Applied Proof Theory: Proof Interpretations and Their Use in Mathematics

Transcription

1 Applied Proof Theory: Proof Interpretations and Their Use in Mathematics Ulrich Kohlenbach Department of Mathematics Technische Universität Darmstadt ASL Logic Colloquium, Sofia, July 31-Aug 5, 2009 Corrected version Nov.20: a confused slide on the functional interpretation of weak compactness as well as a slide stating a bound on Browder s theorem have been deleted as the latter has been superseded meanwhile: weak compactess can be bypassed resulting in a primitive recursive bound.

2 Overview of Contents Lecture I: General Introduction to the Unwinding of Proofs ( Proof Mining ) and the Proof-Theoretic Methods Used.

3 Overview of Contents Lecture I: General Introduction to the Unwinding of Proofs ( Proof Mining ) and the Proof-Theoretic Methods Used. Lecture II: Logical Metatheorems for Proof Mining:

4 Overview of Contents Lecture I: General Introduction to the Unwinding of Proofs ( Proof Mining ) and the Proof-Theoretic Methods Used. Lecture II: Logical Metatheorems for Proof Mining: The case of compact spaces and continuous functions.

5 Overview of Contents Lecture I: General Introduction to the Unwinding of Proofs ( Proof Mining ) and the Proof-Theoretic Methods Used. Lecture II: Logical Metatheorems for Proof Mining: The case of compact spaces and continuous functions. Application in numerical analysis.

6 Overview of Contents Lecture I: General Introduction to the Unwinding of Proofs ( Proof Mining ) and the Proof-Theoretic Methods Used. Lecture II: Logical Metatheorems for Proof Mining: The case of compact spaces and continuous functions. Application in numerical analysis. General metatheorems for noncompact and nonseparable abstract metric, hyperbolic and normed spaces.

7 Overview of Contents Lecture I: General Introduction to the Unwinding of Proofs ( Proof Mining ) and the Proof-Theoretic Methods Used. Lecture II: Logical Metatheorems for Proof Mining: The case of compact spaces and continuous functions. Application in numerical analysis. General metatheorems for noncompact and nonseparable abstract metric, hyperbolic and normed spaces. Lecture III: Applications in Nonlinear Analysis and Topological Dynamics.

8 Lecture I

9 Extractive Proof Theory (G. Kreisel): New results by logical analysis of proofs Input: Ineffective proof P of C

10 Extractive Proof Theory (G. Kreisel): New results by logical analysis of proofs Input: Ineffective proof P of C Goal: Additional information on C:

11 Extractive Proof Theory (G. Kreisel): New results by logical analysis of proofs Input: Ineffective proof P of C Goal: Additional information on C: effective bounds,

12 Extractive Proof Theory (G. Kreisel): New results by logical analysis of proofs Input: Ineffective proof P of C Goal: Additional information on C: effective bounds, algorithms,

13 Extractive Proof Theory (G. Kreisel): New results by logical analysis of proofs Input: Ineffective proof P of C Goal: Additional information on C: effective bounds, algorithms, continuous dependency or full independence from certain parameters,

14 Extractive Proof Theory (G. Kreisel): New results by logical analysis of proofs Input: Ineffective proof P of C Goal: Additional information on C: effective bounds, algorithms, continuous dependency or full independence from certain parameters, generalizations of proofs: weakening of premises.

15 Extractive Proof Theory (G. Kreisel): New results by logical analysis of proofs Input: Ineffective proof P of C Goal: Additional information on C: effective bounds, algorithms, continuous dependency or full independence from certain parameters, generalizations of proofs: weakening of premises. E.g. Let C x IN y IN F (x, y)

16 Extractive Proof Theory (G. Kreisel): New results by logical analysis of proofs Input: Ineffective proof P of C Goal: Additional information on C: effective bounds, algorithms, continuous dependency or full independence from certain parameters, generalizations of proofs: weakening of premises. E.g. Let C x IN y IN F (x, y) Naive Attempt: try to extract an explicit computable function realizing (or bounding) y : x IN F (x, f (x)).

17 Naive attempt fails Proposition There exist a sentence A x y z A qf (x, y, z) in the language of arithmetic (A qf quantifier-free and hence decidable), such A is logical valid,

18 Naive attempt fails Proposition There exist a sentence A x y z A qf (x, y, z) in the language of arithmetic (A qf quantifier-free and hence decidable), such A is logical valid, there is no recursive bound f s.t. x y f (x) z A qf (x, y, z).

19 Naive attempt fails Proposition There exist a sentence A x y z A qf (x, y, z) in the language of arithmetic (A qf quantifier-free and hence decidable), such A is logical valid, there is no recursive bound f s.t. x y f (x) z A qf (x, y, z). Proof: Take A : x y z ( T (x, x, y) T (x, x, z)), where T is the (primitive recursive) Kleene-T-predicate.

20 Naive attempt fails Proposition There exist a sentence A x y z A qf (x, y, z) in the language of arithmetic (A qf quantifier-free and hence decidable), such A is logical valid, there is no recursive bound f s.t. x y f (x) z A qf (x, y, z). Proof: Take A : x y z ( T (x, x, y) T (x, x, z)), where T is the (primitive recursive) Kleene-T-predicate. Any bound g on y, i.e. no computable g such that x y g(x) z (T (x, x, y) T (x, x, z)) since this would solve the halting problem!

21 However, one can obtain such witness candidates and bounds (and even realizing function(al)s) for a weakened version A H of A: Definition A x 1 y 1 x 2 y 2 A qf (x 1, y 1, x 2, y 2 ). Then the Herbrand normal form of A is defined as A H : x 1, x 2 A qf (x 1, f (x 1 ), x 2, g(x 1, x 2 )), where f, g are new function symbols, called index functions.

22 However, one can obtain such witness candidates and bounds (and even realizing function(al)s) for a weakened version A H of A: Definition A x 1 y 1 x 2 y 2 A qf (x 1, y 1, x 2, y 2 ). Then the Herbrand normal form of A is defined as A H : x 1, x 2 A qf (x 1, f (x 1 ), x 2, g(x 1, x 2 )), where f, g are new function symbols, called index functions. A and A H are equivalent with respect to logical validity, i.e. but are not logically equivalent. = A = A H,

23 We now consider again the sentence A x y z(p(x, y) P(x, z)),

24 We now consider again the sentence A x y z(p(x, y) P(x, z)), In contrast to A, the Herbrand normal form A H of A A H y ( P(x, y) P(x, g(y)) ) allows to construct a list of candidates (uniformly in x, g) for y, namely (c, g(c)) (and also (x, g(x))) for any constant c):

25 We now consider again the sentence A x y z(p(x, y) P(x, z)), In contrast to A, the Herbrand normal form A H of A A H y ( P(x, y) P(x, g(y)) ) allows to construct a list of candidates (uniformly in x, g) for y, namely (c, g(c)) (and also (x, g(x))) for any constant c): A H,D : ( P(x, c) P(x, g(c)) ) ( P(x, g(c)) P(x, g(g(c))) )

26 We now consider again the sentence A x y z(p(x, y) P(x, z)), In contrast to A, the Herbrand normal form A H of A A H y ( P(x, y) P(x, g(y)) ) allows to construct a list of candidates (uniformly in x, g) for y, namely (c, g(c)) (and also (x, g(x))) for any constant c): A H,D : ( P(x, c) P(x, g(c)) ) ( P(x, g(c)) P(x, g(g(c))) ) is a tautology. } {{ } TAUT

27 J. Herbrand s Theorem ( Théorème fondamental, 1930) Theorem Let A x 1 y 1 x 2 y 2 A qf (x 1, y 1, x 2, y 2 ). Then: PL A iff there are terms s 1,..., s k, t 1,..., t n (built up out of the constants and variables of A and the index functions used for the formation of A H ) such that A H,D : k i=1 j=1 n ( A qf si, f (s i ), t j, g(s i, t j ) ) is a tautology. A H,D is called Herbrand Disjunction.

