Descriptive complexity on non-Polish spaces Mathieu Hoyrup joint - - PowerPoint PPT Presentation

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

Descriptive complexity on non-Polish spaces

Mathieu Hoyrup joint work with Antonin Callard

Loria - Inria, Nancy (France)

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

DST outside Polish spaces

Descriptive Set Theory (DST): ‚ Mainly on Polish spaces (completely metrizable spaces). Theoretical Computer Science induces other spaces: ‚ Partial functions, ‚ Higher-order functionals, e.g. pN Ñ Nq Ñ N, ‚ Computation with advice, ‚ etc.

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

DST outside Polish spaces

Descriptive Set Theory (DST): ‚ Mainly on Polish spaces (completely metrizable spaces). Theoretical Computer Science induces other spaces: ‚ Partial functions, ‚ Higher-order functionals, e.g. pN Ñ Nq Ñ N, ‚ Computation with advice, ‚ etc. Need to develop DST outside Polish spaces: ‚ Domains [Selivanov], quasi-Polish spaces [de Brecht] ‚ Represented spaces [Brattka, de Brecht, Pauly, Schröder, Selivanov]

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

Two measures of complexity

In an admissibly represented space X, two measures of complexity of a set A Ď X. Topological complexity Complexity of describing A from open sets. Symbolic complexity Complexity of testing whether a point belongs to A.

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

Two measures of complexity

In an admissibly represented space X, two measures of complexity of a set A Ď X. Topological complexity Complexity of describing A from open sets. Symbolic complexity Complexity of testing whether a point belongs to A. Theorem (de Brecht, 2013) They coincide on countably-based spaces. What about other spaces?

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

Motivating example

Let A “ " 1 n : n P N * Ď R. How complicated is A? Two approaches: ‚ How to test x P A it with an algorithm? ‚ How to describe A in terms of simpler sets?

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

Motivating example

Let A “ " 1 n : n P N * Ď R. Algorithm No Yes No Dn,

1 n`1 ă x ă 1 n´1?

x ‰ 1

n?

Description A “ p0, `8qz ď

n

´ 1 n ` 1, 1 n ¯ .

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

Motivating example

These approaches are equivalent: for any A Ď R, A is decidable with ď 2 mind changes No-Yes-No ð ñ A is a difference of two effective open sets (A P D2pRq).

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

More generally

Are these two approaches always equivalent? ‚ Algorithms make sense on represented spaces, ‚ Descriptions using open sets make sense on topological spaces. So let’s work on topological spaces with an admissible representation.

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

Polynomials

Representation A polynomial P P RrXs is represented by: ‚ Some n ě degpPq, ‚ The coefficients of P “ p0 ` p1X ` . . . ` pnXn.

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

Polynomials

Representation A polynomial P P RrXs is represented by: ‚ Some n ě degpPq, ‚ The coefficients of P “ p0 ` p1X ` . . . ` pnXn. How complicated is A “ " P P RrXs : p0 “ 0 or p0 ą 1 degpPq * ?

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

Polynomials

Representation A polynomial P P RrXs is represented by: ‚ Some n ě degpPq, ‚ The coefficients of P “ p0 ` p1X ` . . . ` pnXn. How complicated is A “ " P P RrXs : p0 “ 0 or p0 ą 1 degpPq * ? ‚ Decidable with 2 mind-changes, ‚ But not a difference of two open sets!

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

Polynomials

A “ " P P RrXs : p0 “ 0 or p0 ą 1 degpPq * . Algorithm Given P and n ě degpPq, Yes No No Yes p0 » 0 p0 ‰ 0? p0 fi 0 p0 ą

1 degpPq?

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

Polynomials

A “ " P P RrXs : p0 “ 0 or p0 ą 1 degpPq * . Descriptive complexity A is not a difference of 2 open sets:

1 n ` Xn`1 p

Ý Ý Ý Ñ

pÑ8 1 n

Ý Ý Ý Ñ

nÑ8

P A R A P A

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

The problem

Algorithms and topology induce the same complexity on R but not on RrXs. ‚ Why? ‚ What about other spaces? ‚ What about other complexity levels (

• Σ0

α, etc.)

