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

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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

• rigin and has many different 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.

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Computable Analysis

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Pioneers in Computable Analysis

Alan M. Turing 1912–1954 Ernst Specker 1920–2011 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 different varieties

(Markov, Sanin, Orevkov, Kushner in the Russian school and Bishop, Bridges, Ishihara in the western school).

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The Monotone Convergence Theorem

Theorem (Monotone Convergence Theorem) Every monotone increasing and bounded sequence of real numbers (xn)n has a least upper bound supn2N xn. Proposition (Turing 1937, Specker 1949) There is a computable monotone increasing and bounded sequence (xn)n of real numbers such that x = supn2N xn is not computable.

• Proof. Consider an injective computable enumeration (an)n of the

halting problem K ✓ N to construct a suitable computable sequence xn := Pn

i=0 2ai. Then

x = supn2N xn = P

i2K 2i

is non-computable. ⇤ Such sequences are nowadays called Specker sequences.

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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 (fn)n of functions fn : [0, 1] ! R with fn(0) · fn(1) < 0 for all n 2 N and such that there is no computable sequence (xn)n with fn(xn) = 0. Pour-El and Richards used two computably inseparable c.e. sets.

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Realizers and Representations

I A representation of X is a surjective map X :✓ NN ! X. I F :✓ NN ! NN 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). NN

• X
• F

f

?

Y X NN Y

?

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¨

• der) and we know what suitable

representations of spaces such as R and C[0, 1] are.

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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.

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Weak K˝

• nig’s Lemma

Theorem (Weak K˝

• nig’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 2N is called low if p0 T ;0.

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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.

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The Bolzano-Weierstraß Theorem

Theorem (Bolzano-Weierstraß Theorem) Every sequence (xn)n in the unit cube [0, 1]k has a cluster point. Proposition (Kreisel 1952, Rice 1954) There exists a computable sequence (xn)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 (xn)n in [0, 1] without a limit computable cluster point. Proposition (B., Gherardi and Marcone 2012) Every computable sequence (xn)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.

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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 M0 T ;00.

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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.

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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

• n 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 (Tn)n of computable,

bijective and linear operators Tn : `2 ! `2 such that (T 1

n )n

is not a computable sequence. Is is necessary to use infinite dimensional spaces here.

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Instancewise Complexity of Theorems

Theorem non-uniform parallelized Montone Convergence B T A0 halting problem B T A0 halting problem Maximum B0 T A0 low B0 T A0 low Intermediate Value B T A computable B0 T A0 low Bolzano-Weierstraß B0 T A00 low rel. to halting p. B0 T A00 low rel. to halting p. Banach Inverse Mapping B T A computable B T A0 halting problem

I Upper Turing bound that a solution B satisfies for a given instance A.

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Instancewise Complexity of Theorems

Theorem non-uniform parallelized Montone Convergence Fr´ echet-Riesz Radon-Nikodym B T A0 halting problem B T A0 halting problem Maximum Hahn-Banach Brouwer Fixed Point Weak K˝

• nig’s Lemma

B0 T A0 low B0 T A0 low Intermediate Value B T A computable B0 T A0 low Bolzano-Weierstraß K˝

• nig’s Lemma

B0 T A00 low rel. to halting p. B0 T A00 low rel. to halting p. Banach Inverse Mapping Baire Category Open Mapping B T A computable B T A0 halting problem

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Reverse Mathematics

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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 RCA0: recursive comprehension, I WKL0: Weak K˝

• nig’s Lemma,

I ACA0: arithmetical comprehension, I ATR0: arithmetical transfinite recursion

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Theorems in Reverse Mathematics

Theorem non-uniform parallelized reverse math. Montone Convergence B T A0 halting problem B T A0 halting problem ACA0 Weak K˝

• nig’s Lemma

B0 T A0 low B0 T A0 low WKL0 Intermediate Value B T A computable B0 T A0 low RCA0 Bolzano-Weierstraß B0 T A00 low rel. to hp. B0 T A00 low rel. to hp. ACA0 Baire Category B T A computable B T A0 halting problem RCA0

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Weihrauch Complexity

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Is there a Space of Theorems?

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The Weihrauch Lattice

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Mathematical Problems and Solutions

Definition A mathematical problem is a partial multi-valued f :✓ X ◆ Y .

I There are a certain sets of potential inputs X and outputs Y . I D = dom(f ) contains the valid instances of the problem. I f (x) is the set of solutions of the problem f for instance x.

Any theorem T of the Π2 form (8x 2 X)(x 2 D = ) (9y 2 Y ) P(x, y)) is identified with F :✓ X ◆ Y with dom(F) := D and F(x) := {y 2 Y : P(x, y)}.

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Theorems as Problems

I IVT :✓ C[0, 1] ◆ [0, 1], f 7! f 1{0}, dom(IVT) := {f : f (0) · f (1) < 0}. I BIMX,Y :✓ C(X, Y ) ! C(Y , X), T 7! T 1, restricted to bijective, linear,

bounded T and for computable Banach spaces X, Y .

