Computability and ergodic theory Mathieu Hoyrup Ergodic - - PowerPoint PPT Presentation

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Computability and ergodic theory Mathieu Hoyrup Ergodic - - PowerPoint PPT Presentation

Sep 25, 2022 265 likes •1.03k views

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Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Computability and ergodic theory

Mathieu Hoyrup

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Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Ergodic decomposition

  • Let P be a shift-invariant measure over Ω = {0, 1}N:

P[w] = P[0w] + P[1w].

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SLIDE 3

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Ergodic decomposition

  • Let P be a shift-invariant measure over Ω = {0, 1}N:

P[w] = P[0w] + P[1w].

  • [Birkhoff, 1931] For P-almost every x ∈ Ω, and every w ∈ {0, 1}∗,

µx[w] := limiting frequency of w along x exists.

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SLIDE 4

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Ergodic decomposition

  • Let P be a shift-invariant measure over Ω = {0, 1}N:

P[w] = P[0w] + P[1w].

  • [Birkhoff, 1931] For P-almost every x ∈ Ω, and every w ∈ {0, 1}∗,

µx[w] := limiting frequency of w along x exists.

  • µx is itself a shift-invariant probability measure.
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SLIDE 5

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Ergodic decomposition

  • Let P be a shift-invariant measure over Ω = {0, 1}N:

P[w] = P[0w] + P[1w].

  • [Birkhoff, 1931] For P-almost every x ∈ Ω, and every w ∈ {0, 1}∗,

µx[w] := limiting frequency of w along x exists.

  • µx is itself a shift-invariant probability measure.

Question

Reading more and more bits of x, can one compute µx from x?

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SLIDE 6

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Ergodic decomposition

There are two cases:

  • µx is the same for almost all x’s. In that case, µx = P almost
  • surely. P is said to be ergodic.
  • µx depends on x.
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SLIDE 7

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Ergodic decomposition

There are two cases:

  • µx is the same for almost all x’s. In that case, µx = P almost
  • surely. P is said to be ergodic.
  • µx depends on x.

P is not ergodic ⇐ ⇒ it can be decomposed into P = λP1 + (1 − λ)P2 with P1 = P2 shift-invariant and 0 < λ < 1.

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SLIDE 8

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Ergodic decomposition

Observation

In general µx cannot be uniformly computed from x.

Example

Let P = 1

2(Bp + Bq) with 0 < p = q < 1.

Every finite sequence is compatible with Bp and Bq so one can never determine whether µx = Bp or µx = Bq.

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SLIDE 9

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Ergodic decomposition

Observation

In general µx cannot be uniformly computed from x.

Example

Let P = 1

2(Bp + Bq) with 0 < p = q < 1.

Every finite sequence is compatible with Bp and Bq so one can never determine whether µx = Bp or µx = Bq.

Definition

Let P be a computable shift-invariant measure. P is effectively decomposable if there is a machine M such that for every ε > 0, P{x : Mx(ε) computes µx} ≥ 1 − ε.

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Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Ergodic decomposition

The Pólya urn

run 0 . . .

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SLIDE 11

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Ergodic decomposition

The Pólya urn

run 0 . . .

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SLIDE 12

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Ergodic decomposition

The Pólya urn

run 0 . . .

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Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Ergodic decomposition

The Pólya urn

run 0 . . .

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Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Ergodic decomposition

The Pólya urn

run 0 . . .

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Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Ergodic decomposition

The Pólya urn

run 0 . . .

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Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Ergodic decomposition

The Pólya urn

run 0 . . .

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Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Ergodic decomposition

The Pólya urn

run 0 . . .

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Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Ergodic decomposition

The Pólya urn

run 0 . . .

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SLIDE 19

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Ergodic decomposition

The Pólya urn

run 0 . . .

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SLIDE 20

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Ergodic decomposition

The Pólya urn

run 0 . . . . . .