28 J. Herbrand s Theorem ( Théorème fondamental, 1930) Theorem Let A x 1 y 1 x 2 y 2 A qf (x 1, y 1, x 2, y 2 ). Then: PL A iff there are terms s 1,..., s k, t 1,..., t n (built up out of the constants and variables of A and the index functions used for the formation of A H ) such that A H,D : k i=1 j=1 n ( A qf si, f (s i ), t j, g(s i, t j ) ) is a tautology. A H,D is called Herbrand Disjunction. Note that the length of this disjunction is fixed: k n.

29 J. Herbrand s Theorem ( Théorème fondamental, 1930) Theorem Let A x 1 y 1 x 2 y 2 A qf (x 1, y 1, x 2, y 2 ). Then: PL A iff there are terms s 1,..., s k, t 1,..., t n (built up out of the constants and variables of A and the index functions used for the formation of A H ) such that A H,D : k i=1 j=1 n ( A qf si, f (s i ), t j, g(s i, t j ) ) is a tautology. A H,D is called Herbrand Disjunction. Note that the length of this disjunction is fixed: k n. The terms s i, t j can be extracted from a given PL-proof of A.

30 Herbrand s Theorem continued Replacing in A H,D all terms g(s i, t j ), f (s i ), by new variables (treating larger terms first) results in another tautological disjunction A D s.t. A can be inferred from A by a direct proof.

31 Remark For sentences A x y z A qf (x, y, z), A D can be written in the form A qf (x, t 1, b 1 ) A qf (x, t 2, b 2 )... A qf (x, t k, b k ), where the b i are new variables and t i does not contain any b j with i j (used by Luckhardt s analysis of Roth s theorem, see below).

32 Remark For sentences A x y z A qf (x, y, z), A D can be written in the form A qf (x, t 1, b 1 ) A qf (x, t 2, b 2 )... A qf (x, t k, b k ), where the b i are new variables and t i does not contain any b j with i j (used by Luckhardt s analysis of Roth s theorem, see below). Herbrand s theorem immediately extends to first-order theories T whose non-logical axioms G 1,..., G n are all purely universal.

33 Theorem (Roth 1955) An algebraic irrational number α has only finitely many exceptionally good rational approximations, i.e. for ε > 0 there are only finitely many q IN such that R(q) : q > 1!p ZZ : (p, q) = 1 α pq 1 < q 2 ε.

34 Theorem (Roth 1955) An algebraic irrational number α has only finitely many exceptionally good rational approximations, i.e. for ε > 0 there are only finitely many q IN such that R(q) : q > 1!p ZZ : (p, q) = 1 α pq 1 < q 2 ε. Theorem (Luckhardt 1985/89) The following upper bound on #{q : R(q)} holds: #{q : R(q)} < 7 3 ε 1 log N α ε 5 log 2 d log(50ε 2 log d), where N α < max(21 log 2h(α), 2 log(1 + α )) and h is the logarithmic absolute homogeneous height and d = deg(α). Independently: Bombieri and van der Poorten 1988.

35 Limitations Techniques work only for restricted formal contexts: mainly purely universal ( algebraic ) axioms, restricted use of induction, no higher analytical principles.

36 Limitations Techniques work only for restricted formal contexts: mainly purely universal ( algebraic ) axioms, restricted use of induction, no higher analytical principles. Require that one can guess the correct Herbrand terms: in general procedure results in proofs of length 2 P n, where 2 k n+1 = 22k n (n cut complexity).

37 Towards generalizations of Herbrand s theorem Allow functionals Φ(x, f ) instead of just Herbrand terms: Let s consider again the example A x y z ( T (x, x, y) T (x, x, z)) ).

38 Towards generalizations of Herbrand s theorem Allow functionals Φ(x, f ) instead of just Herbrand terms: Let s consider again the example A x y z ( T (x, x, y) T (x, x, z)) ). A H can be realized by a computable functional of type level 2 which is defined by cases: Φ(x, g) := { x if T (x, x, g(x)) g(x) otherwise.

39 Towards generalizations of Herbrand s theorem Allow functionals Φ(x, f ) instead of just Herbrand terms: Let s consider again the example A x y z ( T (x, x, y) T (x, x, z)) ). A H can be realized by a computable functional of type level 2 which is defined by cases: Φ(x, g) := From this definition it easily follows that { x if T (x, x, g(x)) g(x) otherwise. x, g ( T (x, x, Φ(x, g)) T (x, x, g(φ(x, g)) ). Φ satisfies G. Kreisel s no-counterexample interpretation!

40 If A is not provable in PL but e.g. in PA more complicated functionals are needed (Kreisel 1951):

41 If A is not provable in PL but e.g. in PA more complicated functionals are needed (Kreisel 1951): Let (a n ) be a nonincreasing sequence in [0, 1]. Then, clearly, (a n ) is convergent and so a Cauchy sequence which we write as: (1) k IN n IN m IN i, j [n; n + m] ( a i a j 2 k ), where [n; n + m] := {n, n + 1,..., n + m}.

42 If A is not provable in PL but e.g. in PA more complicated functionals are needed (Kreisel 1951): Let (a n ) be a nonincreasing sequence in [0, 1]. Then, clearly, (a n ) is convergent and so a Cauchy sequence which we write as: (1) k IN n IN m IN i, j [n; n + m] ( a i a j 2 k ), where [n; n + m] := {n, n + 1,..., n + m}. Then the (partial) Herbrand normal form of this statement is (2) k IN g IN IN n IN i, j [n; n + g(n)] ( a i a j 2 k ).

43 By E. Specker 1949 there exist computable such sequences (a n ) even in Q [0, 1] without computable bound on n in (1).

44 By E. Specker 1949 there exist computable such sequences (a n ) even in Q [0, 1] without computable bound on n in (1). By contrast, there is a simple (primitive recursive) bound Φ (g, k) on (2) (also referred to as metastability by T.Tao):

45 By E. Specker 1949 there exist computable such sequences (a n ) even in Q [0, 1] without computable bound on n in (1). By contrast, there is a simple (primitive recursive) bound Φ (g, k) on (2) (also referred to as metastability by T.Tao): Proposition Let (a n ) be any nonincreasing sequence in [0, 1] then k IN g IN IN n Φ (g, k) i, j [n; n + g(n)] ( a i a j 2 k ), where Φ (g, k) := g (2k ) (0) with g(n) := n + g(n). Moreover, there exists an i < 2 k such that n can be taken as g (i) (0).

