Descriptive Set Theory, endofunctors and hypercomputation 1 Arno - - PowerPoint PPT Presentation

Descriptive Set Theory, endofunctors and hypercomputation1

Arno Pauly

Swansea University

Computability- and Category Theoretic Perspectives on DST, Swansea 2018

1Based on joint work with Matthew de Brecht

The talk in a nutshell

◮ General observation: Concepts in descriptive set theory correspond to certain computable endofunctors, ◮ concepts linked by classic theorems are generally derived from the same endofunctor in different ways, ◮ and many properties of the concepts can be derived from simple properties of the associated endofunctor. ◮ The endofunctors for the standard concepts are tied to models of hypercomputation.

Some guiding principles

◮ Hypercomputation is a special kind of computation, not a generalization of computation. ◮ Everything relevant should live in some category. ◮ Generalize for the sake of simplification.

Background Computable endofunctors and derived concepts Examples of endofunctors The representability conjecture

Represented spaces and computability

Definition

A represented space X is a pair (X, δX) where X is a set and δX :⊆ NN → X a surjective partial function.

Definition

F :⊆ NN → NN is a realizer of f : X → Y, iff δY(F(p)) = f(δX(p)) for all p ∈ δ−1

X (dom(f)). Abbreviate: F ⊢ f.

NN

F

− − − − → NN   δX   δY X

f

− − − − → Y

Definition

f : X → Y is called computable (continuous), iff it has a computable (continuous) realizer.

Type-2 Turing machines

Figure: The core model

The various classes of spaces

Represented spaces QCB0-spaces ∼ = admissibly represented spaces Quasi-Polish spaces Polish spaces

Cartesian closure

Observation

We can form function spaces (to be denoted by C(−, −)) in the category of represented spaces by the UTM-theorem/

Definition

Let S = ({⊤, ⊥}, δS) be defined via δS(p) = ⊥ iff p = 0N.

Definition

The space O(X) of open subsets of X is obtained from C(X, S) via identification.

Definition

We call X admissible, if the canonic computable map κX : X → C(C(X, S), S) is computably invertible.

Theorem (Schröder)

If Y is admissible, then for functions f : X → Y topological continuity and realizer continuity coincide.

Some definitions from DST

Definition

A set is Σ0

1 iff it is open. A set is Π0 n, if it is the complement of a

Σ0

n-set. A set U is Σ0 n+1, iff it is of the form U = n∈N An with

Π0

n-sets An. A set is ∆0 n iff it is both Σ0 n and Π0 n.

Definition

A function f is Σ0

n-measurable, if f −1(U) is Σ0 n for any open U.

Definition

A function is Baire class 0, if it is continuous. A function is Baire class n + 1, if it is the pointwise limit of Baire class n functions.

The Banach-Lebesgue-Hausdorff theorem

Theorem

On Polish spaces, the Σ0

n+1-measurable functions are just the

Baire class n functions. (Conditions apply)

The Jayne-Rogers theorem

Definition

Call f : X → Y Π0

1-piecewise continuous, iff ∃(An)n∈N, An is Π0 1,

X =

n∈N An, f|An is continuous.

Theorem

On Polish spaces, a function is ∆0

2-measurable iff it is

Π0

1-piecewise continuous.

Defining computable endofunctors

Definition

An endofunctor on the category of represented spaces is an

• peration d that
• 1. maps represented spaces to represented spaces,
• 2. maps continuous functions from X to Y to continuous

functions from dX to dY,

• 3. and is compatible with composition.

◮ An endofunctor d is called computable, if there is a matching computable function d : C(X, Y) → C(dX, dY) for any spaces X, Y.

The basic derived concepts

Definition

Call f : X → Y d-continuous, iff f : X → dY is continuous. (Keyword: Kleisli-category)

Definition

Call U ⊆ X d-open, iff χU : X → dS is continuous. The space of d-opens is Od(X).

Definition

Call f : X → Y d-measurable, iff f −1 : O(Y) → Od(X) is continuous.

A first observation

Proposition

Any d-continuous function is d-measurable.

Definition

Call Y d-admissible, if the canonic map κd : dY → C(C(Y, S), dS) is computably invertible.

Theorem

If Y is d-admissible, then for functions f : X → Y d-continuity and d-admissibility coincide.

Some structural properties

Theorem

Let d satisfy (d(X × X) ∼ = dX × dX) (dC(N, X) = C(N, dX)) for all represented spaces X. We may conclude:

• 1. (f, U) → f −1(U) : C(X, Y) × Od(Y) → Od(X) is well-defined

and computable.

• 2. ∩, ∪ : Od(X) × Od(X) → Od(X) are well-defined and

computable.

• 3. Any countably based admissible space X is d-admissible.
• 4. : C(N, Od(X)) → Od(X) is well-defined and computable.

Lifting further properties

Definition

Call a space X d-Hausdorff, iff = : X × X → dS is computable.

Definition

Call a space X d-compact, iff IsFull : Od(X) → dS is computable.