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SLIDE 21

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Ergodic decomposition

The Pólya urn

run 0 . . . run 1 . . . run 2 . . . run 3 . . . run 4 . . . run 5 . . . run 6 . . . . . .

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Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Ergodic decomposition

The Pólya urn

P( ) = 1 2 · 2 3 · 1 4 · 3 5 · 2 6 · 4 7 = 4! · 2! 7!

1convention: 0! = 1

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Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Ergodic decomposition

The Pólya urn

P( ) = 1 2 · 2 3 · 1 4 · 3 5 · 2 6 · 4 7 = 4! · 2! 7! and more generally for w ∈ { , }∗, P(w) = R! · B! (R + B + 1)! = R! · B! (|w| + 1)! where R is the number of ’s and B the number of ’s in w.

1

1convention: 0! = 1

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Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Ergodic decomposition

The Pólya urn

P( ) = 1 2 · 2 3 · 1 4 · 3 5 · 2 6 · 4 7 = 4! · 2! 7! and more generally for w ∈ { , }∗, P(w) = R! · B! (R + B + 1)! = R! · B! (|w| + 1)! where R is the number of ’s and B the number of ’s in w.

1

P is a computable shift-invariant measure, so P-almost every sequence x induces a measure µx.

1convention: 0! = 1

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SLIDE 25

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Ergodic decomposition

The Pólya urn

P( ) = 1 2 · 2 3 · 1 4 · 3 5 · 2 6 · 4 7 = 4! · 2! 7! and more generally for w ∈ { , }∗, P(w) = R! · B! (R + B + 1)! = R! · B! (|w| + 1)! where R is the number of ’s and B the number of ’s in w.

1

P is a computable shift-invariant measure, so P-almost every sequence x induces a measure µx.

Question

What does µx look like? Can it be computed from x?

1convention: 0! = 1

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Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Ergodic decomposition

The Pólya urn

run 0 . . . run 1 . . . run 2 . . . run 3 . . . run 4 . . . run 5 . . . run 6 . . . . . .

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SLIDE 27

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Ergodic decomposition

The Pólya urn

run 0 . . . run 1 . . . run 2 . . . run 3 . . . run 4 . . . run 5 . . . run 6 . . . . . . Each run is equivalent to tossing a coin with some particular bias p, chosen uniformly at random in [0, 1]. P(w) = R! · B! (R + B + 1)! = 1 pR(1 − p)B dp

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Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Effective topology on measures

  • The space of probability measures over Ω with the metric

d(P, Q) =

  • w∈{0,1}∗

2−|w||P[w] − Q[w]| is a compact metric space, hence a Baire space.

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SLIDE 29

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Effective topology on measures

  • The space of probability measures over Ω with the metric

d(P, Q) =

  • w∈{0,1}∗

2−|w||P[w] − Q[w]| is a compact metric space, hence a Baire space.

  • The subset of shift-invariant measures is closed:

P ∈ S ⇐ ⇒ ∀w, P[0w] + P[1w] = P[w] so it is a compact metric subspace, hence a Baire space.

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Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Effective topology on measures

  • The space of probability measures over Ω with the metric

d(P, Q) =

  • w∈{0,1}∗

2−|w||P[w] − Q[w]| is a compact metric space, hence a Baire space.

  • The subset of shift-invariant measures is closed:

P ∈ S ⇐ ⇒ ∀w, P[0w] + P[1w] = P[w] so it is a compact metric subspace, hence a Baire space.

  • In S , the set E of ergodic measures is a dense Π0

2-set:

  • P /

∈ E ⇐ ⇒ ∃P0, P1 ∈ S such that P0 = P1 and P = P0 + P1 2

  • The Markovian ergodic measures are dense in S

E is co-meager in S

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Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Ergodic decomposition

Theorem (V’yugin, 1997)

There exists a computable shift-invariant measure P which is not effectively decomposable.

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Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

  • First step: pick i ∈ {1, 2, 3, . . .} with probability 2−i.

