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) 'e 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 'e 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 'e m1;:::; mk and (k);f Ψ f ; e; m1;:::; mk 'e 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 'f (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 'e 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 'e 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 'e 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 'e 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 (!) 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.