Definition

Call a space X d-overt, iff IsNonEmpty : Od(X) → dS is computable.

From hypercomputation to endofunctors

Observation

Consider a notion C of hypercomputation admitting universal

• machines. Then we can define an operation c on represented

spaces such that the following are equivalent:

• 1. f : X → Y is C-computable.
• 2. f : X → cY is computable.

Observation

If C is closed under composition with computable functions in a uniform way, then c is a computable endofunctor.

Game characterizations

Game characterizations give endofunctors, too. Here cY corresponds to moves that Player 2 makes to indicate some value f(x).

The lim operator

Definition

Consider lim ⊆: NN → NN defined via lim(p)(n) = limi→∞ p(n, i). Now define an endofunctor ′ by (X, δX)′ = (X, δX ◦ lim).

Proposition

′ is a computable endofunctor satisfying C(N, X)′ ∼

= C(N, X′).

Definition

Let X(0) = X and X(n+1) = (X(n))′.

Proposition

(n) is a computable endofunctor satisfying

C(N, X)(n) ∼ = C(N, X(n)).

Limit machines

Figure: Limit machine

f : X → Y is computable by a limit machine iff f : X → Y′ is computable.

The correspondence

Classic DST Synthetic DST Σ0

n+1-sets (n)-open sets

Σ0

n+1-measurable functions (n)-measurable functions

Baire class n + 1

(n)-continuous functions

Banach-Lebesgue-Hausdorff Theorem

(n)-admissibility

′-overtness and ′-compactness

Proposition

A Polish space is′-overt iff it is Kσ.

Theorem

For a Quasi-Polish space, the following are equivalent:

• 1. Noetherian
• 2. ′-compactness
• 3. ∇-compactness
• 4. ∇-overtness

The ∇-endofunctor

Definition

Define ∇ :⊆ NN → NN via ∇(w0p)(n) = p(n) − 1 iff p contains no 0. Define an operator ∇ via (X, δX)∇ = (X, δX ◦ ∇).

Proposition

∇ is a computable endofunctor satisfying (X × X)∇ ∼

= X∇ × X∇. Classic DST Synthetic DST ∆0

2-sets ∇-open sets

∆0

2-measurable functions ∇-measurable functions

Π0

1-piecewise continuous ∇-continuous functions

Jayne-Rogers Theorem

∇-admissibility

Turing machines changing their minds

Figure: Computation with mindchanges

The unique choice endofunctor

Definition

Consider UCNN :⊆ A(NN) → NN defined by UC({p}) = p. Define an operation b by b(X, δX) = (X, δX ◦ UCNN ◦ ψ−

NN).

Proposition

b is a computable endofunctor satisfying bC(N, X) ∼ = C(N, bX). Classic DST Synthetic DST Borel sets b-open sets Borel-measurable functions b-measurable functions ?? b-continuous functions Semmes’ tree game characterization b-admissibility

Turing machines changing their minds

Figure: Non-deterministic computation

An inherent constructive perspective

◮ The notion of d-measurability has an inherent constructive flavour: The preimage map is required to be continuous. ◮ In classic DST, such a requirement is alien. ◮ We may relax the requirement, but we cannot avoid it entirely. ◮ Luckily, we have:

Theorem (BRATTKA)

Let X, Y be Polish, and let f : X → Y be Σ0

n+1-measurable. Then

f −1 : O(Y) → O(n)(X) is continuous.

Theorem (GREGORIADES)

Let X, Y be Polish, and let f : X → Y be (Σ0

m+1, Σ0 n+1)-measurable. Then f −1 : O(m)(Y) → bO(n)(X) is

continuous.

The decomposability conjecture

Conjecture

Let X, Y be Polish and n ≤ m ≤ 2n. Then f : X → Y is (Σ0

n+1, Σ0 m+1)-measurable, iff there is a Π0 m partition of X s.t. any

restriction of f to a piece is Σ0

m−n+1-measurable.

Theorem (KIHARA)

For countably dimensional spaces, the decomposability conjecture is true iff any (Σ0

n+1, Σ0 m+1)-measurable function has

a continuous preimage map.

Conjecture (Strong representability conjecture)

id : C−1(O(n)(Y), bO(m)(X)) → bC−1(O(n)(Y), O(m)(X)) is computable.

There is more

◮ Left-adjoint endofunctors correspond to retopologizing. ◮ but for adjoints, we need to use Markov-computability instead of computability ◮ this leads to e.g the Gandy-Harrington space ◮ The induced monads capture notions like low-computability.

The sources

• A. Pauly & M. de Brecht.

Non-deterministic computability and the Jayne Rogers theorem.

• Proc. DCM 2012, EPTCS 143 , 2014.
• A. Pauly & M. de Brecht.

Descriptive Set Theory in the Category of Represented Spaces. LICS, 2015.

• A. Pauly.

Computability on the countable ordinals and the Hausdorff-Kuratowski theorem. MFCS, 2015.

• M. de Brecht & A. Pauly.

Noetherian Quasi-Polish spaces CSL, 2017.