Background on Turing Degrees and Jump Operators

Hayden Jananthan

Vanderbilt University

March 18, 2019

Hayden Jananthan (Vanderbilt University) Background on Turing Degrees and Jump Operators March 18, 2019 1 / 38 1 Basic Definitions

2 Arithmetical Theory

3 Hyperarithmetical Theory

Hayden Jananthan (Vanderbilt University) Background on Turing Degrees and Jump Operators March 18, 2019 2 / 38 Basic Notation – Topological Spaces

∗ Baire Space: NN with topology generated by subbasis, for σ N ,

N Nσ f N σ f ∈

∶= { ∈ S ⊂ } Cantor Space:2 N 0, 1 N with subspace topology.

Proposition = { } 1 NN is homeomorphic to R Q with the subspace topology. 2 2N is compact (and is homeomorphic to any compact Hausdorff space without isolated points having∖ a countable basis of clopen sets).

Elements of 2N are freely identified with subsets of N via their characteristic functions.

Hayden Jananthan (Vanderbilt University) Background on Turing Degrees and Jump Operators March 18, 2019 4 / 38 Basic Notation – Partial Functions

Notation: f A B is a function f dom f B where dom f A. Convergence: f x if x dom f (f converges or is defined at x) f ∶x⊆ →otherwise (f diverges∶ or is undefined→ at x). ⊆ Strong Equality:( f) ↓ g if and∈ only if f and g converge on same inputs and are( ) equal↑ when they converge. ≃

Hayden Jananthan (Vanderbilt University) Background on Turing Degrees and Jump Operators March 18, 2019 5 / 38 Partial Recursive Functions

Definition (Partial Recursive) k Suppose f N N is given.

f is∶⊆partial→ recursive f is algorithmically computable

where ‘algorithm’ is interpreted⇐⇒ in your favorite programming language.

If e is the G¨odelnumber of such an algorithm, write

(k) ϕe m1,..., mk f m1,..., mk

We call e an index of f .( ) ≃ ( )

Hayden Jananthan (Vanderbilt University) Background on Turing Degrees and Jump Operators March 18, 2019 6 / 38 Equivalent Characterizations of Partial Recursiveness

Being a bit more precise: Theorem Suppose f N N is given. The following are equivalent: 1 f is Turing machine computable. ∶⊆ → 2 f is register machine program computable. 3 f is µ-recursive. 4 f is λ-computable.

Church-Turing Thesis A partial function f N N is computable by a digital computer (ignoring resource limitations) if and only if it is computable by any of the above equivalent definitions.∶⊆ →

Hayden Jananthan (Vanderbilt University) Background on Turing Degrees and Jump Operators March 18, 2019 7 / 38 Partial Recursive Functionals

Definition (Partial Recursive) k Suppose Ψ NN N N is given.

Ψ is ∶⊆partial× recursive→ Ψ is algorithmically computable

where ‘algorithm’ now includes⇐⇒ oracle/black-box computations that make use of the function parameter.

If e is the G¨odelnumber of such an algorithm, write

(k),f ϕe m1,..., mk Ψ f , m1,..., mk

We call e an index of Ψ.( ) ≃ ( )

Hayden Jananthan (Vanderbilt University) Background on Turing Degrees and Jump Operators March 18, 2019 8 / 38 Some Basic Results

Theorem (Enumeration Theorem) (k) The partial functions F e, m1,..., mk ϕe m1,..., mk and (k),f Ψ f , e, m1,..., mk ϕe m1,..., mk are partial recursive. ( ) ≃ ( ) Theorem( (Parametrization) ≃ ( Theorem) ) k+1 Suppose F N N is partial recursive. Then there exists a primitive recursive f N N such that ∶⊆ → (k) ∶ → F e, m1,..., mk ϕf (e) m1,..., mk ( ) ≃ ( )

Hayden Jananthan (Vanderbilt University) Background on Turing Degrees and Jump Operators March 18, 2019 9 / 38 Turing Reducibility, Equivalence, and Degrees Definition (Turing Reducibility)

