Kenneth Harris Characterizing low n Degrees 0 A C - PDF document

✬ ✩ Kenneth Harris Characterizing low n Degrees 0 A C    low n D  K  H  Department of Computer Science University of Chicago http://people.cs.uchicago.edu/ ✒ kaharris kaharris@uchicago.edu ✫ ✪ kaharris@uchicago.edu Presented @ SEALS, Univ. of Florida 03 / 06

✬ ✩ Kenneth Harris Characterizing low n Degrees 1 Dominate and Escape Let f , g : N Ñ N . • f dominates g if ♣❅ ✽ x q ✏ ✘ f ♣ x q → g ♣ x q f is a dominant function if f dominates every computable function. • g escapes (domination from) f if ♣❉ ✽ x q ✏ ✘ f ♣ x q ↕ g ♣ x q f has an escape function if there is a computable g which escapes domination by f . ✫ ✪ kaharris@uchicago.edu Presented @ SEALS, Univ. of Florida 03 / 06

✬ ✩ Kenneth Harris Characterizing low n Degrees 2 Martin’s Characterization Theorem ( Martin , 1966) Let a be a Turing degree. a ✶ ➙ 0 ✷ ✟ � • a is high i ff there is an a -computable dominant function: ✏ ✘ ♣❉ f ↕ a q♣❅ g ↕ 0 q f dominates g a ✶ ➔ 0 ✷ ✟ � • a is non-high i ff every a -computable function has an escape function: ✏ ✘ ♣❅ f ↕ a q♣❉ g ↕ 0 q g escapes f ✫ ✪ kaharris@uchicago.edu Presented @ SEALS, Univ. of Florida 03 / 06

✬ ✩ Kenneth Harris Characterizing low n Degrees 3 Uniform Escape Property Question : For what non-high degrees can escape functions be e ff ectively produced? Definition : A degree a has the Uniform Escape Property (UEP), or (1-UEP), when for any set A P a : There is a partial computable λ ex . h e ♣ x q such that whenever Φ A e is total, then h e total and escapes Φ A e Recall, h e escapes Φ A e if ♣❉ ✽ x q ✏ Φ A ✘ e ♣ x q ↕ h e ♣ x q ✫ ✪ kaharris@uchicago.edu Presented @ SEALS, Univ. of Florida 03 / 06

✬ ✩ Kenneth Harris Characterizing low n Degrees 4 UEP Equivalent to low 1 Theorem : For all degrees a TFAE a ✶ ↕ 0 ✶ ✟ � (A) a is low 1 . (B) a has the Uniform Escape Property. ✫ ✪ kaharris@uchicago.edu Presented @ SEALS, Univ. of Florida 03 / 06

✬ ✩ Kenneth Harris Characterizing low n Degrees 5 low n Degrees and Escape Functions There is a hierarchy of properties characterized by progressively less e ff ective procedures, n - Uniform Escape Property ( n -UEP), starting with (1-UEP) = (UEP), such that Theorem : For all degrees a and all n ➙ 1 TFAE a ♣ n q ↕ 0 ♣ n q ✟ � (A) a is low n . (B) a has ( n -UEP). ✫ ✪ kaharris@uchicago.edu Presented @ SEALS, Univ. of Florida 03 / 06

✬ ✩ Kenneth Harris Characterizing low n Degrees 6 Quantifiers on Steroids ♣❅ ✽ x q : For almost every x . Reduces to ❉❅ and behaves like ❅ . ♣❉ ✽ x q : There exists infinitely many x . Reduces to ❅❉ and behaves like ❉ . Fundamental Relations ❉ ✽ P ðñ ✥❅ ✽ ✥ P ❅ ùñ ❅ ✽ ùñ ❉ ✽ ùñ ❉ Theorem ( Strong Normal Form ): The arithmetic hierarchy is characterized by alternations of the two strongest quantifiers, ❅ and ❅ ✽ . ✫ ✪ kaharris@uchicago.edu Presented @ SEALS, Univ. of Florida 03 / 06

✬ ✩ Kenneth Harris Characterizing low n Degrees 7 low 1 D    1-U  E  P  ✫ ✪ kaharris@uchicago.edu Presented @ SEALS, Univ. of Florida 03 / 06

✬ ✩ Kenneth Harris Characterizing low n Degrees 8 low 1 implies Uniform Escape Property Theorem : All low 1 sets A have (1-UEP): There is a partial computable function λ ex . h e ♣ x q such that whenever Φ A e is total, then h e total and escapes Φ A e ✫ ✪ kaharris@uchicago.edu Presented @ SEALS, Univ. of Florida 03 / 06

✬ ✩ Kenneth Harris Characterizing low n Degrees 9 The Key Idea Let A be low 1 , so Π A 2 ⑨ Π 2 . Want : Computable g such that for each total Φ A e , W g ♣ e q satisfies ♣❉ ✽ x q♣❉ s q Φ A ✏ ✘ ♣ escape q e , s ♣ x qÓ↕ s & x � W g ♣ e q , s ♣ total q W g ♣ e q ✏ ω Define : ✏ ✘ h e ♣ x q ✏ ♣ µ s q x P W g ♣ e q , s Problem : How to match ( escape ) with ( total )? ✫ ✪ kaharris@uchicago.edu Presented @ SEALS, Univ. of Florida 03 / 06