46 Remark The previous result can be viewed as a polished form of a Herbrand disjunction of variable (in k) length: 2 k 1 i=0 ( a g (i) (0) a g( g (i) (0)) 2 k).

47 Remark The previous result can be viewed as a polished form of a Herbrand disjunction of variable (in k) length: 2 k 1 i=0 ( a g (i) (0) a g( g (i) (0)) 2 k). Corollary (T. Tao s finite convergence principle) k IN, g : IN IN M IN 1 a 0... a M 0 N IN ( N + g(n) M n, m [N, N + g(n)]( an a m 2 k)). One may take N := Φ (g, k) as above.

48 An Example from Ergodic Theory X Hilbert space, f : X X linear and f (x) x for all x X. A n (x) := 1 n + 1 S n(x), where S n (x) := n f i (x) (n 0). i=0

49 An Example from Ergodic Theory X Hilbert space, f : X X linear and f (x) x for all x X. A n (x) := 1 n + 1 S n(x), where S n (x) := n f i (x) (n 0). i=0 Theorem (von Neumann Mean Ergodic Theorem) For every x X, the sequence (A n (x)) n converges.

50 An Example from Ergodic Theory X Hilbert space, f : X X linear and f (x) x for all x X. A n (x) := 1 n + 1 S n(x), where S n (x) := n f i (x) (n 0). i=0 Theorem (von Neumann Mean Ergodic Theorem) For every x X, the sequence (A n (x)) n converges. Avigad/Gerhardy/Towsner (TAMS to appear): in general no computable rate of convergence.

51 An Example from Ergodic Theory X Hilbert space, f : X X linear and f (x) x for all x X. A n (x) := 1 n + 1 S n(x), where S n (x) := n f i (x) (n 0). i=0 Theorem (von Neumann Mean Ergodic Theorem) For every x X, the sequence (A n (x)) n converges. Avigad/Gerhardy/Towsner (TAMS to appear): in general no computable rate of convergence. Theorem (Garrett Birkhoff 1939) Mean Ergodic Theorem holds for uniformly convex Banach spaces.

52 Based on logical metatheorem to be discussed in the 2nd lecture: Theorem (K./Leuştean, to appear in Ergodic Theor. Dynam. Syst.) X uniformly convex Banach space, η a modulus of uniform convexity and f : X X as above, b > 0. Then for all x X with x b, all ε > 0, all g : IN IN : n Φ(ε, g, b, η) i, j [n; n + g(n)] ( A i (x) A j (x) < ε ),

53 Based on logical metatheorem to be discussed in the 2nd lecture: Theorem (K./Leuştean, to appear in Ergodic Theor. Dynam. Syst.) X uniformly convex Banach space, η a modulus of uniform convexity and f : X X as above, b > 0. Then for all x X with x b, all ε > 0, all g : IN IN : where n Φ(ε, g, b, η) i, j [n; n + g(n)] ( A i (x) A j (x) < ε ), Φ(ε, g, b, η) := M h K (0), with M := 16b ε, γ := ε 16 η ( ) ε 8b, K := h, h : IN IN, h(n) := 2(Mn + g(mn)), h(n) := maxi n h(i). b γ,

54 Based on logical metatheorem to be discussed in the 2nd lecture: Theorem (K./Leuştean, to appear in Ergodic Theor. Dynam. Syst.) X uniformly convex Banach space, η a modulus of uniform convexity and f : X X as above, b > 0. Then for all x X with x b, all ε > 0, all g : IN IN : where n Φ(ε, g, b, η) i, j [n; n + g(n)] ( A i (x) A j (x) < ε ), Φ(ε, g, b, η) := M h K (0), with M := 16b ε, γ := ε 16 η ( ) ε 8b, K := h, h : IN IN, h(n) := 2(Mn + g(mn)), h(n) := maxi n h(i). b γ, Special Hilbert case: treated prior by Avigad/Gerhardy/Towsner (again based on logical metatheorem).

55 Problems of the no-counterexample interpretation For principles F n.c.i. no longer correct. C n := {0, 1,..., n}.

56 Problems of the no-counterexample interpretation For principles F n.c.i. no longer correct. C n := {0, 1,..., n}. Direct example: Infinitary Pigeonhole Principle (IPP): n IN f : IN C n i n k IN m k ( f (m) = i ).

57 Problems of the no-counterexample interpretation For principles F n.c.i. no longer correct. C n := {0, 1,..., n}. Direct example: Infinitary Pigeonhole Principle (IPP): n IN f : IN C n i n k IN m k ( f (m) = i ). IPP causes arbitrary primitive recursive complexity, but (IPP) H n IN f : IN C n F : C n IN i n m F (i) ( f (m) = i )

58 Problems of the no-counterexample interpretation For principles F n.c.i. no longer correct. C n := {0, 1,..., n}. Direct example: Infinitary Pigeonhole Principle (IPP): n IN f : IN C n i n k IN m k ( f (m) = i ). IPP causes arbitrary primitive recursive complexity, but (IPP) H n IN f : IN C n F : C n IN i n m F (i) ( f (m) = i ) has trivial n.c.i.-solution for i, m : M(n, f, F ) := max{f (i) : i n} and I (n, f, F ) := f (M(n, f, F )).

59 Problems of the no-counterexample interpretation For principles F n.c.i. no longer correct. C n := {0, 1,..., n}. Direct example: Infinitary Pigeonhole Principle (IPP): n IN f : IN C n i n k IN m k ( f (m) = i ). IPP causes arbitrary primitive recursive complexity, but (IPP) H n IN f : IN C n F : C n IN i n m F (i) ( f (m) = i ) has trivial n.c.i.-solution for i, m : M(n, f, F ) := max{f (i) : i n} and I (n, f, F ) := f (M(n, f, F )). M, I do not reflect true complexity of IPP!

60 Problems of the no-counterexample interpretation For principles F n.c.i. no longer correct. C n := {0, 1,..., n}. Direct example: Infinitary Pigeonhole Principle (IPP): n IN f : IN C n i n k IN m k ( f (m) = i ). IPP causes arbitrary primitive recursive complexity, but (IPP) H n IN f : IN C n F : C n IN i n m F (i) ( f (m) = i ) has trivial n.c.i.-solution for i, m : M(n, f, F ) := max{f (i) : i n} and I (n, f, F ) := f (M(n, f, F )). M, I do not reflect true complexity of IPP! Related problem: bad behavior w.r.t. modus ponens!

61 A Modular Approach: Proof Interpretations Interpret the formulas A in P : A A I,

62 A Modular Approach: Proof Interpretations Interpret the formulas A in P : A A I, Interpretation C I contains the additional information,

63 A Modular Approach: Proof Interpretations Interpret the formulas A in P : A A I, Interpretation C I contains the additional information, Construct by recursion on P a new proof P I of C I.

64 A Modular Approach: Proof Interpretations Interpret the formulas A in P : A A I, Interpretation C I contains the additional information, Construct by recursion on P a new proof P I of C I. Our approach is based on novel forms and extensions of: K. Gödel s functional interpretation!