N Suppose f , g N . The Turing reducibility preorder T is defined by

f T g∈ f is algorithmically computable using≤ oracle g (1),g ≤ ⇐⇒ f ϕe for some e

f and g are Turing⇐⇒ equivalent= , f T g, if and only if f T g and g T f . T is an equivalence relation. ≡ ≤ ≤ Can≡ similarly define g-computability (g NN) for partial functions or predicates. ∈ Definition (Turing Degree) Suppose f NN. The Turing degree associated with f is

N ∈ degT f g N f T g

Hayden Jananthan (Vanderbilt University) Background( ) on∶= Turing{ ∈ Degrees andS Jump≡ Operators} March 18, 2019 10 / 38 Basic Results Proposition f m if n 2m Suppose f , g NN and define f g n . Then ⎧g m if n 2m 1 ⎪ ( ) = ∈ ( ⊕ )( ) ∶= ⎨ ⎪ sup degT f , degT g degT f ⎩⎪deg(T )g deg=T f + g Proof. ( ( ) ( )) = ( ) ∨ ( ) = ( ⊕ ) Straight-forward.

Proposition N N Suppose f N . Then there exists X 2 such that f T X .

Proof. ∈ ∈ ≡ Let X be the characteristic function for the graph of f (under some suitable recursive pairing function of N).

Hayden Jananthan (Vanderbilt University) Background on Turing Degrees and Jump Operators March 18, 2019 11 / 38 Existence of Non-Recursive Functions

Proposition There exists f NN which is non-recursive.

Proof. ∈ (1) The set of recursive 1-place functions is countable (e ϕe yields a surjection of N onto class of recursive functions), but NN is uncountable. ↦

Hayden Jananthan (Vanderbilt University) Background on Turing Degrees and Jump Operators March 18, 2019 12 / 38 Existence of Non-Recursive Functions – Halting Problem A particular example: Definition (Halting Problem)

′ (1) 0 e N ϕe e

Proposition ∶= { ∈ S ( ) ↓} 0′ is non-recursive.

Proof. 1 if e 0′ f e ′ ⎧undefined if e 0 ⎪ ∉ ′ ( ) ≃ ⎨ ′ If 0 is recursive, so is f . Let e⎪be an index for f . e 0 ? ⎩⎪ ∈ Case 1: e 0′. Then f e by definition, but f e by hypothesis. ∈ Case 2: e 0′. Then f e by definition, but f e by hypothesis. ∈ ( ) ↑ ( ) ↓ ∉ ( ) ↓ ( ) ↑ Hayden Jananthan (Vanderbilt University) Background on Turing Degrees and Jump Operators March 18, 2019 13 / 38 Turing Jump

Relativizing the halting problem to an arbitrary oracle results in the following definition: Definition (Turing Jump) Suppose f NN. The Turing jump of f is defined by

′ (1),f ∈ f e N ϕe e

Proposition ∶= { ∈ S ( ) ↓} ′ f T f

Proposition< ′ ′ If f T g, then f T g . Consequently, the Turing jump is well-defined on Turing degrees. ≤ ≤

Hayden Jananthan (Vanderbilt University) Background on Turing Degrees and Jump Operators March 18, 2019 15 / 38 Iterated Turing Jumps

Can define iterated Turing jumps:

f (0) f f (n+1) f (n) ′ ∶= To extend past the finite ordinals, we∶= set( )

∞ f (ω) f (n) n, e e f (n) n=0 ∶= ? = {⟨ ⟩S ∈ } In general, can define f (α) for any recursive ordinal. (Requires some care to ensure well-definedness up to Turing equivalence.)

Hayden Jananthan (Vanderbilt University) Background on Turing Degrees and Jump Operators March 18, 2019 16 / 38 Theorems about the Turing Jump

Theorem (Friedberg’s Jump Theorem/Jump Inversion Theorem) ′ ′ ′ If 0 T A, then there exists B such that A T B T B 0 .

′ Consequently,≤ every A T 0 is (Turing equivalent≡ ≡ to) a⊕ Turing jump.