Let pi = 2−ti where ti is the halting time of Turing machine Mi (pi = 0 when Mi does not halt).

  • Following steps: run the following Markov chain

1/2

  • 1/2
  • pi
  • 1−pi
  • 1

pi

  • 1−pi
  • Let ε < /2. Run the test M0(ε)[0] > 3/4. It eventually halts and has

read a finite number n of bits of 0. P{0} < P[0n] ≤ P{0} + ε so P[0n] is an ε-approximation of /2.

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Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

  • V’yugin’s example is a countably infinite combination of ergodic

measures.

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Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

  • V’yugin’s example is a countably infinite combination of ergodic

measures.

  • What about the finite case?

Let P = Q be ergodic. Assume R = P+Q

2

is computable.

Proposition

R is effectively decomposable ⇐ ⇒ P and Q are computable.

Proof.

  • If P and Q are computable, the speed of convergence can be

computed for P and Q.

  • If R is effectively decomposable then using M one can compute P

and Q.

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Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

  • V’yugin’s example is a countably infinite combination of ergodic

measures.

  • What about the finite case?

Question

If P, Q are ergodic and 1

2(P + Q) is computable, are P and Q

computable?

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Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

  • V’yugin’s example is a countably infinite combination of ergodic

measures.

  • What about the finite case?

Question

If P, Q are ergodic and 1

2(P + Q) is computable, are P and Q

computable?

Remark

Without the assumption that P, Q are ergodic, it is (too) easy. Take a non-computable λ ∈ (0, 1) and let P = λδ0 + (1 − λ)δ1, Q = (1 − λ)δ0 + λδ1.

1 2(P + Q) = 1 2(δ0 + δ1) is computable, contrary to P and Q.

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Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Theorem (H., 2011)

There exist ergodic measures P, Q that are not computable relative to

1 2(P + Q).

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SLIDE 38

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Theorem (H., 2011)

There exist ergodic measures P, Q that are not computable relative to

1 2(P + Q).

Theorem (H., 2012)

There exist ergodic measures P, Q that are not computable while

1 2(P + Q) is computable.

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Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

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Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

First of all,

Proposition

P cannot be uniformly computable from P+Q

2

(P and Q varying among the ergodic measures).

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Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

First of all,

Proposition

P cannot be uniformly computable from P+Q

2

(P and Q varying among the ergodic measures). The reason is topological: the splitting map P+Q

2

→ {P, Q} is not continuous.

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Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

First of all,

Proposition

P cannot be uniformly computable from P+Q

2

(P and Q varying among the ergodic measures). The reason is topological: the splitting map P+Q

2

→ {P, Q} is not continuous. The splitting map is even discontinuous at every P+Q

2

with P = Q (and P, Q ergodic).

... and can be proved to be continuous at every ergodic measure P = P+P

2

.

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Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Proposition

The splitting map is discontinuous at every P+Q

2

with P = Q (and P, Q ergodic).

Proof.

P Q (P + Q)/2 Start with P = Q are ergodic.

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Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Proposition

The splitting map is discontinuous at every P+Q

2

with P = Q (and P, Q ergodic).

Proof.

P Q (P + Q)/2 P ′ Q′ = (P ′ + Q′)/2 Let 0 < λ < 1 and P′ := λP + (1 − λ)Q, Q′ := (1 − λ)P + λQ.

P′+Q′ 2

= P+Q

2

but P′ and Q′ are not ergodic!

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Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Proposition

The splitting map is discontinuous at every P+Q

2

with P = Q (and P, Q ergodic).

Proof.

P Q (P + Q)/2 P ′ Q′ = (P ′ + Q′)/2 P ′′ Q′′ (P ′′ + Q′′)/2 Take P′′, Q′′ ergodic such that P′′ ≈ P′, Q′′ ≈ Q′. Hence P′′+Q′′

2

≈ P+Q

2 .

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Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

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Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Theorem

There exist ergodic measures P, Q that are not computable relative to

1 2(P + Q).