65 Gödel s functional interpretation in five minutes Gödel s functional interpretation D combined with Krivine s negative translation N results in an interpretation Sh = D N (Streicher/K.07) such that A A Sh (Shoenfield variant)

66 Gödel s functional interpretation in five minutes Gödel s functional interpretation D combined with Krivine s negative translation N results in an interpretation Sh = D N (Streicher/K.07) such that A A Sh A Sh x y A Sh (x, y), where A qf (Shoenfield variant) is quantifier-free,

67 Gödel s functional interpretation in five minutes Gödel s functional interpretation D combined with Krivine s negative translation N results in an interpretation Sh = D N (Streicher/K.07) such that A A Sh A Sh x y A Sh (x, y), where A qf (Shoenfield variant) For A x y A qf (x, y) one has A Sh A. is quantifier-free,

68 Gödel s functional interpretation in five minutes Gödel s functional interpretation D combined with Krivine s negative translation N results in an interpretation Sh = D N (Streicher/K.07) such that A A Sh A Sh x y A Sh (x, y), where A qf (Shoenfield variant) For A x y A qf (x, y) one has A Sh A. is quantifier-free, A A Sh by classical logic and quantifier-free choice in all types QF-AC : a b F qf (a, b) B a F qf (a, B(a)).

69 Gödel s functional interpretation in five minutes Gödel s functional interpretation D combined with Krivine s negative translation N results in an interpretation Sh = D N (Streicher/K.07) such that A A Sh A Sh x y A Sh (x, y), where A qf (Shoenfield variant) For A x y A qf (x, y) one has A Sh A. is quantifier-free, A A Sh by classical logic and quantifier-free choice in all types QF-AC : a b F qf (a, b) B a F qf (a, B(a)). x, y are tuples of functionals of finite type over the base types of the system at hand,

70 A Sh u x A Sh (u, x), B Sh v y B Sh (v, y).

71 A Sh u x A Sh (u, x), B Sh v y B Sh (v, y). (Sh1) P Sh P P Sh for atomic P

72 A Sh u x A Sh (u, x), B Sh v y B Sh (v, y). (Sh1) P Sh P P Sh for atomic P (Sh2) ( A) Sh f u A Sh (u, f (u))

73 A Sh u x A Sh (u, x), B Sh v y B Sh (v, y). (Sh1) P Sh P P Sh for atomic P (Sh2) ( A) Sh f u A Sh (u, f (u)) (Sh3) (A B) Sh u, v x, y ( A Sh (u, x) B Sh (v, y) )

74 A Sh u x A Sh (u, x), B Sh v y B Sh (v, y). (Sh1) P Sh P P Sh for atomic P (Sh2) ( A) Sh f u A Sh (u, f (u)) (Sh3) (A B) Sh u, v x, y ( A Sh (u, x) B Sh (v, y) ) (Sh4) ( z A) Sh z, u x A Sh (z, u, x)

75 A Sh u x A Sh (u, x), B Sh v y B Sh (v, y). (Sh1) P Sh P P Sh for atomic P (Sh2) ( A) Sh f u A Sh (u, f (u)) (Sh3) (A B) Sh u, v x, y ( A Sh (u, x) B Sh (v, y) ) (Sh4) ( z A) Sh z, u x A Sh (z, u, x) (Sh5) (A B) Sh f, v u, y ( A Sh (u, f (u)) B Sh (v, y) )

76 A Sh u x A Sh (u, x), B Sh v y B Sh (v, y). (Sh1) P Sh P P Sh for atomic P (Sh2) ( A) Sh f u A Sh (u, f (u)) (Sh3) (A B) Sh u, v x, y ( A Sh (u, x) B Sh (v, y) ) (Sh4) ( z A) Sh z, u x A Sh (z, u, x) (Sh5) (A B) Sh f, v u, y ( A Sh (u, f (u)) B Sh (v, y) ) (Sh6) ( za) Sh U z, f A Sh (z, U(z, f ), f (U(z, f )))

77 A Sh u x A Sh (u, x), B Sh v y B Sh (v, y). (Sh1) P Sh P P Sh for atomic P (Sh2) ( A) Sh f u A Sh (u, f (u)) (Sh3) (A B) Sh u, v x, y ( A Sh (u, x) B Sh (v, y) ) (Sh4) ( z A) Sh z, u x A Sh (z, u, x) (Sh5) (A B) Sh f, v u, y ( A Sh (u, f (u)) B Sh (v, y) ) (Sh6) ( za) Sh U z, f A Sh (z, U(z, f ), f (U(z, f ))) (Sh7) (A B) Sh n, u, v x, y (n=0 A Sh (u, x)) (n 0 B Sh (v, y)) u, v x, y ( A Sh (u, x) B Sh (v, y) ).

78 Sh extracts from a given proof p p x y A qf (x, y) an explicit effective functional Φ realizing A Sh, i.e. x A qf (x, Φ(x)).

79 3. Monotone functional interpretation (K.1996) Monotone Sh extracts Φ such that Y ( Φ Y x A Sh (x, Y (x)) ),

80 3. Monotone functional interpretation (K.1996) Monotone Sh extracts Φ such that Y ( Φ Y x A Sh (x, Y (x)) ), where is some suitable notion of being a bound that applies to higher order function spaces (W.A. Howard) { x IN x : x x, x ρ τ x : y, y(y ρ y x (y ) τ x(y)). Also relevant: bounded functional interpretation (F. Ferreira, P. Oliva)

81 Monotone interpretation of PCM and IPP The monotone functional interpretation of PCM coincides with the interpretation given above (Tao s finitary PCM). The monotone functional interpretation yields a version of the finitary IPP propososed by T. Tao.

82 Monotone interpretation of PCM and IPP The monotone functional interpretation of PCM coincides with the interpretation given above (Tao s finitary PCM). The monotone functional interpretation yields a version of the finitary IPP propososed by T. Tao. Tao s original formulation was wrong as shown by Jaime Gaspar by a counterexample (see Tao s correction on his Blog and the acknowledment in his recent book)!

83 Monotone interpretation of PCM and IPP The monotone functional interpretation of PCM coincides with the interpretation given above (Tao s finitary PCM). The monotone functional interpretation yields a version of the finitary IPP propososed by T. Tao. Tao s original formulation was wrong as shown by Jaime Gaspar by a counterexample (see Tao s correction on his Blog and the acknowledment in his recent book)! Full story in: Gaspar/K. On Tao s finitary infinite pigeonhole principle (JSL, to appear).

84 Summarizing the discussion so far To exhibit the finitary combinatorial/computational content (f.c.c.) of ineffective principles P requires nontrivial transformations P MFI of P as provided by monotone functional interpretation (MFI). For P -sentence, P MFI provides uniform bound.

85 Summarizing the discussion so far To exhibit the finitary combinatorial/computational content (f.c.c.) of ineffective principles P requires nontrivial transformations P MFI of P as provided by monotone functional interpretation (MFI). For P -sentence, P MFI provides uniform bound. MFI provides a general method for carrying out the extraction of this f.c.c. throughout a given proof all the way down to the conclusion (already for IPP this is nontrivial).

86 Summarizing the discussion so far To exhibit the finitary combinatorial/computational content (f.c.c.) of ineffective principles P requires nontrivial transformations P MFI of P as provided by monotone functional interpretation (MFI). For P -sentence, P MFI provides uniform bound. MFI provides a general method for carrying out the extraction of this f.c.c. throughout a given proof all the way down to the conclusion (already for IPP this is nontrivial). Specialized to the two prime examples of infinitary principles discussed in Tao s essay Soft analysis, hard analysis and the finite convergence principle, MFI yields the finitary reformulations (with explicit bounds) suggested by Tao.

