XIV. Arithmetic Hierarchy Yuxi Fu BASICS, Shanghai Jiao Tong - PowerPoint PPT Presentation

XIV. Arithmetic Hierarchy Yuxi Fu BASICS, Shanghai Jiao Tong University

We introduce a hierarchy of sets in terms of logical formula and prove its relationship to the hierarchy 0 , 0 ′ , 0 ′′ , . . . of Turing degree. Computability Theory, by Y. Fu XIV. Arithmetic Hierarchy 1 / 32

Synopsis 1. Arithmetic Hierarchy 2. Post Theorem 3. Σ n -Complete Set 4. Relative Arithmetic Hierarchy Computability Theory, by Y. Fu XIV. Arithmetic Hierarchy 2 / 32

Arithmetic Hierarchy Computability Theory, by Y. Fu XIV. Arithmetic Hierarchy 3 / 32

Arithmetic Hierarchy A set B is in Σ 0 (Π 0 ) if B is recursive. A set B is in Σ n , where n ≥ 1, if there is a recursive relation R ( x , y 1 , y 2 , . . . , y n ) such that x ∈ B iff ∃ y 1 . ∀ y 2 . ∃ y 3 . . . . Q n y n . R ( x , y 1 , y 2 , . . . , y n ) . A set B is in Π n , where n ≥ 1, if there is a recursive relation R ( x , y 1 , y 2 , . . . , y n ) such that x ∈ B iff ∀ y 1 . ∃ y 2 . ∀ y 3 . . . . Q n y n . R ( x , y 1 , y 2 , . . . , y n ) . ∆ n = Σ n ∩ Π n . Computability Theory, by Y. Fu XIV. Arithmetic Hierarchy 4 / 32

Arithmetic Set B is arithmetical if B ∈ � n ∈ ω (Σ n ∪ Π n ). Computability Theory, by Y. Fu XIV. Arithmetic Hierarchy 5 / 32

Basic Property Theorem . The following hold: (i) A ∈ Σ n iff A ∈ Π n . (ii) If A ∈ Σ n (Π n ) then ∀ m > n . A ∈ Σ m ∩ Π m . (iii) If A , B ∈ Σ n (Π n ) then A ∪ B , A ∩ B ∈ Σ n (Π n ). (iv) If R ∈ Σ n ∧ n > 0 ∧ A = { x : ( ∃ y ) R ( x , y ) } then A ∈ Σ n . (v) If B ≤ m A ∧ A ∈ Σ n then B ∈ Σ n . (vi) If R ∈ Σ n (Π n ) and A , B are defined by � x , y � ∈ A ⇔ ∀ z < y . R ( x , y , z ) , � x , y � ∈ B ⇔ ∃ z < y . R ( x , y , z ) , then A , B ∈ Σ n (Π n ). Computability Theory, by Y. Fu XIV. Arithmetic Hierarchy 6 / 32

Fin ∈ Σ 2 Fact . Fin ∈ Σ 2 . x ∈ Fin ⇔ W x is finite ⇔ ∃ s . ∀ t . ( t ≤ s ∨ W x , t = W x , s ) . Fact . Inf ∈ Π 2 . Computability Theory, by Y. Fu XIV. Arithmetic Hierarchy 7 / 32

Cof ∈ Σ 3 Fact . Cof ∈ Σ 3 . x ∈ Cof ⇔ W x is finite ⇔ ∃ y . ∀ z . ( z ≤ y ∨ z ∈ W x ) ⇔ ∃ y . ∀ z . ∃ s . ( z ≤ y ∨ z ∈ W x , s ) . Computability Theory, by Y. Fu XIV. Arithmetic Hierarchy 8 / 32

Tot ∈ Π 2 Fact . {� x , y � | W x ⊆ W y } ∈ Π 2 . W x ⊆ W y ⇔ ∀ z . ( z ∈ W x ⇒ z ∈ W y ) ⇔ ∀ z . ( z / ∈ W x ∨ z ∈ W y ) ⇔ ∀ z . ( ∀ s . z / ∈ W x , s ∨ ∃ t . z ∈ W y , t ) ⇔ ∀ z . ∀ s . ∃ t . ( z / ∈ W x , s ∨ z ∈ W y , t ) � � ⇔ ∀ w . ∃ t . ( w ) 0 / ∈ W x , ( w ) 1 ∨ ( w ) 0 ∈ W y , t . Fact . {� x , y � | W x = W y } ∈ Π 2 . Fact . Tot = { x | W x = ω } ∈ Π 2 . Computability Theory, by Y. Fu XIV. Arithmetic Hierarchy 9 / 32

