Descriptive Set Theory, endofunctors and hypercomputation 1 Arno Pauly Swansea University Computability- and Category Theoretic Perspectives on DST, Swansea 2018 1 Based 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 , δ X ) where X is a set and δ X : ⊆ N N → X a surjective partial function. Definition F : ⊆ N N → N N is a realizer of f : X → Y , iff δ Y ( F ( p )) = f ( δ X ( p )) for all p ∈ δ − 1 X (dom( f )) . Abbreviate: F ⊢ f . F N N → N N − − − − � δ X � δ Y f X − − − − → 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 QCB 0 -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 = ( {⊤ , ⊥} , δ S ) be defined via δ S ( p ) = ⊥ iff p = 0 N . 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 κ X : 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 Σ 0 1 iff it is open. A set is Π 0 n , if it is the complement of a Σ 0 n -set. A set U is Σ 0 n + 1 , iff it is of the form U = � n ∈ N A n with Π 0 n -sets A n . A set is ∆ 0 n iff it is both Σ 0 n and Π 0 n . Definition A function f is Σ 0 n -measurable, if f − 1 ( U ) is Σ 0 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 Σ 0 n + 1 -measurable functions are just the Baire class n functions. (Conditions apply)
The Jayne-Rogers theorem Definition Call f : X → Y Π 0 1 -piecewise continuous, iff ∃ ( A n ) n ∈ N , A n is Π 0 1 , X = � n ∈ N A n , f | A n is continuous. Theorem On Polish spaces, a function is ∆ 0 2 -measurable iff it is Π 0 1 -piecewise continuous.
Defining computable endofunctors Definition An endofunctor on the category of represented spaces is an operation d that 1. maps represented spaces to represented spaces, 2. maps continuous functions from X to Y to continuous functions from d X to d Y , 3. and is compatible with composition. ◮ An endofunctor d is called computable , if there is a matching computable function d : C ( X , Y ) → C ( d X , d Y ) for any spaces X , Y .
The basic derived concepts Definition Call f : X → Y d -continuous, iff f : X → d Y is continuous. (Keyword: Kleisli-category) Definition Call U ⊆ X d -open, iff χ U : X → d S is continuous. The space of d -opens is O d ( X ) . Definition Call f : X → Y d -measurable, iff f − 1 : O ( Y ) → O d ( X ) is continuous.
A first observation Proposition Any d-continuous function is d-measurable. Definition Call Y d -admissible, if the canonic map κ d : d Y → C ( C ( Y , S ) , d S ) 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 ) ∼ = d X × d X ) (d C ( N , X ) = C ( N , d X ) ) for all represented spaces X . We may conclude: 1. ( f , U ) �→ f − 1 ( U ) : C ( X , Y ) × O d ( Y ) → O d ( X ) is well-defined and computable. 2. ∩ , ∪ : O d ( X ) × O d ( X ) → O d ( X ) are well-defined and computable. 3. Any countably based admissible space X is d-admissible. 4. � : C ( N , O d ( X )) → O d ( X ) is well-defined and computable.
Lifting further properties Definition Call a space X d -Hausdorff, iff � = : X × X → d S is computable. Definition Call a space X d -compact, iff IsFull : O d ( X ) → d S is computable. Definition Call a space X d -overt, iff IsNonEmpty : O d ( X ) → d S 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 → c Y 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 c Y corresponds to moves that Player 2 makes to indicate some value f ( x ) .
The lim operator Definition Consider lim ⊆ : N N → N N defined via lim( p )( n ) = lim i →∞ p ( � n , i � ) . Now define an endofunctor ′ by ( X , δ X ) ′ = ( X , δ X ◦ 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 Σ 0 ( n ) -open sets n + 1 -sets Σ 0 ( n ) -measurable functions n + 1 -measurable functions ( n ) -continuous functions Baire class n + 1 ( n ) -admissibility Banach-Lebesgue-Hausdorff Theorem
′ -overtness and ′ -compactness Proposition A Polish space is ′ -overt iff it is K σ . Theorem For a Quasi-Polish space, the following are equivalent: 1. Noetherian 2. ′ -compactness 3. ∇ -compactness 4. ∇ -overtness
The ∇ -endofunctor Definition Define ∇ : ⊆ N N → N N via ∇ ( w 0 p )( n ) = p ( n ) − 1 iff p contains no 0. Define an operator ∇ via ( X , δ X ) ∇ = ( X , δ X ◦ ∇ ) . Proposition ∇ is a computable endofunctor satisfying ( X × X ) ∇ ∼ = X ∇ × X ∇ . Classic DST Synthetic DST ∆ 0 ∇ -open sets 2 -sets ∆ 0 ∇ -measurable functions 2 -measurable functions Π 0 ∇ -continuous functions 1 -piecewise continuous ∇ -admissibility Jayne-Rogers Theorem
Turing machines changing their minds Figure: Computation with mindchanges
The unique choice endofunctor Definition Consider UC N N : ⊆ A ( N N ) → N N defined by UC ( { p } ) = p . Define an operation b by b ( X , δ X ) = ( X , δ X ◦ UC N N ◦ ψ − N N ) . Proposition b is a computable endofunctor satisfying b C ( N , X ) ∼ = C ( N , b X ) . 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 (B RATTKA ) Let X , Y be Polish, and let f : X → Y be Σ 0 n + 1 -measurable. Then f − 1 : O ( Y ) → O ( n ) ( X ) is continuous. Theorem (G REGORIADES ) Let X , Y be Polish, and let f : X → Y be n + 1 ) -measurable. Then f − 1 : O ( m ) ( Y ) → b O ( n ) ( X ) is (Σ 0 m + 1 , Σ 0 continuous.
The decomposability conjecture Conjecture Let X , Y be Polish and n ≤ m ≤ 2 n. Then f : X → Y is (Σ 0 n + 1 , Σ 0 m + 1 ) -measurable, iff there is a Π 0 m partition of X s.t. any restriction of f to a piece is Σ 0 m − n + 1 -measurable. Theorem (K IHARA ) For countably dimensional spaces, the decomposability conjecture is true iff any (Σ 0 n + 1 , Σ 0 m + 1 ) -measurable function has a continuous preimage map. Conjecture (Strong representability conjecture) id : C − 1 ( O ( n ) ( Y ) , b O ( m ) ( X )) → b C − 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.
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