‚ What do algorithms measure?

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

The problem

Algorithms and topology induce the same complexity on R but not on RrXs. ‚ Why? ‚ What about other spaces? ‚ What about other complexity levels (

• Σ0

α, etc.)

‚ What do algorithms measure? Guess Algorithms reflect the sequential rather than topological aspects of the space.

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

Algorithms prefer sequences

Only sequential spaces can be handled by representations (Schröder). Franklin 65 Sequential spaces ” quotients of metric spaces, Schröder 02 Adm. rep. spaces ” quotients of countably-based metric spaces. ‚ A subspace of a represented space is not a topological subspace but its sequentialization, ‚ A product of represented spaces is not the topological product but its sequentialization.

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

Topology vs sequences

Countably-based First-countable Fréchet-Urysohn Sequential

• Seq. closure “ closure
• Seq. continuity “ continuity

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

Symbolic complexity Hardness CoPolish spaces Spaces of open sets

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

Topological complexity

Borel

• Σ0

α

• Π0

α

. . .

• Σ0

3

• Π0

3

• ∆0

3

• Σ0

2

• Π0

2

• ∆0

2

• Σ0

1

• Π0

1

(a) Borel hierarchy

• ∆0

2

• Dα

ˇ

• Dα

. . .

• D3

ˇ

• D3
• D3 X ˇ
• D3
• D2

ˇ

• D2
• D2 X ˇ
• D2
• Σ0

1

• Π0

1

(b) Hausdorff

difference hierarchy

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

Symbolic complexity

We work on represented spaces with a total admissible representation pX, δXq. Let Γ be some complexity class (

• Σ0

α, Σ0 α, etc.).

Definition (Symbolic complexity) A set A Ď X belongs to rΓs if δ´1

X pAq P ΓpNq.

One always has Γ Ď rΓs, (δX is continuous)

• Σ0

1 “ r

• Σ0

1s

(final topology).

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

Countably-based spaces

Symbolic and topological complexity coincide on countably-based spaces. Theorem (De Brecht, 2013) If X is countably-based, then rΓs “ Γ. Already in (Brattka 2005), (Saint-Raymond 2007) for Polish spaces. Theorem The following are equivalent: ‚ X is countably-based, ‚ r

• D2s “
• D2 in a uniform way.

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

Symbolic complexity

We have seen that on RrXs, r

• D2s Ę
• D2,

and even rD2s Ę

• D2,

witnessed by A “ " P P RrXs : p0 “ 0 or p0 ą 1 degpPq * .

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

We mainly study two classes of spaces: ‚ CoPolish spaces ” inductive limits of compact metric spaces, ‚ Spaces of open subsets of Polish spaces.

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

Symbolic complexity Hardness CoPolish spaces Spaces of open sets

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

Hardness

‚ In a Polish space, how to show that a set A is not

• Σ0

2?

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

Hardness

‚ In a Polish space, how to show that a set A is not

• Σ0

2?

§ Prove that it is

• Π0

2-hard: every

• Π0

2-subset of NN is

continuously reducible to A.

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

Hardness

‚ In a Polish space, how to show that a set A is not

• Σ0

2?

§ Prove that it is

• Π0

2-hard: every

• Π0

2-subset of NN is

continuously reducible to A. Theorem (Wadge) Let Γ ‰ ˇ Γ. For any Borel subset A of a Polish space, A R Γ ð ñ A is ˇ Γ-hard. Applies to Γ “

• Dα,
• Σ0

α,

• Π0

α, but not

• ∆0

α.

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

Hardness outside Polish spaces

‚ What about non-Polish spaces?

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

Hardness outside Polish spaces

‚ What about non-Polish spaces? § Hardness actually reflects symbolic complexity.