87 Tao on a finitary approach to analysis it is common to make a distinction between hard, quantitative, or finitary analysis on the one hand, and soft, qualitative, or infinitary analysis on the other hand.... It is fairly well known that the results obtained by hard and soft analysis resp. can be connected to each other by various correspondence principles or compactness principles. It is however my belief that the relationship between the two types of analysis is much deeper.... There are rigorous results from proof theory which can allow one to automatically convert certain types of qualitative arguments into quantitative ones... (T. Tao: Soft analysis, hard analysis, and the finite convergence principle, 2007)

88 Literature 1) Kohlenbach, U., Applied Proof Theory: Proof Interpretations and their Use in Mathematics. Springer Monographs in Mathematics. xx+536pp., Springer Heidelberg-Berlin, ) Kreisel, G., Macintyre, A., Constructive logic versus algebraization I. In: Troelstra, A.S., van Dalen, D. (eds.), Proc. L.E.J. Brouwer Centenary Symposium (Noordwijkerhout 1981), North-Holland (Amsterdam), pp (1982). 3) Luckhardt, H., Herbrand-Analysen zweier Beweise des Satzes von Roth: Polynomiale Anzahlschranken. J. Symbolic Logic 54, pp (1989). 4) Tao, T., Soft analysis, hard analysis, and the finite convergence principle. In: Structure and Randomness. AMS, 298pp., ) Special issue of Dialectica on Gödel s interpretation with contributions e.g. by Ferreira, Kohlenbach, Oliva, 2008.

89 Lecture II

90 General logical metatheorems I Context: continuous functions between constructively represented Polish spaces.

91 General logical metatheorems I Context: continuous functions between constructively represented Polish spaces. Uniformity w.r.t. parameters from compact Polish spaces.

92 General logical metatheorems I Context: continuous functions between constructively represented Polish spaces. Uniformity w.r.t. parameters from compact Polish spaces. Extraction of bounds from ineffective existence proofs.

93 K., : P Polish space, K a compact P-space, A existential. BA:= basic arithmetic, HBC Heine/Borel compactness (SEQ restricted sequential compactness).

94 K., : P Polish space, K a compact P-space, A existential. BA:= basic arithmetic, HBC Heine/Borel compactness (SEQ restricted sequential compactness). From a proof BA + HBC(+SEQ ) x P y K m INA (x, y, m)

95 K., : P Polish space, K a compact P-space, A existential. BA:= basic arithmetic, HBC Heine/Borel compactness (SEQ restricted sequential compactness). From a proof BA + HBC(+SEQ ) x P y K m INA (x, y, m) one can extract a closed term Φ of BA (+iteration) BA (+ IA ) x P y K m Φ(f x )A (x, y, m).

96 K., : P Polish space, K a compact P-space, A existential. BA:= basic arithmetic, HBC Heine/Borel compactness (SEQ restricted sequential compactness). From a proof BA + HBC(+SEQ ) x P y K m INA (x, y, m) one can extract a closed term Φ of BA (+iteration) Important: BA (+ IA ) x P y K m Φ(f x )A (x, y, m). Φ(f x ) does not depend on y K but on a representation f x of x!

97 Logical comments Heine-Borel compactness = WKL (binary König s lemma). WKL strict-σ 1 1 Π0 1 (see applications in algebra by Coquand, Lombardi, Roy...)

98 Logical comments Heine-Borel compactness = WKL (binary König s lemma). WKL strict-σ 1 1 Π0 1 (see applications in algebra by Coquand, Lombardi, Roy...) Restricted sequential compactness = restricted arithmetical comprehension.

99 Limits of Metatheorem for concrete spaces Compactness means constructively: completeness and total boundedness.

100 Limits of Metatheorem for concrete spaces Compactness means constructively: completeness and total boundedness. Necessity of completeness: The set [0, 2] Q is totally bounded and constructively representable and BA q [0, 2] Q n IN( q 2 > IR 2 n ). However: no uniform bound on n IN!

101 Necessity of total boundedness: Let B be the unit ball C[0, 1]. B is bounded and constructively representable. By Weierstraß theorem BA f B n IN ( n code of p Q[X ] s.t. p f < 1 2 but no uniform bound on n : take f n := sin(nx). )

102 Necessity of A -formula : Let (f n ) be the usual sequence of spike-functions in C[0, 1], s.t. (f n ) converges pointwise but not uniformly towards 0. Then BA x [0, 1] k IN n IN m IN( f n+m (x) 2 k ), but no uniform bound on n (proof based on Σ 0 1 -LEM).

103 Necessity of A -formula : Let (f n ) be the usual sequence of spike-functions in C[0, 1], s.t. (f n ) converges pointwise but not uniformly towards 0. Then BA x [0, 1] k IN n IN m IN( f n+m (x) 2 k ), but no uniform bound on n (proof based on Σ 0 1 -LEM). Uniform bound only if (f n (x)) monotone (Dini): m IN superfluous!

104 Necessity of Φ(f x ) depending on a representative of x : Consider BA x IR n IN((n) IR > IR x). Suppose there would exist an = IR -extensional computable Φ : IN IN IN producing such a n. Then Φ would represent a continuous and hence constant function IR IN which gives a contradiction.

105 Unique existence X, K Polish, K compact, f : X K IR (BA-definable).

106 Unique existence X, K Polish, K compact, f : X K IR (BA-definable). MFI transforms uniqueness statements x X, y 1, y 2 K ( 2 f (x, y i ) = IR 0 d K (y 1, y 2 ) = IR 0 ) i=1 into moduli of uniqueness Φ : Q + Q + x X, y 1, y 2 K, ε > 0 ( 2 f (x, y i ) < Φ(x, ε) d K (y 1, y 2 ) < ε ). i=1

107 Unique existence X, K Polish, K compact, f : X K IR (BA-definable). MFI transforms uniqueness statements x X, y 1, y 2 K ( 2 f (x, y i ) = IR 0 d K (y 1, y 2 ) = IR 0 ) i=1 into moduli of uniqueness Φ : Q + Q + x X, y 1, y 2 K, ε > 0 ( 2 f (x, y i ) < Φ(x, ε) d K (y 1, y 2 ) < ε ). i=1 Let ŷ K be the unique root of f (x, ), y ε an ε-root f (x, y n ) < ε.

108 Unique existence X, K Polish, K compact, f : X K IR (BA-definable). MFI transforms uniqueness statements x X, y 1, y 2 K ( 2 f (x, y i ) = IR 0 d K (y 1, y 2 ) = IR 0 ) i=1 into moduli of uniqueness Φ : Q + Q + x X, y 1, y 2 K, ε > 0 ( 2 f (x, y i ) < Φ(x, ε) d K (y 1, y 2 ) < ε ). i=1 Let ŷ K be the unique root of f (x, ), y ε an ε-root f (x, y n ) < ε. Then d K (ŷ, y Φ(x,ε) ) < ε).

109 Case study: strong unicity in L 1 -approximation P n space of polynomials of degree n, f C[0, 1], f 1 := 1 0 f, dist 1(f, P n ) := inf p P n f p 1.