Theorem (Posner-Robinson≥ Theorem) ′ Suppose 0 T Z T A and 0 T A. Then there exists B such that

′ < ≤ A≤ T B T B Z

Consequently, every non-recursive≡ Z is,≡ relative⊕ to some B, a Turing jump.

Hayden Jananthan (Vanderbilt University) Background on Turing Degrees and Jump Operators March 18, 2019 17 / 38 Relativized Arithmetical Hierarchy Suppose f NN.

Definition ∈ 0,f 0,f 0,f k Σ0 Π0 ∆0 R R N (k N) a f -recursive predicate Σ0,f S S m n R n, m for R Π0,f = = n+1 ∶= { S ⊆ ∈ n } Π0,f S S m n R n, m for R Σ0,f n+1 ∶= { S ( ⃗) ≡ ∃ ( ⃗) ∈ n } ∆0,f Σ0,f Π0,f n ∶= { n S ( ⃗n) ≡ ∀ ( ⃗) ∈ }

Proposition ∶= ∩ 0,f 0,f 1 Σn (Πn ) is closed under conjunction, disjunction, and bounded quantification. k 0,f c 0,f 2 S N is Σn if and only if S is Πn . 3 For each n, ⊆ 0,f 0,f 0,f 0,f 0,f Σn Πn ∆n+1 Σn+1 Πn+1

Hayden Jananthan (Vanderbilt University) Background∪ on Turing⊊ Degrees= and Jump Operators∩ March 18, 2019 18 / 38 Recursively-Enumerable relative to an Oracle

Definition N 0,f S N is recursively enumerable relative to f N if S is Σ1 .

Proposition⊆ ∈ Suppose f NN and S N. The following are equivalent: 0,f 1 S is Σ . ∈1 ⊆ 2 S is the domain of some partial function partial recursive relative to f .

3 S or S is the range of some g T f .

Proposition= ∅ ≤

N 0,f Suppose f N and S N. Then S T f if and only if S is ∆1 .

∈ ⊆ ≤

Hayden Jananthan (Vanderbilt University) Background on Turing Degrees and Jump Operators March 18, 2019 19 / 38 0,f Σ1 and f ′

An important interpretation of f ′ is as a uniform complete relativized r.e. set: Theorem ′ 0,f 0,f f is a complete Σ1 set, i.e. if S is Σ1 , there exists a total recursive function g such that g −1 f ′ S.

Iterated Turing jumps are[ closely] = related to higher levels of the relativized arithmetical hierarchy: Theorem (Post’s Theorem)

0,f (n) 0,f S is Σ1 if and only if S is Σn+1.

Hayden Jananthan (Vanderbilt University) Background on Turing Degrees and Jump Operators March 18, 2019 20 / 38 Arithmetical Hierarchy of Subsets of NN Nk Definition 0 0 0 N k × Σ0 Π0 ∆0 R R N N (k N) a recursive predicate Σ0 S S f , m n R n, f , m for R Π0 = = n+1 ∶= { S ⊆ × ∈ n } Π0 S S f , m n R n, f , m for R Σ0 n+1 ∶= { S ( ⃗) ≡ ∃ ( ⃗) ∈ n} ∆0 Σ0 Π0 n ∶= { n S (n ⃗) ≡ ∀ ( ⃗) ∈ } 0 S is arithmetical if∶=S Σ∩n for some n.

Proposition ∈ 0 0 1 Σn (Πn) is closed under conjunction, disjunction, and bounded quantification. N k 0 c 0 2 S N N is Σn if and only if S is Πn. 3 For each n, ⊆ × 0 0 0 0 0 Σn Πn ∆n+1 Σn+1 Πn+1

Hayden Jananthan (Vanderbilt University) Background∪ on⊊ Turing Degrees= and Jump∩ Operators March 18, 2019 21 / 38 Universal Predicates

Theorem 0 0 For each n and k, there exists a universal Σn (resp. Πn) predicate k+1 0 0 k U N , i.e. for each Σn (resp. Πn) predicate S N there exists e N such that ⊆ S m U e, m ⊆ ∈

Proof (Sketch). ( ⃗) ≡ ( ⃗) (k) For n 1, U e, m ϕe m .