I MCT :✓ RN ! R, (xn)n 7! supn2N xn restricted to monotone bounded

sequences.

I WKL :✓ Tr ◆ 2N, T 7! [T] restricted to infinite binary trees. I BFTn : C([0, 1]n, [0, 1]n) ◆ [0, 1]n, f 7! {x : f (x) = x} for n 1. I BWTX :✓ X N ◆ X, (xn)n 7! {x : x cluster point of (xn)n}, restricted to

sequences that are in a compact subset of X.

I MAXX :✓ C(X) ◆ R, f 7! {x 2 X : f (x) = max f (X)} for computably

compact computable metric spaces X and, in particular, for X = [0, 1].

I RTn

k : Cn,k ◆ 2N, c 7! {H ✓ N : H infinite homogeneous for c}, where

Cn,k denotes the set of colorings c : [N]n ! k.

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Weihrauch Reducibility

Let f :✓ X ◆ Y and g :✓ Z ◆ W be two multi-valued functions. K H G F p F(p)

I f is Weihrauch reducible to g, f W g, if there are computable

H, K :✓ NN ! NN such that Hhid, GKi ` f whenever G ` g.

I f ⌘W g : (

) f W g and g W f .

I It is easy to see that W is reflexive and transitive, i.e., ⌘W is

an equivalence relation and one obtains a distributive lattice.

I Weihrauch reducibility was originally defined by K. Weihrauch

(around 1990) and has been reinvented several times.

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Choice

Definition For every computable metric space X CX :✓ C(X) ! X, f 7! f 1{0} is called the choice problem of X.

I CX is the problem to find solutions x 2 X of equations of type

f (x) = 0 for continuous f : X ! R.

I We can calibrate the complexity of CX by choosing X among

spaces such as 2, N, 2N, R and NN.

I We can also restrict the problem to such f with a zero set

that is connected (CCX), of positive measure (PCX), or that satisfies any other meaningful additional property.

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Theorem of the Maximum

Theorem For X = [0, 1] (and more generally for computably compact X) MAXX ⌘W CX.

• Proof. We prove CX W MAXX. Given a continuous function

f : X ! R with A = f 1{0} 6= ;, we can compute the function g : X ! R with g := |f |. Then MAX(g) = f 1{0} = A. This proves the claim. We now prove MAXX W CX. Given a continuous function f : X ! R with MAX(f ) = A 6= ;, we can compute g : X ! R with g := f max f (X), since X is computably compact. Now we

• btain g1{0} = MAX(f ) = A. This proves the claim.

⇤

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Two Operations in the Weihrauch Lattice

The Weihrauch lattice has a rich algebraic structure. For instance, for f :✓ X ◆ Y we consider:

I Parallelization:

b f :✓ X N ◆ Y N, (xn)n 7! X

n2N f (xn).

Parallelization is a closure operator in the Weihrauch lattice.

I Jump:

f 0 :✓ X 0 ◆ Y , x 7! f (x), where X 0 is the set X represented in a different way: a name is now a sequence that converges to a name in the sense of X.

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Choice on Natural Numbers

Theorem (B. and Gherardi 2011) The following problems and theorems are Weihrauch equivalent:

I The choice problem CN on natural numbers I The Baire Category Theorem BCT1 I The Banach Inverse Mapping Theorem BIM`2,`2 I The Open Mapping Theorem for `2 I The Closed Graph Theorem for `2 I The Uniform Boundedness Theorem for `2

All members of the equivalence class share the following features:

I They map computable inputs to (some) computable outputs. I They are computable with finitely many mind changes and even

complete for this class.

I They have parallelizations that are equivalent to the limit map. I They are closed under composition.

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Choice on Cantor Space

Theorem The following problems and theorems are Weihrauch equivalent:

I The choice problem C2N on Cantor space 2N I Weak K˝

• nig’s Lemma WKL

I The Theorem of the Maximum MAX[0,1] I The parallelization d

IVT of the Intermediate Value Theorem (B. and Gherardi 2011)

I The Hahn-Banach Theorem HBT (Gherardi and Marcone 2009) I The Brouwer-Fixed Point Theorem BFTn for dimension n 2

(B., Le Roux, J.S. Miller and Pauly 2016) All members of the equivalence class share the following features:

I They map computable inputs to (some) low outputs. I They are non-deterministically computable and even complete

for this class.

I They are closed under composition and parallelization.

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Parallelized Choice on Natural Numbers

Theorem The following problems and theorems are Weihrauch equivalent:

I The limit problem lim :✓ NN ! NN, hp0, p1, p2, ...i 7! limn!1 pn I The parallelization

\ BIM`2,`2 of the Banach Inverse Mapping Theorem

I The Monotone Convergence Theorem MCT I The Fr´

echet-Riesz Theorem for Hilbert spaces (follows from B. and Yoshikawa 2006)

I The Radon-Nikodym Theorem (Hoyrup, Rojas, Weihrauch 2012)

All members of the equivalence class share the following features:

I They map computable inputs to (some) limit computable outputs. I They are limit computable and complete for this class. I They are closed under parallelization, but not under composition.