110 Case study: strong unicity in L 1 -approximation P n space of polynomials of degree n, f C[0, 1], f 1 := 1 0 f, dist 1(f, P n ) := inf p P n f p 1. Best approximation in the mean of f C[0, 1]: f C[0, 1]!p b P n ( f pb 1 = dist 1 (f, P n ) ) (existence and uniqueness use: WKL!)

111 Theorem (K./Paulo Oliva, APAL 2003) Let dist 1 (f, P n ) := inf f p 1 and ω a modulus of uniform continuity p P n for f. Ψ(ω, n, ε) := min{ cnε 8(n+1), cnε 2 2 ω n( cnε 2 )}, where c n := n/2! n/2! 2 4n+3 (n+1) and 3n+1 ω n (ε) := min{ω( ε 4 ), ε 40(n+1) 4 1 }. ω(1)

112 Theorem (K./Paulo Oliva, APAL 2003) Let dist 1 (f, P n ) := inf f p 1 and ω a modulus of uniform continuity p P n for f. Ψ(ω, n, ε) := min{ cnε 8(n+1), cnε 2 2 ω n( cnε 2 )}, where c n := n/2! n/2! 2 4n+3 (n+1) and 3n+1 ω n (ε) := min{ω( ε 4 ), ε 40(n+1) 4 1 }. ω(1) Then n IN, p 1, p 2 P n ε Q ( 2 + ( f p i 1 dist 1 (f, P n ) Ψ(ω, n, ε)) p 1 p 2 1 ε ). i=1

113 Comments on the result in the L 1 -case Ψ provides the first effective version of results due to Bjoernestal (1975) and Kroó ( ).

114 Comments on the result in the L 1 -case Ψ provides the first effective version of results due to Bjoernestal (1975) and Kroó ( ). Kroó (1978) implies that the ε-dependency in Ψ is optimal.

115 Comments on the result in the L 1 -case Ψ provides the first effective version of results due to Bjoernestal (1975) and Kroó ( ). Kroó (1978) implies that the ε-dependency in Ψ is optimal. Ψ allows the first complexity upper bound for the sequence of best L 1 -approximations (p n ) in P n of poly-time functions f C[0, 1] (P. Oliva, MLQ 2003).

116 The nonseparable/noncompact case Proposition Let (X, ) be a strictly convex normed space and C X a convex subset. Then any point x X has at most one point c C of minimal distance, i.e. x c =dist(x, C).

117 The nonseparable/noncompact case Proposition Let (X, ) be a strictly convex normed space and C X a convex subset. Then any point x X has at most one point c C of minimal distance, i.e. x c =dist(x, C). Hence: if X is separable and complete and provably strictly convex and C compact, then one can extract a modulus of uniqueness.

118 The nonseparable/noncompact case Proposition Let (X, ) be a strictly convex normed space and C X a convex subset. Then any point x X has at most one point c C of minimal distance, i.e. x c =dist(x, C). Hence: if X is separable and complete and provably strictly convex and C compact, then one can extract a modulus of uniqueness. Observation: compactness only used to extract uniform bound on strict convexity (= modulus of uniform convexity) from proof of strict convexity.

119 Assume that X is uniformly convex with modulus η.

120 Assume that X is uniformly convex with modulus η. Then for d dist(x, C) we have the following modulus of uniqueness (K.1990):

121 Assume that X is uniformly convex with modulus η. Then for d dist(x, C) we have the following modulus of uniqueness (K.1990): ( Φ(ε) := min 1, ε 4, ε ) 4 η(ε/(d + 1)). 1 η(ε/(d + 1)) Conclusion: neither compactness nor separability required!

122 General logical metatheorems II Many abstract types of metric structures can be added as atoms: metric, hyperbolic, CAT(0), δ-hyperbolic, normed, uniformly convex, Hilbert,... spaces or IR-trees X : add new base type X, all finite types over IN, X and a new constant d X representing d etc.

123 General logical metatheorems II Many abstract types of metric structures can be added as atoms: metric, hyperbolic, CAT(0), δ-hyperbolic, normed, uniformly convex, Hilbert,... spaces or IR-trees X : add new base type X, all finite types over IN, X and a new constant d X representing d etc. Condition: Defining axioms must have a monotone functional interpretation.

124 General logical metatheorems II Many abstract types of metric structures can be added as atoms: metric, hyperbolic, CAT(0), δ-hyperbolic, normed, uniformly convex, Hilbert,... spaces or IR-trees X : add new base type X, all finite types over IN, X and a new constant d X representing d etc. Condition: Defining axioms must have a monotone functional interpretation. Counterexamples (to extractibility of uniform bounds): for the classes of strictly convex ( uniformly convex) or separable ( totally bounded) spaces!

125 A formal system for analysis Types: (i) IN, X are types, (ii) with ρ, τ also ρ τ is a type. Functionals of type ρ τ map type-ρ objects to type-τ objects.

126 A formal system for analysis Types: (i) IN, X are types, (ii) with ρ, τ also ρ τ is a type. Functionals of type ρ τ map type-ρ objects to type-τ objects. PA ω,x is the extension of Peano Arithmetic to all types. A ω,x :=PA ω,x +DC, where DC: axiom of dependent choice for all types Implies full comprehension for numbers (higher order arithmetic).

127 A formal system for analysis Types: (i) IN, X are types, (ii) with ρ, τ also ρ τ is a type. Functionals of type ρ τ map type-ρ objects to type-τ objects. PA ω,x is the extension of Peano Arithmetic to all types. A ω,x :=PA ω,x +DC, where DC: axiom of dependent choice for all types Implies full comprehension for numbers (higher order arithmetic). A ω [X, d,...] results by adding constants d X,... with axioms expressing that (X, d,...) is a nonempty metric, hyperbolic... space.

128 A warning concerning equality Extensionality rule (only!): s = ρ t r(s) = τ r(t), where only x = IN y primitive equality predicate but for ρ τ s X = X t X : d X (x, y) = IR 0 IR, s = ρ τ t : v ρ (s(v) = τ t(v)).

129 A novel form of majorization y, x functionals of types ρ, ρ := ρ[in/x ] and a X of type X : x IN a IN y IN : x y x IN a X y X : x d(y, a).

130 A novel form of majorization y, x functionals of types ρ, ρ := ρ[in/x ] and a X of type X : x IN a IN y IN : x y x IN a X y X : x d(y, a). For complex types ρ τ this is extended in a hereditary fashion.

131 A novel form of majorization y, x functionals of types ρ, ρ := ρ[in/x ] and a X of type X : x IN a IN y IN : x y x IN a X y X : x d(y, a). For complex types ρ τ this is extended in a hereditary fashion. Example: f a X X f n IN, x X [n d(a, x) f (n) d(a, f (x))].

132 A novel form of majorization y, x functionals of types ρ, ρ := ρ[in/x ] and a X of type X : x IN a IN y IN : x y x IN a X y X : x d(y, a). For complex types ρ τ this is extended in a hereditary fashion. Example: f a X X f n IN, x X [n d(a, x) f (n) d(a, f (x))]. f : X X is nonexpansive (n.e.) if d(f (x), f (y)) d(x, y). Then λn.n + b a X X f, if d(a, f (a)) b.