Corollary= ( ⃗) ∶= ( ⃗) ↓ 0 0 For each n, k, there is an effective enumeration of the Σn (resp. Πn) k predicates S N .

k Analogous results⊆ for predicates NN N .

Hayden Jananthan (Vanderbilt University) Background on⊆ Turing Degrees× and Jump Operators March 18, 2019 22 / 38 0 Uniformization of Σ1 Sets

Theorem N 0 ˆ 0 Suppose S N N is Σ1. Then there exists S S which is Σ1 and for which if there exists n such that S f , n , then there exists a unique n such that Sˆ f , n⊆. × ⊆ ( ) Proof.( ) Suppose S f , n m R f , n, m where R is recursive. Define

Sˆ f , n (m R) f≡,∃m 0(, m 1 ) n m 0 k m R f , k 0, k 1

which( ) has≡ ∃ the( desired( ( ) properties.( ) ) ∧ ( = ( ) ) ∧ ¬∃( < ) ( ( ) ( ) )) (Here m m 0, m 1 is some recursive pairing function.)

↦ (( ) ( ) )

Hayden Jananthan (Vanderbilt University) Background on Turing Degrees and Jump Operators March 18, 2019 23 / 38 0 Π1 Sets and Recursive Trees Definition (Recursive Tree) ∗ A recursive tree is a recursive T N closed under initial segments. A path through T is an f NN such that f n T for all n. ⊆ Proposition ∈ ↾ ∈ N N 0 P 2 (resp. P N ) is Π1 if and only if P is the set of paths through a ∗ ∗ recursive subtree of 2 (resp. N ). ⊆ ⊆ Proof. N X ,(1) P X 2 ϕe e for some e N X ↾n,(1) = {X ∈ 2 S nϕe ( ) ↑}e X 2N X a path through T = { ∈ S ∀ ( ) ↑}

= { ∗∈ σ,S (1) } N ∗ where T σ 2 ϕe e is recursive. Analogous for N and N .

= { ∈ S ( ) ↑} Hayden Jananthan (Vanderbilt University) Background on Turing Degrees and Jump Operators March 18, 2019 24 / 38 0 Properties of Π1 Classes

N (1),X Let Pe X 2 ϕe e .

Proposition= { ∈ S ( ) ↑} N ′ 1 For every X 2 , X e N X Pe . 2 N k 0 k If Q 2 N is Π1, there is α N N primitive recursive such that N∈ = { ∈ S ∈ } for all f 2 and m1,..., mk N, ⊆ × ∶ →

∈ X Pα(m1,...,mk∈) Q X , m1,..., mk

Corollary ∈ ⇐⇒ ( ) 2 There is a primitive recursive function v N N such that

Pv(n,m) Pn∶ Pm→

= ∩

Hayden Jananthan (Vanderbilt University) Background on Turing Degrees and Jump Operators March 18, 2019 25 / 38 0 Basis Theorems for Π1 Classes 0 N Let P be a non-empty Π1 subset of 2 . Theorem (Kleene Basis Theorem) ′ There exists B P such that B T 0 .

This can be substantially∈ strengthened:≤ Theorem (Jump Inversion) Suppose P is special, i.e. contains no recursive elements. Then for every ′ A T 0 , there exists B P such that

′ ′ ≥ ∈ A T B T B 0

In a different direction: ≡ ≡ ⊕ Theorem (Kreisel Basis Theorem)

If 0 T Z, then there exists B P such that Z T B.