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SLIDE 48

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Theorem

There exist ergodic measures P, Q that are not computable relative to

1 2(P + Q).

Let S be the subspace of shift-invariant measures. We actually prove that the set T :=

  • (P, Q) ∈ S 2 : P and Q are ergodic

and not computable relative to P + Q 2

  • is co-meager in S 2, i.e. contains a dense Gδ-set.
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SLIDE 49

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Theorem

There exist ergodic measures P, Q that are not computable relative to

1 2(P + Q).

Let S be the subspace of shift-invariant measures. We actually prove that the set T :=

  • (P, Q) ∈ S 2 : P and Q are ergodic

and not computable relative to P + Q 2

  • is co-meager in S 2, i.e. contains a dense Gδ-set.

As S × S is a Baire space, it implies that T is non-empty.

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Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Lemma

Let M be a Turing machine. The set CM := {(P, Q) ∈ S 2 : M

P+Q 2

computes P} is nowhere dense in S 2.

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Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Lemma

Let M be a Turing machine. The set CM := {(P, Q) ∈ S 2 : M

P+Q 2

computes P} is nowhere dense in S 2.

Proof.

Let U, V be two open sets of measures. We prove that CM is not dense in U × V .

  • If U × V is disjoint from CM, we are done;
  • otherwise let (P, Q) ∈ CM ∩ (U × V ).
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SLIDE 52

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Lemma

Let M be a Turing machine. The set CM := {(P, Q) ∈ S 2 : M

P+Q 2

computes P} is nowhere dense in S 2.

Proof (cont’d).

  • Let (P, Q) ∈ CM ∩ (U × V ).

P Q (P + Q)/2

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Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Lemma

Let M be a Turing machine. The set CM := {(P, Q) ∈ S 2 : M

P+Q 2

computes P} is nowhere dense in S 2.

Proof (cont’d).

  • Let (P, Q) ∈ CM ∩ (U × V ).

P Q (P + Q)/2

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SLIDE 54

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Lemma

Let M be a Turing machine. The set CM := {(P, Q) ∈ S 2 : M

P+Q 2

computes P} is nowhere dense in S 2.

Proof (cont’d).

  • Let (P, Q) ∈ CM ∩ (U × V ).

M P Q (P + Q)/2

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SLIDE 55

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Lemma

Let M be a Turing machine. The set CM := {(P, Q) ∈ S 2 : M

P+Q 2

computes P} is nowhere dense in S 2.

Proof (cont’d).

  • Let (P, Q) ∈ CM ∩ (U × V ).
  • M

P′+Q′ 2

does not compute P′ M P Q (P + Q)/2 P ′ Q′ (P ′ + Q′)/2 =

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SLIDE 56

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Lemma

Let M be a Turing machine. The set CM := {(P, Q) ∈ S 2 : M

P+Q 2

computes P} is nowhere dense in S 2.

Proof (cont’d).

  • Let (P, Q) ∈ CM ∩ (U × V ).
  • M

P′+Q′ 2

does not compute P′

  • M

P′′+Q′′ 2

does not compute P′′ for all P′′ ≈ P′ and Q′′ ≈ Q′. M P Q (P + Q)/2 P ′ Q′ (P ′ + Q′)/2 =

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SLIDE 57

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Lemma

Let M be a Turing machine. The set CM := {(P, Q) ∈ S 2 : M

P+Q 2

computes P} is nowhere dense in S 2. As a result, the set {(P, Q) ∈ S 2 : P is computable relative to P + Q 2 } is meager.

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SLIDE 58

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Lemma

Let M be a Turing machine. The set CM := {(P, Q) ∈ S 2 : M

P+Q 2

computes P} is nowhere dense in S 2. As a result, the set {(P, Q) ∈ S 2 : P is computable relative to P + Q 2 } is meager. Symmetrically, {(P, Q) ∈ S 2 : P and Q are not computable relative to P + Q} is co-meager.

