is finite.

       Work in progress: also for functions f :⊆ Σω → Σω ?
Level of Discontinuity        Tests in Computation Trees      Degrees of Discontinuity



   Continuous Degrees: Finite Refinement of the Level


       Theorem
       With any function f :⊆ Σω → Y , Y an arbitrary discrete space,
       one can associate a forest (= set of trees) F (f ) describing the
       discontinuities of f and define a relation ≤2 on trees and forests
       so that
                           f ≤2 g ⇐⇒ F (f ) ≤2 G (g)
       for any f :⊆ Σω → Y , f :⊆ Σω → Z (Y , Z discrete spaces) of
       level < ω.

       Work in progress: also for functions f :⊆ Σω → Σω ?
Level of Discontinuity             Tests in Computation Trees         Degrees of Discontinuity



             The Degrees are invariant under admissible
                         representations


       Theorem
       Let δ :⊆ Σω → X be an admissible representation,
       Y a discrete space,
       f :⊆ X → Y a function.
            •
                                       F (f ) ≡2 F (f δ).


            • Let Y be countable, ν :⊆ {0, 1}∗ → Y be a notation of Y .
                • For every (δ, ν)-realization F of f one has F (f ) ≤2 F (F ).
                • There is a (δ, ν)-realization F of f with F (f ) ≡2 F (F ).
Level of Discontinuity         Tests in Computation Trees    Degrees of Discontinuity



                   A Stronger Version of Smale’s Problem


       Smale’s Problem
       How many tests does an algorithm need which, given a
       complex polynomial of degree n with pairwise different zeros,
       computes a vector of zeros of the polynomial?
Level of Discontinuity         Tests in Computation Trees     Degrees of Discontinuity



                   A Stronger Version of Smale’s Problem


       Smale’s Problem
       How many tests does an algorithm need which, given a
       complex polynomial of degree n with pairwise different zeros,
       computes a vector of zeros of the polynomial?

       Equivalent to:
       What is the minimum level of discontinuity of any function
       f : {z ∈ Cn | (∀i, j)(i = j ⇒ zi = zj } → Sn
       with (∀z ∈ dom(f )) (∀γ ∈ Sn ) f (γz)γz = f (z)z?
Level of Discontinuity         Tests in Computation Trees     Degrees of Discontinuity



                   A Stronger Version of Smale’s Problem


       Smale’s Problem
       How many tests does an algorithm need which, given a
       complex polynomial of degree n with pairwise different zeros,
       computes a vector of zeros of the polynomial?

       Equivalent to:
       What is the minimum level of discontinuity of any function
       f : {z ∈ Cn | (∀i, j)(i = j ⇒ zi = zj } → Sn
       with (∀z ∈ dom(f )) (∀γ ∈ Sn ) f (γz)γz = f (z)z?

       More difficult problem
       Determine the possible ≤2 -minimal forests of such functions!
Level of Discontinuity          Tests in Computation Trees   Degrees of Discontinuity



                         Summary and open problems
Level of Discontinuity            Tests in Computation Trees   Degrees of Discontinuity



                         Summary and open problems

            • Closely related:
                • Level of discontinuity
                • Schwarz genus
                • number of tests in computation trees
Level of Discontinuity                Tests in Computation Trees   Degrees of Discontinuity



                         Summary and open problems

            • Closely related:
                • Level of discontinuity
                • Schwarz genus
                • number of tests in computation trees
            • These notions give rise to questions in set theory and
                algebraic topology,
Level of Discontinuity                Tests in Computation Trees   Degrees of Discontinuity



                         Summary and open problems

            • Closely related:
                • Level of discontinuity
                • Schwarz genus
                • number of tests in computation trees
            • These notions give rise to questions in set theory and
                algebraic topology,
            • Number of tests in computation trees with restricted set of
                arithmetic operations: algebraic complexity theory.
Level of Discontinuity                Tests in Computation Trees   Degrees of Discontinuity



                         Summary and open problems

            • Closely related:
                • Level of discontinuity
                • Schwarz genus
                • number of tests in computation trees
            • These notions give rise to questions in set theory and
                algebraic topology,
            • Number of tests in computation trees with restricted set of
                arithmetic operations: algebraic complexity theory.
            • Can also be analyzed for algorithms in numerical
                mathematics (information-based complexity).
Level of Discontinuity                Tests in Computation Trees   Degrees of Discontinuity



                         Summary and open problems

            • Closely related:
                • Level of discontinuity
                • Schwarz genus
                • number of tests in computation trees
            • These notions give rise to questions in set theory and
                algebraic topology,
            • Number of tests in computation trees with restricted set of
                arithmetic operations: algebraic complexity theory.
            • Can also be analyzed for algorithms in numerical
                mathematics (information-based complexity).
            • Continuous reducibilities lead to natural refinements of the
                level.
Level of Discontinuity                Tests in Computation Trees   Degrees of Discontinuity



                         Summary and open problems

            • Closely related:
                • Level of discontinuity
                • Schwarz genus
                • number of tests in computation trees
            • These notions give rise to questions in set theory and
                algebraic topology,
            • Number of tests in computation trees with restricted set of
                arithmetic operations: algebraic complexity theory.
            • Can also be analyzed for algorithms in numerical
                mathematics (information-based complexity).
            • Continuous reducibilities lead to natural refinements of the
                level.
            • Develop practical theory for handling discontinuities!
                E.g., in computational geometry.

				
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Description: Institut fAr Theoretische Informatik und Mathematik. UniversitAt der ... Ininity in Logic and Computation. Cape Town, 3a5 November 2007. Level of Discontinuity ...