Hayden Jananthan< (Vanderbilt University) Background∈ on Turing Degrees and Jump≰ Operators March 18, 2019 26 / 38 Analytical Hierarchy

Definition 1 1 1 N k Σ0 Π0 ∆0 R R N N (k N) an arithmetical predicate Σ1 S S f , m g R f g, m for R Π1 = = n+1 ∶= { S ⊆ × ∈ n } Π1 S S f , m g R f g, m for R Σ1 n+1 ∶= { S ( ) ≡ ∃ ( ⊕ ) ∈ n} ∆1 Σ1 Π1 n ∶= { n S (n ) ≡ ∀ ( ⊕ ) ∈ } 1 S is analytical if∶=S Σ∩n for some n.

k Analogous definition∈ for subsets of N , as well as a relativized analytical hierarchy. 1 1 As with the arithmetical hierarchy, there are complete Σn and Πn sets for each n.

Hayden Jananthan (Vanderbilt University) Background on Turing Degrees and Jump Operators March 18, 2019 28 / 38 Hyperarithmetical Sets

Suppose X 2N.

Theorem (Kleene)∈ The following are equivalent: (α) 1 X T 0 for some recursive ordinal α.

2 1 X N is ∆1. ≤ 3 X is an element of any ω-model of ZFC. ⊆ Definition X 2N is hyperarithmetical if any of the equivalent conditions above N hold. If f N and f T X , then f is hyperarithmetical exactly when X is. ∈ ∈ HYP≡ f NN f is hyperarithmetical

∶= { ∈ S }

Hayden Jananthan (Vanderbilt University) Background on Turing Degrees and Jump Operators March 18, 2019 29 / 38 Hyperarithmetical Reducibility Relativizing the equivalent definitions of hyperarithmeticity yields the following: Theorem Suppose X , Y 2N. The following are equivalent: (α) 1 X T Y for some ordinal α recursive in Y . ∈ 1,Y 2 X N is ∆ . ≤ 1 3 X is an element of any ω-model of ZFC which contains Y . ⊆ Definition N Suppose X , Y 2 . The hyperarithmetical reducibility preorder HYP is defined by declaring X HYP Y if any of the equivalent conditions above hold. ∈ ≤ X and Y are hyperarithmetically≤ equivalent, X HYP Y , if and only if X HYP Y and Y HYP X . HYP is an equivalence relation. ≡

Hayden≤ Jananthan (Vanderbilt University)≤ Background≡ on Turing Degrees and Jump Operators March 18, 2019 30 / 38 Existence of Non-Hyperarithmetical Functions – Kleene’s

O A cardinality argument shows that existence of non-hyperarithmetical functions. As a particular example: Definition (Kleene’s ) is a fixed complete Π1 subset of . O1 N TheoremO is not hyperarithmetical.

Proof.O 1 1 Being a complete Π1 set implies that it is not ∆1.

Hayden Jananthan (Vanderbilt University) Background on Turing Degrees and Jump Operators March 18, 2019 31 / 38 1 0 Σ1 Sets as the Analogue for Π1 Sets

1 0 One might expect that Σ1 sets correspond to Σ1 sets, but that is not the case. 0 1 Σ1 sets instead correspond more closely to Π1 sets.

1 Theorem (Π1 Uniformization) N 1 ˆ 1 Suppose S N N is Π1. Then there exists S S which is Π1 and for which if there exists n such that S f , m, n , then there exists a unique n such that Sˆ⊆f , m×, n . ⊆ ( ⃗ ) 1 N N (Kondo’s Theorem( ⃗ ) shows that Π1 Uniformization holds for S N N (much harder).) 1 ⊆ × Σ1 Uniformization fails badly!

Hayden Jananthan (Vanderbilt University) Background on Turing Degrees and Jump Operators March 18, 2019 32 / 38 Hyperjumps

Our characterizations of the Turing jump motivates the existence of a similar operation with analogous use of the analytical hierarchy (with the 0,X 1,X twist that Σ1 corresponds to Π1 ). Definition 1,X X The hyperjump of X is a fixed complete Π1 set (uniform in X ).