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

Hardness outside Polish spaces

‚ What about non-Polish spaces? § Hardness actually reflects symbolic complexity.

Proposition Let Γ ‰ ˇ Γ. For any Borel set A, A R rΓs ð ñ A is ˇ Γ-hard. How to capture topological complexity?

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

Hardness*

Definition A Ď X is Γ-hard* if for every countably-based weaker topology τ, A is Γ-hard in pX, τq. Hardness reflects topological complexity: Theorem For any Borel set A Ď X, and Γ ‰ ˇ Γ, A R Γ ð ñ A is ˇ Γ-hard*.

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

Hardness*

Definition A Ď X is Γ-hard* if for every countably-based weaker topology τ, A is Γ-hard in pX, τq. Hardness reflects topological complexity: Theorem For any Borel set A Ď X, and Γ ‰ ˇ Γ, A R Γ ð ñ A is ˇ Γ-hard*, A R rΓs ð ñ A is ˇ Γ-hard

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

Symbolic complexity Hardness CoPolish spaces Spaces of open sets

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

CoPolish spaces

CoPolish space: inductive limit of (locally) compact metrizable spaces (Schröder, 2004). X “ ď

nPN

Xn with Xn Ď Xn`1. Example The space X of real polynomials, with Xn “ tpolynomials of degree ď nu.

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

CoPolish spaces

Topology A set U Ď X is open if each U X Xn is open in Xn. Representation A name for x P X is given by: ‚ Any n P N such that x P Xn, ‚ A Cauchy name of x in Xn.

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

CoPolish spaces

. . . . . . r

• Σ0

3s

• Σ0

3

r

• Σ0

2s

• Σ0

2

r

• ∆0

2s

• ∆0

2

r

• D2s
• D2

r

• Σ0

1s

• Σ0

1

Figure: On a CoPolish space

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

CoPolish spaces

For all k P N, r

• Σ0

ks “

• Σ0

k,

r

• ∆0

ks “

• ∆0

k,

rΣ0

ks “ Σ0 k,

r∆0

ks “ ∆0 k.

Proof. The degree is limit-computable from the coefficients. So for levels k ě 2, the space is like a countably-based space. But in RrXs, some A P rD2s is

• ∆0

2-complete*.

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

Polynomials

Let B “ " 1 k1 ` Xk1 k2 ` Xk2 k3 ` . . . ` Xkn´2 kn´1 ` Xkn´1 kn : k1 ă k2 ă . . . ă kn and n even * . One has A P r

• D2s but A is
• ∆0

2-complete*.

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

CoPolish spaces

Theorem If X is coPolish, then r

• D2s “
• D2 ð

ñ X is Fréchet-Urysohn.

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

CoPolish spaces

Theorem If X is coPolish, then r

• D2s “
• D2 ð

ñ X is Fréchet-Urysohn. If X is Hausdorff admissibly represented, then r

• D2s “
• D2 ù

ñ X is Fréchet-Urysohn. We will see that it fails for some non-Hausdorff X.

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

CoPolish spaces

Theorem If X is coPolish, then r

• D2s “
• D2 ð

ñ X is Fréchet-Urysohn. If X is Hausdorff admissibly represented, then r

• D2s “
• D2 ù

ñ X is Fréchet-Urysohn. We will see that it fails for some non-Hausdorff X. Proof of ù ñ. If X is not Fréchet-Urysohn, then it contains the Arens’ space, where r

• D2s ‰
• D2.

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

CoPolish spaces

Theorem If X is coPolish, then r

• D2s “
• D2 ð

ñ X is Fréchet-Urysohn. If X is Hausdorff admissibly represented, then r

• D2s “
• D2 ù

ñ X is Fréchet-Urysohn. We will see that it fails for some non-Hausdorff X. Open question Is it an equivalence?