Rec ∈ Σ 3 Fact . Rec ∈ Σ 3 . x ∈ Rec ⇔ W x is recursive ⇔ ∃ y . � � W x = W y ⇔ ∃ y . ( W x ∩ W y = ∅ ∧ W x ∪ W y = ω ) ⇔ ∃ y . (( ∀ s . W x , s ∩ W y , s = ∅ ) ∧ ( ∀ z . ∃ s . z ∈ W x , s ∪ W y , s )) . Computability Theory, by Y. Fu XIV. Arithmetic Hierarchy 10 / 32

Ext ∈ Σ 3 Fact . Ext ∈ Σ 3 . x ∈ Ext ⇔ ∃ y . ( φ x ⊆ φ y ∧ W y = ω ) ⇔ ∃ y . ∀ z . ∃ s . ∃ t . (( z / ∈ W x , s ∨ φ x , s ( z ) = φ y , s ( z )) ∧ z ∈ W y , t ) . Computability Theory, by Y. Fu XIV. Arithmetic Hierarchy 11 / 32

Crt ∈ Σ 3 Fact . Crt = { x | W x is creative } ∈ Σ 3 . x ∈ Crt ⇔ W x is productive � � ⇔ ∃ y . ∀ z . W z ⊆ W x ⇒ ( φ y ( z ) ↓ ∧ φ y ( z ) ∈ W x \ W z ) ⇔ ∃ y . ∀ z . ( W z ∩ W x = ∅ ⇒ ( φ y ( z ) ↓ ∧ φ y ( z ) / ∈ W x ∪ W z )) ⇔ ∃ y . ∀ z . ( W z ∩ W x � = ∅ ∨ ( φ y ( z ) ↓ ∧ φ y ( z ) / ∈ W x ∪ W z )) Now W z ∩ W x � = ∅ iff ∃ s . W z , s ∩ W x , s � = ∅ , and φ y ( z ) ↓ ∧ φ y ( z ) / ∈ W x ∪ W z iff ∃ s . z ∈ W y , s ∧ ∀ s . ( z / ∈ W y , s ∨ φ y , s ( z ) / ∈ W x , s ∪ W z , s ) . Computability Theory, by Y. Fu XIV. Arithmetic Hierarchy 12 / 32

Let P TM be { x | P x runs in polynomial time } . ∃ c . ∀ z . ( P x ( z ) terminates in cz c ) x ∈ P TM ⇔ Hence P TM ∈ Σ 2 . Computability Theory, by Y. Fu XIV. Arithmetic Hierarchy 13 / 32

Post Theorem Computability Theory, by Y. Fu XIV. Arithmetic Hierarchy 14 / 32

Completeness A set A ∈ Σ n is Σ n -complete if B ≤ 1 A for every B ∈ Σ n . A set A ∈ Π n is Π n -complete if B ≤ 1 A for every B ∈ Π n . Computability Theory, by Y. Fu XIV. Arithmetic Hierarchy 15 / 32

Post Theorem (i) B ∈ Σ n +1 iff B is r.e. in a Π n set iff B is r.e. in a Σ n set. Proof. If B ∈ Σ n +1 , then x ∈ B iff ∃ y . R ( x , y ) for some R ∈ Π n . So B is r.e. in {� x , y � | R } ∈ Π n . Suppose B is r.e. in some C ∈ Π n . Then for some e , x ∈ B iff x ∈ W C e iff ∃ s . ∃ σ. ( σ ⊂ C ∧ x ∈ W σ e , s ) . Now x ∈ W σ e , s is recursive, and σ ⊂ C is C -recursive since σ ⊂ C ∀ y < | σ | . ( σ ( y ) = 1 ∧ y ∈ C ∨ σ ( y ) = 0 ∧ y / ∈ C ) . iff Hence B ∈ Σ n +1 . Computability Theory, by Y. Fu XIV. Arithmetic Hierarchy 16 / 32

Post Theorem (ii) ∅ ( n ) is Σ n -complete for all n > 0. Proof. ∅ ′ = K is Σ 1 -complete. Now assume ∅ ( n ) is Σ n -complete. Then B ∈ Σ n +1 iff B is r . e . in some Σ n set B is r . e . in ∅ ( n ) iff B ≤ 1 ∅ ( n +1) . iff Hence ∅ ( n +1) is Σ n +1 -complete. Computability Theory, by Y. Fu XIV. Arithmetic Hierarchy 17 / 32

Post Theorem (iii) B ∈ Σ n +1 iff B is r.e. in ∅ ( n ) . (iv) B ∈ ∆ n +1 iff B ≤ T ∅ ( n ) . Proof. We have the following equivalence: B ∈ ∆ n +1 B , B ∈ Σ n +1 iff B , B are r . e . in ∅ ( n ) iff B ≤ T ∅ ( n ) . iff Computability Theory, by Y. Fu XIV. Arithmetic Hierarchy 18 / 32