Proposition O X Y If X HYP Y , then T . Consequently, the hyperjump is well-defined on hyper degrees. ≤ O ≤ O

Hayden Jananthan (Vanderbilt University) Background on Turing Degrees and Jump Operators March 18, 2019 33 / 38 The Analogy between Arithmetical and Hyperarithmetical Theory

Arithmetical Theory Hyperarithmetical Theory

Arithmetical Hierarchy Analytical Hierarchy

0 1 Recursive ∆1 Hyperarithmetical ∆1

Turing reducibility= Hyperarithmetical reducibility=

0 1 Π1 Σ1

Turing jump Hyperjump

First-Order Logic ω-logic

Hayden Jananthan (Vanderbilt University) Background on Turing Degrees and Jump Operators March 18, 2019 34 / 38 1 Properties of Σ1 Classes

The analogy holds well in terms of the relation between the hyperjump 1 1 and Σ1 sets, as well as the structural properties of Σ1 sets. Proposition

1 For every X 2N, X ∗ T e N X Pe ∈ 2 N k 1 k If K 2 N is Σ1, thereO ≡ is α{ N∈ S N∈primitive} recursive such that N for all X 2 and m1,..., mk N, ⊆ × ∶ → ∈ X P∗ ∈ K X , m ,..., m α(m1,...,mk ) 1 k ∈ ⇐⇒ ( )

Hayden Jananthan (Vanderbilt University) Background on Turing Degrees and Jump Operators March 18, 2019 35 / 38 1 Basis Theorems for Σ1 Classes

1 N Let K be a non-empty Σ1 subset of 2 . Theorem (Gandy Basis Theorem) B CK There exists B K such that B HYP . Consequently, ω1 ω1 .

Theorem (Hyperjump∈ Inversion)< O = Suppose K is special, i.e. contains no hyperarithmetical elements. Then for every A T , there exists B K such that

B ≥ O A T ∈ T B

Theorem (Kreisel Basis Theorem)≡ O ≡ ⊕ O

If Z is not hyperarithmetical, then there exists B K such that Z HYP B.

∈ ≰

Hayden Jananthan (Vanderbilt University) Background on Turing Degrees and Jump Operators March 18, 2019 36 / 38 Next time – Posner-Robinson for Turing Degrees of Hyperjumps

Theorem (Posner-Robinson for Turing Degrees of Hyperjumps)

Suppose 0 HYP Z T A and T A. Then there exists B such that

B < ≤ AOT≤ T B Z

Proof will involve Kumabe-Slaman≡ O Forcing≡ ⊕ and a stronger combined basis 1 theorem for special Σ1 sets.

Hayden Jananthan (Vanderbilt University) Background on Turing Degrees and Jump Operators March 18, 2019 37 / 38 Thank you!

Hayden Jananthan (Vanderbilt University) Background on Turing Degrees and Jump Operators March 18, 2019 38 / 38 Another Characterization of the Turing Jump

Suppose B 2N. Consider the language of arithmetic with an additional unary predicate X , i.e. 0, S, ,∈ , X where ar 0 0, ar S 1, ar ar 2, and ar X 1. Define{ + ⋅ } ( ) = ( ) = (+) = (⋅) = ( ) = PA X B PA X n n B X n n B

+ ( = ) ∶= ∪ { ( )S ∈ } ∪ {¬ ( )S ∉ } Theorem ′ B T G¨odelnumbers of theorems of PA X B

In particular, the≡ { halting problem is Turing equivalent+ to( the= problem)} of deciding whether a given sentence is a theorem of PA.

Hayden Jananthan (Vanderbilt University) Background on Turing Degrees and Jump Operators March 18, 2019 39 / 38 Hyperjumps - Alternative Characterization

Let Z2 be the second-order theory of arithmetic. A characterization of the hyperjump which is analogous to that of the Turing jump is in terms of theorems of Z2 X B . Say that ϕ is an ω-theorem of Φ if it true in every ω-model of Φ. + ( = ) Theorem

B T G¨odelnumbers of ω-theorems of Z2 X B

Another characterizationO ≡ { of B makes precise the notion+ ( of= “notations)} for ordinals recursive in B”. O

Hayden Jananthan (Vanderbilt University) Background on Turing Degrees and Jump Operators March 18, 2019 40 / 38