On the Computational Content of Theorems Vasco Brattka Universit - PowerPoint PPT Presentation

On the Computational Content of Theorems Vasco Brattka Universit¨ at der Bundeswehr M¨ unchen, Germany University of Cape Town, South Africa Logic Colloquium 2018 Udine, Italy, 23–28 July 2018

Facets of the Topic ; I Computability theory early on was combined with the goal to characterize the computational content of theorems. I Computable analysis is a theory that emerged from this origin and has many di ff erent facets (e.g., complexity). I Reverse mathematics is a proof theoretic approach to capture the computational content of theorems. I Weihrauch complexity can be seen as a uniform computability theoretic refinement of reverse mathematics.

; Computable Analysis

Pioneers in Computable Analysis ; Ernst Specker Alan M. Turing 1920–2011 1912–1954 I Turing ’s famous article “On computable numbers ...” actually treated computability on real numbers (1936). I Specker was the first who took up this subject and continued to study computability properties of theorems (1949). I Computable analysis is the theory of computability on the reals in this tradition (Grezgorczyk, Lacombe, Hauck, Pour-El, Richards, Weihrauch). I Constructive analysis is a related subject and has di ff erent varieties (Markov, Sanin, Orevkov, Kushner in the Russian school and Bishop, Bridges, Ishihara in the western school).

The Monotone Convergence Theorem ; Theorem (Monotone Convergence Theorem) Every monotone increasing and bounded sequence of real numbers ( x n ) n has a least upper bound sup n 2 N x n . Proposition (Turing 1937, Specker 1949) There is a computable monotone increasing and bounded sequence ( x n ) n of real numbers such that x = sup n 2 N x n is not computable. Proof. Consider an injective computable enumeration ( a n ) n of the halting problem K ✓ N to construct a suitable computable sequence x n := P n i =0 2 � a i . Then x = sup n 2 N x n = P i 2 K 2 � i is non-computable. ⇤ Such sequences are nowadays called Specker sequences.

The Intermediate Value Theorem ; Theorem (Intermediate Value Theorem) Every continuous function f : [0 , 1] ! R with f (0) · f (1) < 0 has a zero x 2 [0 , 1] . Proposition (Turing 1937, Specker 1959) Every computable function f : [0 , 1] ! R with f (0) · f (1) < 0 has a computable zero x 2 [0 , 1] . The proof requires a non-constructive case distinction (according to whether the function has a nowhere dense zero set or not). Proposition (Specker 1959, Pour-El and Richards 1989) There is a computable sequence ( f n ) n of functions f n : [0 , 1] ! R with f n (0) · f n (1) < 0 for all n 2 N and such that there is no computable sequence ( x n ) n with f n ( x n ) = 0 . Pour-El and Richards used two computably inseparable c.e. sets.

Realizers and Representations ; I A representation of X is a surjective map � X : ✓ N N ! X . I F : ✓ N N ! N N is a realizer of f : ✓ X ◆ Y , in symbols F ` f , if � Y F ( p ) 2 f � X ( p ) for all p 2 dom ( f � X ). F N N - N N � X � Y ? ? X - Y f I f is continuous, computable, polynomial-time computable or Borel measurable, if it admits a corresponding realizer F . I There is a well-developed theory of representations (Hauck, Kreitz, Weihrauch, Schr¨ oder) and we know what suitable representations of spaces such as R and C [0 , 1] are.

The Theorem of the Maximum ; Theorem (Theorem of the Maximum) For every continuous function f : [0 , 1] ! R there exists a point x 2 [0 , 1] such that f ( x ) = max f [0 , 1] . Grzegorczyk (1955) raised the question whether every computable function f : [0 , 1] ! R attains its maximum at a computable point. Proposition (Lacombe 1957, Specker 1959) There exists a computable function f : [0 , 1] ! R such that there is no computable x 2 [0 , 1] with f ( x ) = max f [0 , 1] . Specker used a Kleene tree for his construction.

Weak K˝ onig’s Lemma ; Theorem (Weak K˝ onig’s Lemma 1936) Every infinite binary tree has an infinite path. Proposition (Kleene 1952) There exists a computable infinite binary tree without computable paths. Kleene used the set of separators of two computably inseparable c.e. sets. Theorem (Low Basis Theorem of Jockusch and Soare 1972) Every computable infinite binary tree has a low path. Here p 2 2 N is called low if p 0  T ; 0 .

The Brouwer Fixed Point Theorem ; Theorem (Brouwer Fixed Point Theorem) For every continuous function f : [0 , 1] k ! [0 , 1] k there exists a point x 2 [0 , 1] k such that f ( x ) = x . Proposition (Orevkov 1963, Baigger 1985) There is a computable function f : [0 , 1] 2 ! [0 , 1] 2 without a computable x 2 [0 , 1] 2 with f ( x ) = x . Such a counterexample exists for dimension k � 2.