133 A novel form of majorization y, x functionals of types ρ, ρ := ρ[in/x ] and a X of type X : x IN a IN y IN : x y x IN a X y X : x d(y, a). For complex types ρ τ this is extended in a hereditary fashion. Example: f a X X f n IN, x X [n d(a, x) f (n) d(a, f (x))]. f : X X is nonexpansive (n.e.) if d(f (x), f (y)) d(x, y). Then λn.n + b a X X f, if d(a, f (a)) b. Normed linear case: a := 0 X.

134 Hyperbolic spaces Definition (Takahashi,Kirk,Reich) A hyperbolic space is a triple (X, d, W ) where (X, d) is metric space and W : X X [0, 1] X s.t. (i) d(z, W (x, y, λ)) (1 λ)d(z, x) + λd(z, y), (ii) d(w (x, y, λ), W (x, y, λ)) = λ λ d(x, y), (iii) W (x, y, λ) = W (y, x, 1 λ), (iv) d(w (x, z, λ), W (y, w, λ)) (1 λ)d(x, y) + λd(z, w).

135 CAT(0)-spaces (Gromov) are hyperbolic spaces (X, d, W ) which satisfy the CN-inequality of Bruhat-Tits { d(y 0, y 1 ) = 1 2 d(y 1, y 2 ) = d(y 0, y 2 ) d(x, y 0 ) d(x, y 1) d(x, y 2) d(y 1, y 2 ) 2.

136 CAT(0)-spaces (Gromov) are hyperbolic spaces (X, d, W ) which satisfy the CN-inequality of Bruhat-Tits { d(y 0, y 1 ) = 1 2 d(y 1, y 2 ) = d(y 0, y 2 ) d(x, y 0 ) d(x, y 1) d(x, y 2) d(y 1, y 2 ) 2. convex subsets of normed spaces = hyperbolic spaces (X, d, W ) with two additional axioms (Machado (1973).

137 CAT(0)-spaces (Gromov) are hyperbolic spaces (X, d, W ) which satisfy the CN-inequality of Bruhat-Tits { d(y 0, y 1 ) = 1 2 d(y 1, y 2 ) = d(y 0, y 2 ) d(x, y 0 ) d(x, y 1) d(x, y 2) d(y 1, y 2 ) 2. convex subsets of normed spaces = hyperbolic spaces (X, d, W ) with two additional axioms (Machado (1973). Notation: (1 λ)x λy := W (x, y, λ).

138 CAT(0)-spaces (Gromov) are hyperbolic spaces (X, d, W ) which satisfy the CN-inequality of Bruhat-Tits { d(y 0, y 1 ) = 1 2 d(y 1, y 2 ) = d(y 0, y 2 ) d(x, y 0 ) d(x, y 1) d(x, y 2) d(y 1, y 2 ) 2. convex subsets of normed spaces = hyperbolic spaces (X, d, W ) with two additional axioms (Machado (1973). Notation: (1 λ)x λy := W (x, y, λ). Small types (over IN, X ): IN, IN IN, X, IN X, X X.

139 Theorem (Gerhardy/K.,Trans.Amer.Math.Soc. 2008) Let P, K be Polish resp. compact metric spaces, A -formula, τ small. If A ω [X, d, W ] proves x P y K z τ v IN A (x, y, z, v), then one can extract a computable Φ : IN IN IN (IN) IN s.t. the following holds in every nonempty hyperbolic space: for all representatives r x IN IN of x P and all z τ and z IN (IN) s.t. a X (z a τ z): y K v Φ(r x, z ) A (x, y, z, v).

140 Theorem (Gerhardy/K.,Trans.Amer.Math.Soc. 2008) Let P, K be Polish resp. compact metric spaces, A -formula, τ small. If A ω [X, d, W ] proves x P y K z τ v IN A (x, y, z, v), then one can extract a computable Φ : IN IN IN (IN) IN s.t. the following holds in every nonempty hyperbolic space: for all representatives r x IN IN of x P and all z τ and z IN (IN) s.t. a X (z a τ z): y K v Φ(r x, z ) A (x, y, z, v). For the bounded cases: K. Trans.AMS 2005.

141 As special case of general logical metatheorems due to Gerhardy/K. (Trans. Amer. Math. Soc. 2008) one has: Corollary (Gerhardy/K., TAMS 2008) If A ω [X, d, W ] proves x P y K z X f : X X ( f n.e. v IN A ), then one can extract a computable functional Φ : IN IN IN IN s.t. for all x P, b IN y K z X f : X X ( f n.e. d X (z, f (z)) b v Φ(r x, b)a ) holds in all nonempty hyperbolic spaces (X, d, W ).

142 As special case of general logical metatheorems due to Gerhardy/K. (Trans. Amer. Math. Soc. 2008) one has: Corollary (Gerhardy/K., TAMS 2008) If A ω [X, d, W ] proves x P y K z X f : X X ( f n.e. v IN A ), then one can extract a computable functional Φ : IN IN IN IN s.t. for all x P, b IN y K z X f : X X ( f n.e. d X (z, f (z)) b v Φ(r x, b)a ) holds in all nonempty hyperbolic spaces (X, d, W ). Normed case: also z b.

143 Mean Ergodic Theorem again Since Birkhoff s proof formalizes in A ω [X,, η] the following is guaranteed:

144 Mean Ergodic Theorem again Since Birkhoff s proof formalizes in A ω [X,, η] the following is guaranteed: X uniformly convex Banach space with modulus η and f : X X nonexpansive linear operator. Let b > 0. Then there is an effective functional Φ in ε, g, b, η s.t. for all x X with x b, all ε > 0, all g : IN IN : n Φ(ε, g, b, η) i, j [n, n + g(n)] ( A i (x) A j (x) < ε ). (see Lecture I)

145 Tao also established (without bound) a uniform version (in a special case) of the Mean Ergodic Theorem as base step for a generalization to commuting families of operators.

146 Tao also established (without bound) a uniform version (in a special case) of the Mean Ergodic Theorem as base step for a generalization to commuting families of operators. We shall establish Theorem 1.6 by finitary ergodic theory techniques, reminiscent of those used in [Green-Tao]... The main advantage of working in the finitary setting... is that the underlying dynamical system becomes extremely explicit... In proof theory, this finitisation is known as Gödel functional interpretation...which is also closely related to the Kreisel no-counterexample interpretation (T. Tao: Norm convergence of multiple ergodic averages for commuting transformations, Ergodic Theor. and Dynam. Syst. 28, 2008)

147 Projections and Weak Compactness without separability (to appear in: Festschrift for G. Mints) A ω [X,, ] does not have nontrivial comprehension over X -type objects but proves (using countable choice for X -objects) schematically

148 Projections and Weak Compactness without separability (to appear in: Festschrift for G. Mints) A ω [X,, ] does not have nontrivial comprehension over X -type objects but proves (using countable choice for X -objects) schematically for definable closed convex subsets (resp. closed linear subspaces) the existence of unique best approximations (resp. orthogonal projections),

149 Projections and Weak Compactness without separability (to appear in: Festschrift for G. Mints) A ω [X,, ] does not have nontrivial comprehension over X -type objects but proves (using countable choice for X -objects) schematically for definable closed convex subsets (resp. closed linear subspaces) the existence of unique best approximations (resp. orthogonal projections), for linear functionals L : X IR with definable graph the Riesz representation theorem,

150 Projections and Weak Compactness without separability (to appear in: Festschrift for G. Mints) A ω [X,, ] does not have nontrivial comprehension over X -type objects but proves (using countable choice for X -objects) schematically for definable closed convex subsets (resp. closed linear subspaces) the existence of unique best approximations (resp. orthogonal projections), for linear functionals L : X IR with definable graph the Riesz representation theorem, the weak compactness of B 1 (0) (here only countable choice for arithmetical formulas needed and restricted induction).