Hierarchy Theorem . ∀ n > 0 . ∆ n ⊂ Σ n ∧ ∆ n ⊂ Π n . Proof. ∅ ( n ) ∈ Σ n \ Π n and ∅ ( n ) ∈ Π n \ Σ n . Computability Theory, by Y. Fu XIV. Arithmetic Hierarchy 19 / 32

A Comment on Completeness B ≤ m ∅ ( n ) ⇒ B ∈ Σ n B is r . e . in ∅ ( n − 1) ⇒ B ≤ 1 ∅ ( n ) ⇒ B ≤ m ∅ ( n ) . ⇒ The following is the relativized version of “ K ≤ m A iff K ≤ 1 A ”: ∅ ( n ) ≤ m A iff ∅ ( n ) ≤ 1 A . Computability Theory, by Y. Fu XIV. Arithmetic Hierarchy 20 / 32

Σ n -Complete Set Computability Theory, by Y. Fu XIV. Arithmetic Hierarchy 21 / 32

Let ( A 1 , A 2 ) and ( B 1 , B 2 ) be two pairs of sets such that A 1 ∩ A 2 = ∅ and B 1 ∩ B 2 = ∅ . Then ( A 1 , A 2 ) ≤ m ( B 1 , B 2 ) if there is a recursive function f such that f ( A 1 ) ⊆ B 1 , f ( A 2 ) ⊆ B 2 and f ( A 1 ∪ A 2 ) ⊆ B 1 ∪ B 2 . We write ( A 1 , A 2 ) ≤ 1 ( B 1 , B 2 ) if f is one-one. For n > 0 we write (Σ n , Π n ) ≤ m ( C , D ) if ( A , A ) ≤ m ( C , D ) for some Σ n -complete set A . The notation (Σ n , Π n ) ≤ 1 ( C , D ) is defined similarly. Computability Theory, by Y. Fu XIV. Arithmetic Hierarchy 22 / 32

Fin is Σ 2 -Complete, Tot is Π 2 -Complete Theorem . (Σ 2 , Π 2 ) ≤ 1 ( Fin , Tot ). Proof. Fin ∈ Σ 2 and Tot ∈ Π 2 . Let A be in Σ 2 . There is a recursive relation R such that x ∈ A iff ∀ y . ∃ z . R ( x , y , z ) . By S-m-n Theorem there is a one-one recursive function s s.t. � 0 , if ∀ y ≤ u . ∃ z . R ( x , y , z ) , φ s ( x ) ( u ) = ↑ , otherwise . Now x ∈ A ⇒ W s ( x ) = ω ⇒ s ( x ) ∈ Tot and x ∈ A ⇒ W s ( x ) is finite ⇒ s ( x ) ∈ Fin . Computability Theory, by Y. Fu XIV. Arithmetic Hierarchy 23 / 32

Cof and Rec are Σ 3 -Complete Let Cmp be { x | W x ≡ T K } , the set of Turing complete r.e. sets. Theorem . (Σ 3 , Π 3 ) ≤ 1 ( Cof , Cmp ) ≤ 1 ( Rec , Cmp ). Corollary . Cof is Σ 3 -complete. Corollary . (Rogers) Rec is Σ 3 -complete. Computability Theory, by Y. Fu XIV. Arithmetic Hierarchy 24 / 32

(Σ 3 , Π 3 ) ≤ 1 ( Cof , Cmp ) Fix an A ∈ Σ 3 . Then some R ∈ Π 2 exists such that x ∈ A iff ∃ y . R ( x , y ) . Since Inf is Π 2 -complete, a one-one recursive function g exists s.t. R ( x , y ) iff W g ( x , y ) is infinite . s ∈ ω W s We will construct an r.e. set W f ( x ) = � f ( x ) in stages s.t. x ∈ A iff W f ( x ) is cofinite . Computability Theory, by Y. Fu XIV. Arithmetic Hierarchy 25 / 32

(Σ 3 , Π 3 ) ≤ 1 ( Cof , Cmp ) Let the elements of the cofinite set W s f ( x ) be denoted by b s x , 0 < b s x , 1 < b s x , 2 < . . . < b s x , k < . . . . Let W 0 f ( x ) := ∅ . Let W s +1 x , k in W s +1 f ( x ) := W s f ( x ) . Additionally put b s f ( x ) if k ≤ s and W g ( x , k ) , s � = W g ( x , k ) , s +1 ∨ k ∈ K s +1 \ K s . So we have constructed some programme P f ( x ) that enumerates W f ( x ) , from which we can calculate f ( x ). Computability Theory, by Y. Fu XIV. Arithmetic Hierarchy 26 / 32