The Bolzano-Weierstraß Theorem ; Theorem (Bolzano-Weierstraß Theorem) Every sequence ( x n ) n in the unit cube [0 , 1] k has a cluster point. Proposition (Kreisel 1952, Rice 1954) There exists a computable sequence ( x n ) n in [0 , 1] without a computable cluster point. Giovanni Lagnese (fom 2006) asked whether this can be improved. Proposition (Le Roux and Ziegler 2008) There exists a computable sequence ( x n ) n in [0 , 1] without a limit computable cluster point. Proposition (B., Gherardi and Marcone 2012) Every computable sequence ( x n ) n in the unit cube [0 , 1] n has a cluster point x 2 [0 , 1] n that is low relative to the halting problem.

Ramsey’s Theorem ; Theorem (Ramsey 1930) Every coloring c : [ N ] n ! k admits an infinite homogeneous set M ✓ N . Theorem (Specker 1969) There exists a computable coloring c : [ N ] 2 ! 2 without a computable infinite homogeneous set M ✓ N . Theorem (Jockusch 1972) I There is a computable c : [ N ] 2 ! 2 without an infinite homogeneous set M ✓ N that is computable in ; 0 . I For every computable coloring c : [ N ] 2 ! 2 there exists an infinite homogeneous set M ✓ N with M 0  T ; 00 .

The Hahn-Banach Theorem ; Theorem (Hahn-Banach Theorem) Let X be a normed space over the field R with a linear subspace Y ✓ X . Then every linear bounded functional f : Y ! R has a linear bounded extension g : X ! R with || g || = || f || . Proposition (Metakides, Nerode and Shore 1985) There exists a computable Banach space X over R with a c.e. closed linear subspace Y ✓ X and a computable linear functional f : Y ! R with a computable norm || f || such that every linear bounded extension g : X ! R with || g || = || f || is non-computable. Necessarily the space X is not a Hilbert space.

The Banach Inverse Mapping Theorem ; Theorem (Banach Inverse Mapping Theorem) If T : X ! Y is a bijective, linear and bounded operator on Banach spaces X , Y , then its inverse T � 1 : Y ! X is bounded. Theorem (B. 2009) I If T : X ! Y is a computable, bijective and linear operator on computable Banach spaces X , Y , then T � 1 is computable. I Inversion BIM : ✓ C ( ` 2 , ` 2 ) ! C ( ` 2 , ` 2 ) , T 7! T � 1 restricted to bijective, linear and bounded T : ` 2 ! ` 2 is not computable. I There is a computable sequence ( T n ) n of computable, bijective and linear operators T n : ` 2 ! ` 2 such that ( T � 1 n ) n is not a computable sequence. Is is necessary to use infinite dimensional spaces here.

Instancewise Complexity of Theorems ; Theorem non-uniform parallelized B  T A 0 B  T A 0 Montone Convergence halting problem halting problem B 0  T A 0 B 0  T A 0 Maximum low low B 0  T A 0 B  T A Intermediate Value computable low B 0  T A 00 B 0  T A 00 Bolzano-Weierstraß low rel. to halting p. low rel. to halting p. B  T A B  T A 0 Banach Inverse Mapping computable halting problem I Upper Turing bound that a solution B satisfies for a given instance A .

Instancewise Complexity of Theorems ; Theorem non-uniform parallelized Montone Convergence B  T A 0 B  T A 0 Fr´ echet-Riesz halting problem halting problem Radon-Nikodym Maximum B 0  T A 0 B 0  T A 0 Hahn-Banach Brouwer Fixed Point low low Weak K˝ onig’s Lemma B 0  T A 0 B  T A Intermediate Value computable low B 0  T A 00 B 0  T A 00 Bolzano-Weierstraß low rel. to halting p. low rel. to halting p. K˝ onig’s Lemma Banach Inverse Mapping B  T A 0 B  T A Baire Category computable halting problem Open Mapping

; Reverse Mathematics

Reverse Mathematics ; I Reverse mathematics is a proof theoretic approach to the classification of theorems. I Theorems are classified according to which axioms are need to prove them in second-order arithmetic . I The basic systems are I RCA 0 : recursive comprehension, I WKL 0 : Weak K˝ onig’s Lemma, I ACA 0 : arithmetical comprehension, I ATR 0 : arithmetical transfinite recursion

Theorems in Reverse Mathematics ; Theorem non-uniform parallelized reverse math. B  T A 0 B  T A 0 Montone Convergence ACA 0 halting problem halting problem B 0  T A 0 B 0  T A 0 Weak K˝ onig’s Lemma WKL 0 low low B 0  T A 0 B  T A Intermediate Value RCA 0 computable low B 0  T A 00 B 0  T A 00 Bolzano-Weierstraß ACA 0 low rel. to hp. low rel. to hp. B  T A B  T A 0 Baire Category RCA 0 computable halting problem

; Weihrauch Complexity

Is there a Space of Theorems? ;

The Weihrauch Lattice ;