151 A theorem of F.E. Browder Using projection to the set of all fixed points of a nonexpansive mapping U : X X (X Hilbert space) and weak compactness, Browder showed in 1967:

152 A theorem of F.E. Browder Using projection to the set of all fixed points of a nonexpansive mapping U : X X (X Hilbert space) and weak compactness, Browder showed in 1967: Theorem[F.E. Browder]: For n IN, v 0 B 1 (0) let u n be the unique fixed point of the contraction U n (x) := (1 1 n )U(x) 1 n v 0. Then (u n ) converges towards the fixed point of U that is closest to v 0.

153 A theorem of F.E. Browder Using projection to the set of all fixed points of a nonexpansive mapping U : X X (X Hilbert space) and weak compactness, Browder showed in 1967: Theorem[F.E. Browder]: For n IN, v 0 B 1 (0) let u n be the unique fixed point of the contraction U n (x) := (1 1 n )U(x) 1 n v 0. Then (u n ) converges towards the fixed point of U that is closest to v 0. Corollary by Metatheorem: There is a functional Φ(k, g) (definable by primitive recursion and bar recursion of lowest type) such that k IN g : IN IN n Φ(k, g) i, j [n; n + g(n)] ( u i u j < 2 k ). Note that Φ does not depend on U, v 0 or X!

154 Lecture III

155 Applications to metric fixed point theory General context: (X, d, W ) is a (non-empty) hyperbolic space.

156 Applications to metric fixed point theory General context: (X, d, W ) is a (non-empty) hyperbolic space. f : X X is a nonexpansive mapping.

157 Applications to metric fixed point theory General context: (X, d, W ) is a (non-empty) hyperbolic space. f : X X is a nonexpansive mapping. (λ n ) is a sequence in [0, 1] that is bounded away from 1 and divergent in sum.

158 Applications to metric fixed point theory General context: (X, d, W ) is a (non-empty) hyperbolic space. f : X X is a nonexpansive mapping. (λ n ) is a sequence in [0, 1] that is bounded away from 1 and divergent in sum. x n+1 = (1 λ n )x n λ n f (x n ) (Krasnoselski-Mann iter.).

159 Applications to metric fixed point theory General context: (X, d, W ) is a (non-empty) hyperbolic space. f : X X is a nonexpansive mapping. (λ n ) is a sequence in [0, 1] that is bounded away from 1 and divergent in sum. x n+1 = (1 λ n )x n λ n f (x n ) (Krasnoselski-Mann iter.). Theorem (Ishikawa 1976, Goebel/Kirk 1983) (Ishikawa I) If (x n ) is bounded, then d(x n, f (x n )) 0.

160 Logical analysis of the proof of Ishikawa s theorem Let K IN and α : IN IN be such that α(n) (λ n ) n IN [0, 1 1 K ]IN and n IN(n λ i ). Logical metatheorem applied to proof of Ishikawa s theorem yields computable Ψ, Φ s.t. for all k IN and n.e. f i, j Ψ(K, α, b, b, k) ( d(x, f (x)) b d(x i, x j ) b ) m Φ(K, α, b, b, k) ( d(x m, f (x m )) < 2 k). holds in any (nonempty) hyperbolic space (X, d, W ). i=0

161 Theorem (K.2007, K./Leustean AAA 2003) (X, d, W ), (λ n ), K, α as above, f : X X nonexpansive the following holds for all ε, b, b > 0 : If d(x, f (x)) b and i Φ j α(φ, M) (d(x i, x i+j ) b) then n Φ ( d(x n, f (x n )) ε ),

162 Theorem (K.2007, K./Leustean AAA 2003) (X, d, W ), (λ n ), K, α as above, f : X X nonexpansive the following holds for all ε, b, b > 0 : If d(x, f (x)) b and i Φ j α(φ, M) (d(x i, x i+j ) b) then n Φ ( d(x n, f (x n )) ε ), where ( b exp K b+3b Φ := Φ(K, α, b, b, ε) := α M := b+3b ε, ε +1 ε α(0, n) := α(0, n), α(i + 1, n) := α( α(i, n), n) with α(i, n) := i + α(i, n) (i, n IN) ) 1, M,

163 Theorem (K.2007, K./Leustean AAA 2003) (X, d, W ), (λ n ), K, α as above, f : X X nonexpansive the following holds for all ε, b, b > 0 : where If d(x, f (x)) b and i Φ j α(φ, M) (d(x i, x i+j ) b) then n Φ ( d(x n, f (x n )) ε ), ( b exp K b+3b Φ := Φ(K, α, b, b, ε) := α M := with α s.t. b+3b ε, ε +1 ε α(0, n) := α(0, n), α(i + 1, n) := α( α(i, n), n) with α(i, n) := i + α(i, n) (i, n IN) i, n IN ( (α(i, n) α(i + 1, n)) (n i+α(i,n) 1 s=i ) 1, M, λ s ) ).

164 Known uniformity results in the bounded case blue = hyperbolic, green = dir.nonex., red = both. Krasnoselski(1955):X unif.convex,c compact,λ k = 1,no uniform. 2 Browder/Petryshyn(1967):X unif.convex,λ k = λ, no uniformity. Groetsch(1972): X unif. convex, general λ k, X, no uniformity Ishikawa (1976): No uniformity Edelstein/O Brien (1978): Uniformity w.r.t. x 0 C (λ k := λ) Goebel/Kirk (1982): Uniformity w.r.t. x 0 and f. General λ k Kirk/Martinez (1990): Uniformity for unif. convex X, λ := 1/2 Goebel/Kirk (1990): Conjecture: no uniformity w.r.t. C Baillon/Bruck (1996): Uniformity w.r.t. x 0, f, C for λ k := λ Kirk (2001): Uniformity w.r.t. x 0, f for constant λ Kohlenbach (2001): Full uniformity for general λ k K./Leustean (2003): Full uniformity for general λ k

165 Corollary (K.2007) (generalizes result by Baillon-Bruck-Reich from 1978) Let (λ n ) in [a, b] (0, 1). then If c(n) lim 0, where c(n) := max{d(x, x j ) : j n}, n n lim d(x n, f (x n )) = 0. n

166 Corollary (K.2007) (generalizes result by Baillon-Bruck-Reich from 1978) Let (λ n ) in [a, b] (0, 1). then If c(n) lim 0, where c(n) := max{d(x, x j ) : j n}, n n lim d(x n, f (x n )) = 0. n Result optimal: c(n) K n not sufficient!

167 Theorem (Ishikawa, Goebel, Kirk) (Ishikawa II) If previous assumptions and X compact, then (x n ) converges towards a fixed point.