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Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Reminder

In S , the set E of ergodic measures is a dense Gδ-set (even Π0

2).

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SLIDE 60

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Reminder

In S , the set E of ergodic measures is a dense Gδ-set (even Π0

2).

To conclude, the set {(P, Q) : P and Q are ergodic and not computable relative to P + Q} is co-meager.

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SLIDE 61

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

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Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Theorem

There exist P, Q ergodic and non-computable such that P + Q is computable.

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SLIDE 63

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Theorem

There exist P, Q ergodic and non-computable such that P + Q is computable. We need to satisfy 3 requirements:

  • P + Q is computable,
  • P is not computable,
  • P and Q are ergodic.
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SLIDE 64

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

Theorem

There exist P, Q ergodic and non-computable such that P + Q is computable. We need to satisfy 3 requirements:

  • P + Q is computable,
  • P is not computable,
  • P and Q are ergodic.

In this proof, we will say that the machine M computes P if for every ball B of measures, M(B) ↓ ⇐ ⇒ P ∈ B.

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Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

The class of ergodic measures is

n Un, where Un are c.e. open sets.

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SLIDE 66

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

The class of ergodic measures is

n Un, where Un are c.e. open sets.

We define a sequence of balls Bn such that

  • Bn+1 ⊆ Bn,
  • Bn ⊆ Un,
  • the radius of Bn tends to 0.
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SLIDE 67

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

The class of ergodic measures is

n Un, where Un are c.e. open sets.

We define a sequence of balls Bn such that

  • Bn+1 ⊆ Bn,
  • Bn ⊆ Un,
  • the radius of Bn tends to 0.

P will be the unique member of

n Bn. It will be automatically ergodic.

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SLIDE 68

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

P B0 M0(B0) B1 M1(B1) B2 M2(B2) B3 M3(B3) . . . . . . P + Q

At each stage, Bn+1 ⊆ Bn and Bn ⊆ Un for all n.

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SLIDE 69

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

P B0 M0(B0) B1 M1(B1) B2 M2(B2) B3 M3(B3) ↓ . . . . . . P + Q

At each stage, Bn+1 ⊆ Bn and Bn ⊆ Un for all n.

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SLIDE 70

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

P B0 M0(B0) B1 M1(B1) B2 M2(B2) B3 M3(B3) . . . . . . P + Q

At each stage, Bn+1 ⊆ Bn and Bn ⊆ Un for all n.

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SLIDE 71

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

P B0 M0(B0) B1 M1(B1) ↓ B2 M2(B2) B3 M3(B3) . . . . . . P + Q

At each stage, Bn+1 ⊆ Bn and Bn ⊆ Un for all n.

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SLIDE 72

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

P B0 M0(B0) B1 M1(B1) B2 M2(B2) B3 M3(B3) . . . . . . P + Q

At each stage, Bn+1 ⊆ Bn and Bn ⊆ Un for all n.

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SLIDE 73

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

P B0 M0(B0) B1 M1(B1) B2 M2(B2) B3 M3(B3) ↓ . . . . . . P + Q

At each stage, Bn+1 ⊆ Bn and Bn ⊆ Un for all n.

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SLIDE 74

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

P B0 M0(B0) B1 M1(B1) B2 M2(B2) B3 M3(B3) . . . . . . P + Q

At each stage, Bn+1 ⊆ Bn and Bn ⊆ Un for all n.

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SLIDE 75

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

P B0 M0(B0) B1 M1(B1) ↓ B2 M2(B2) B3 M3(B3) . . . . . . P + Q

At each stage, Bn+1 ⊆ Bn and Bn ⊆ Un for all n.

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SLIDE 76

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof

P B0 M0(B0) B1 M1(B1) B2 M2(B2) B3 M3(B3) . . . . . . P + Q

At each stage, Bn+1 ⊆ Bn and Bn ⊆ Un for all n.