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

Symbolic complexity Hardness CoPolish spaces Spaces of open sets

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

Spaces of open sets

Admissibly represented spaces have interesting categorical properties (cartesian closed). In particular, if pX, δq is admissibly represented then so is OpXq, the space of open sets, with the Scott topology. We now study symbolic and topological complexity on OpXq.

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

Spaces of open sets

Theorem Let X be admissibly represented. On OpXq, r

• Dns “
• Dn.

r

• Dωs
• Dω

. . . . . . r

• Dns
• Dn

. . . . . . r

• D2s
• D2

r

• Σ0

1s

• Σ0

1

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

Spaces of open sets

Question What about higher complexity levels on OpXq?

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

Spaces of open sets

Question What about higher complexity levels on OpXq? We restrict ourselves to the case when X is Polish.

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

Spaces of open sets

Question What about higher complexity levels on OpXq? We restrict ourselves to the case when X is Polish. Answer The answer depends on the compactness properties of X.

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

Spaces of open sets

Question What about higher complexity levels on OpXq? We restrict ourselves to the case when X is Polish. Answer The answer depends on the compactness properties of X. Best case: X is locally compact, e.g. X “ R. Worst case: X is not σ-compact, e.g. X “ NN (Baire space).

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

Open subsets of Polish spaces

Let Xnk “ tx P X : x has no compact neighborhoodu. Polish spaces are divided in 4 classes Class I: Xnk “ H (i.e., X is locally compact), Class II: Xnk ‰ H is finite, Class III: Xnk is infinite and X is σ-compact, Class IV: X is not σ-compact.

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

Open subsets of Polish spaces

Let Xnk “ tx P X : x has no compact neighborhoodu. Polish spaces are divided in 4 classes Class I: Xnk “ H (i.e., X is locally compact), Ex: R Class II: Xnk ‰ H is finite, Class III: Xnk is infinite and X is σ-compact, Class IV: X is not σ-compact. Ex: NN

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

Open subsets of Polish spaces

If X P Class I then OpXq is countably-based, so (de Brecht 13)

• n OpXq,

. . . . . . r

• Σ0

3s

• Σ0

3

r

• Σ0

2s

• Σ0

2

r

• ∆0

2s

• ∆0

2

r

• Dαs
• Dα

r

• Σ0

1s

• Σ0

1

Symbolic complexity ” Topological complexity.

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

Open subsets of Polish spaces

. . . . . . r

• Σ0

4s

• Σ0

4

r

• Σ0

3s

• Σ0

3

r

• ∆0

3s

• ∆0

3

r

• Dωs
• Dω

r

• Dns
• Dn

r

• Σ0

1s

• Σ0

1

Class II . . . Σ0

6

. . .

• Σ0

5

r

• Σ0

4s

• Σ0

4

r

• Σ0

3s

• Σ0

3

r

• Σ0

2s

• Σ0

2

r

• Dns
• Dn

r

• Σ0

1s

• Σ0

1

Class III Borel r

• Σ0

2s

• Σ0

2

r

• Dns
• Dn

r

• Σ0

1s

• Σ0

1

Class IV

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

Open subsets of Polish spaces

. . . . . . r

• Σ0

4s

• Σ0

4

r

• Σ0

3s

• Σ0

3

r

• ∆0

3s

• ∆0

3

r

• Dωs
• Dω

r

• Dns
• Dn

r

• Σ0

1s

• Σ0

1

Class II . . . Σ0

6

. . .

• Σ0

5

r

• Σ0

4s

• Σ0

4

r

• Σ0

3s

• Σ0

3

r

• Σ0

2s

• Σ0

2

r

• Dns
• Dn

r

• Σ0

1s

• Σ0

1

Class III Borel r

• Σ0

2s

• Σ0

2

r

• Dns
• Dn

r

• Σ0

1s

• Σ0

1

Class IV

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

Open subsets of Polish spaces

Some A P r

• Dωs

is

• ∆0

3-complete*.