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The Jump of Choice on Cantor Space

Theorem The following problems and theorems are Weihrauch equivalent:

I The jump C0

2N of choice on Cantor space 2N

I The jump of Weak K˝

• nig’s Lemma WKL0

I K˝

• nig’s Lemma KL (B. and Rakotoniaina 2015)

I The Bolzano-Weierstraß Theorem BWTR on R

(B., Gherardi, Marcone 2012) All members of the equivalence class share the following features:

I They map computable inputs to (some) outputs that are low

relative to the halting problem.

I They are closed under parallelization, but not under composition.

Theorem (B. and Rakotoniaina 2015) d RTn

k ⌘W WKL(n) ⌘W C(n) 2N for all n 1, k 2.

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Survey on Theorems

Theorem non-uniform parallelized Weihrauch lattice Montone Convergence B T A0 halting problem B T A0 halting problem c CN ⌘W lim Weak K˝

• nig’s Lemma

B0 T A0 low B0 T A0 low C2N ⌘W \ CC[0,1] Intermediate Value B T A computable B0 T A0 low CC[0,1] Bolzano-Weierstraß B0 T A00 low rel. to hp. B0 T A00 low rel. to hp. C0

2N

Baire Category B T A computable B T A0 halting problem CN

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Back to Logic

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Weihrauch Complexity and Reverse Mathematics

CN KN ⌘sW C⇤

2

CC[0,1] C2N ⌘W c C2 CR ⌘W CN ⇥ C2N PC2N lim ⌘W c CN C0

2N

CNN C1 RCA⇤ BΣ0

1

IΣ0

1

ACA0 ATR0 WKL⇤ WKL⇤

0 + IΣ0 1

WWKL⇤

Perfect Subtree Theorem Bolzano-Weierstraß Theorem Monoton Convergence Theorem Frostmann’s Lemma Weak K˝

• nig’s Lemma

Weak Weak K˝

• nig’s Lemma

Intermediate Value Theorem Baire Category Theorem

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Models of Computability and Descriptive Set Theory

CN KN ⌘sW C⇤

2

CC[0,1] C2N ⌘W c C2 CR ⌘W CN ⇥ C2N PC2N lim ⌘W c CN lim0 CNN C1 Σ0

1

∆0

2

Σ0

2

Σ0

3

Effective Borel Measurable Limit Computability Non-Deterministic Las Vegas Finite Mind Change Computability

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Ramsey’s Theorem in the Weihrauch Lattice

lim(3) WKL(3) ⌘W d C(3)

2

RT3

N

... RT3

4

RT3

3

RT3

2

C(3)

2

lim00 WKL00 ⌘W c C00

2

RT2

N

... RT2

4

RT2

3

RT2

2

C00

2

lim0 WKL0 ⌘W c C0

2

RT1

N

... RT1

4

RT1

3

RT1

2

C0

2

lim ⌘W c CN WKL⌘Wc C2 CN C⇤

2

... C4 C3 C2 Σ0

5

Σ0

4

Σ0

3

Σ0

2

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Weihrauch Complexity and Logic

I Weihrauch complexity can be seen as a version of Kolmogorov’s

calculus of problems (1932).

I The Medvedev lattice can already be seen as an attempt to model

this calculus in a lattice structure.

I The Medvedev lattice embeds into the Weihrauch lattice and the

latter can be seen as a parameterized version of the former.

I The emerging algebraic structure of the Weihrauch lattice is similar

to intuitionistic linear logic: logical operation in linear logic algebraic operation ⌦ multiplicative conjunction ⇥ product & additive conjunction t coproduct additive disjunction u infimum & multiplicative disjunction + sum ! bang b parallelization

I However, the Weihrauch lattice is neither a Brouwer nor a Heyting

algebra (Higuchi and Pauly 2013).

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Infima and Suprema

The Weihrauch lattice is not complete and infinite suprema and infima do not always exist. There are some known existent ones. Definition (Compositional product, implication) For two mathematical problem f , g we define the

I f ⇤ g := max{f0 g0 : f0 W f and g0 W g} and I g ! f := min{h : f W g ⇤ h}.

The maximum and minimum is understood with respect to W and they always exist (B. and Pauly 2016). Proposition

I MLR ⌘W(BCT1 ! WWKL)

(B., Pauly 2016)

I PA ⌘W(BCT0 1 ! WKL)

(B., Hendtlass and Kreuzer 2017)

I COH ⌘W(lim ! WKL0)

(B., Hendtlass and Kreuzer 2017)

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Weihrauch Complexity versus Constructive Mathematics

I Weihrauch complexity is uniform and resource sensitive. I Markov’s principle considered as a problem is computable. I We have a (very rough) equation

Weihrauch complexity ⇡ intuitionstic linear logic+Markov’s principle

I Some varieties of constructive mathematics use countable choice:

parallelization ⇡ countable choice

I There are first attempts to make some such equations precise in

different frameworks (Kuyper 2017, Hirst and Mummert 2017).

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A Survey as a Reference

There is a bibliography on Weihrauch complexity with more than 130 items: http : //cca net.de/publications/weibib.php