‚ r

• Σ0

2s Ď

• Σ0

4 and

‚ some A P r

• Σ0

2s

is

• Σ0

3-complete*.

Some A P r

• Σ0

2s is

not Borel. r

• ∆0

3s

• ∆0

3

r

• Dωs
• Dω

Class II r

• Σ0

4s

• Σ0

4

r

• Σ0

3s

• Σ0

3

r

• Σ0

2s

• Σ0

2

Class III Borel r

• Σ0

2s

• Σ0

2

Class IV

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

Open subsets of Polish spaces

Open question For X P Class III, is there a

• Σ0

4-complete* set in r

• Σ0

2s?

• r r
• Σ0

2s Ď

• Σ0

3?

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

Open subsets of the Baire space

Building a set A P rΣ0

2s which is not Borel.

Proof sketch ‚ N is not locally compact,

1Escardo, Heckmann. Topologies on Spaces of Continuous Functions

(2001)

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

Open subsets of the Baire space

Building a set A P rΣ0

2s which is not Borel.

Proof sketch ‚ N is not locally compact, ‚ So1 the product topology on N ˆ OpNq is too weak:

1Escardo, Heckmann. Topologies on Spaces of Continuous Functions

(2001)

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

Open subsets of the Baire space

Building a set A P rΣ0

2s which is not Borel.

Proof sketch ‚ N is not locally compact, ‚ So1 the product topology on N ˆ OpNq is too weak:

‚ It is not sequential,

1Escardo, Heckmann. Topologies on Spaces of Continuous Functions

(2001)

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

Open subsets of the Baire space

Building a set A P rΣ0

2s which is not Borel.

Proof sketch ‚ N is not locally compact, ‚ So1 the product topology on N ˆ OpNq is too weak:

‚ It is not sequential, ‚ It is strictly weaker than the topology of the representation,

1Escardo, Heckmann. Topologies on Spaces of Continuous Functions

(2001)

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

Open subsets of the Baire space

Building a set A P rΣ0

2s which is not Borel.

Proof sketch ‚ N is not locally compact, ‚ So1 the product topology on N ˆ OpNq is too weak:

‚ It is not sequential, ‚ It is strictly weaker than the topology of the representation, ‚ Which is witnessed by the set E “ tpf, Uq P N ˆ OpNq : f P Uu,

1Escardo, Heckmann. Topologies on Spaces of Continuous Functions

(2001)

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

Open subsets of the Baire space

Building a set A P rΣ0

2s which is not Borel.

Proof sketch ‚ N is not locally compact, ‚ So1 the product topology on N ˆ OpNq is too weak:

‚ It is not sequential, ‚ It is strictly weaker than the topology of the representation, ‚ Which is witnessed by the set E “ tpf, Uq P N ˆ OpNq : f P Uu, ‚ And E is not even Borel in the product topology.

1Escardo, Heckmann. Topologies on Spaces of Continuous Functions

(2001)

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

Open subsets of the Baire space

Building a set A P rΣ0

2s which is not Borel.

Proof sketch ‚ N is not locally compact, ‚ So1 the product topology on N ˆ OpNq is too weak:

‚ It is not sequential, ‚ It is strictly weaker than the topology of the representation, ‚ Which is witnessed by the set E “ tpf, Uq P N ˆ OpNq : f P Uu, ‚ And E is not even Borel in the product topology.

‚ It is possible to “embed” E in OpNq as a rΣ0

2s-subset.

1Escardo, Heckmann. Topologies on Spaces of Continuous Functions

(2001)

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Symbolic complexity Hardness CoPolish spaces Spaces of open sets

Conclusion

‚ Several results from DST extend to represented spaces, using symbolic rather than topological complexity.

Hausdorff-Kuratowski theorem, Wadge lemma

‚ Is it possible to better understand symbolic complexity classes?

How to describe

• D2-subsets of RrXs?

‚ Do the current notions of topological complexity make sense when the space is not countably-based?

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