✬ ✩ Kenneth Harris Characterizing low n Degrees 10 Strong Normal Form: Π 2 , Σ 2 Normal Form : For V P Π 2 , there is some v (the Π 2 index for V ) with ✏ ✘ V ♣ e q ðñ ♣❅ y q ① e , y ② P W v Strong Normal Form ( SNF ): There is a computable g , such that for any V P Π 2 with index e ✏ ✘ V ♣ e q ùñ W g ♣ v , e q ✏ ω ✏ ✘ ✥ V ♣ e q ùñ W g ♣ v , e q finite ✫ ✪ kaharris@uchicago.edu Presented @ SEALS, Univ. of Florida 03 / 06

✬ ✩ Kenneth Harris Characterizing low n Degrees 11 Implementation of Key Idea Let A be low 1 (thus Π A 2 ⑨ Π 2 ). Define Π A 2 predicate ( escape ) ♣❉ ✽ x q♣❉ s q Φ A ✏ ✘ e , s ♣ x qÓ↕ s & x � W g ♣ v , e q , s where g is the computable function given by ( SNF ) from a Π 2 index v for ♣ escape q : ✏ ✘ ♣ escape q ùñ W g ♣ v , e q ✏ ω ✏ ✘ ✥♣ escape q ùñ W g ♣ v , e q finite then define ✏ ✘ h e ♣ x q ✏ ♣ µ s q x P W g ♣ v , e q , s ✫ ✪ kaharris@uchicago.edu Presented @ SEALS, Univ. of Florida 03 / 06

✬ ✩ Kenneth Harris Characterizing low n Degrees 12 low 2 D    2-U  E  P  ✫ ✪ kaharris@uchicago.edu Presented @ SEALS, Univ. of Florida 03 / 06

✬ ✩ Kenneth Harris Characterizing low n Degrees 13 2-Uniform Escape Property: First Change Definition : A set A is low 2 if A ✷ ↕ 0 ✷ . With low 2 we add one jump class and one layer of quantifier complexity. Our first change in defining (2-UEP): There are uniformly enumerable (u.e. ) families of ✥ ✭ partial computable functions λ e . h e , y y P ω such that whenever Φ A e is total, then ✒ ✚ ♣❅ ✽ y q h e , y total and escapes Φ A e ✫ ✪ kaharris@uchicago.edu Presented @ SEALS, Univ. of Florida 03 / 06

✬ ✩ Kenneth Harris Characterizing low n Degrees 14 2-Uniform Escape Property Definition : A degree a has the 2-Uniform Escape Property (2-UEP), when for any set A P a : There are uniformly enumerable (u.e. ) families of ✥ ✭ partial computable functions λ e . h e , y y P ω such ✥ Φ A ✭ that for any u.e. family of functions e , y y P ω satisfying ♣❅ ✽ y q Φ A ✏ ✘ e , y total then ✒ ✚ ♣❅ ✽ y q h e , y total and escapes Φ A e , y ✫ ✪ kaharris@uchicago.edu Presented @ SEALS, Univ. of Florida 03 / 06

✬ ✩ Kenneth Harris Characterizing low n Degrees 15 low 2 Equivalent to 2-UEP For all degrees a TFAE a ✷ ↕ 0 ✷ ✟ � (A) a is low 2 . (B) a has the 2-Uniform Escape Property. ✫ ✪ kaharris@uchicago.edu Presented @ SEALS, Univ. of Florida 03 / 06

✬ ✩ Kenneth Harris Characterizing low n Degrees 16 Strong Normal Form: Σ 3 , Π 3 If A is low 2 then Σ A 3 ❸ Σ 3 . Strategy of Proof : Pump ( escape ) property ( Π A 2 ) with strong quantifiers to Σ A 3 and exploit weakness of A . ♣❅ ✽ y q♣❉ ✽ x q♣❉ s q Φ A ✏ ✘ e , y , s ♣ x qÓ↕ s & x � W g ♣ v , e , y q , s Strong Normal Form ( SNF ): There is a computable g , such that for any V P Σ 3 with index e ♣❅ ✽ y q ✏ ✘ ♣ escape q ùñ W g ♣ v , e , y q ✏ ω ✏ ✘ ✥♣ escape q ùñ ♣❅ y q W g ♣ v , e , y q finite ✫ ✪ kaharris@uchicago.edu Presented @ SEALS, Univ. of Florida 03 / 06

✬ ✩ Kenneth Harris Characterizing low n Degrees 17 low 3 D   B  ✫ ✪ kaharris@uchicago.edu Presented @ SEALS, Univ. of Florida 03 / 06

✬ ✩ Kenneth Harris Characterizing low n Degrees 18 3-Uniform Escape Property A degree a is low 3 if a ✸ ✏ 0 ✸ . Definition : A degree a has the 3-Uniform Escape Property (3-UEP) when for any set A P a : There are uniformly enumerable (u.e. ) families of ✥ ✭ partial computable functions λ e . h e , y 1 , y 2 y 1 , y 2 P ω such that for any u.e. family of functions ✥ ✭ Φ A y 1 , y 2 P ω satisfying e , y 1 , y 2 ♣❉ ✽ y 2 q♣❅ ✽ y 1 q Φ A ✏ ✘ e , y 1 , y 2 total then ✒ ✚ ♣❉ ✽ y 2 q♣❅ ✽ y 1 q h e , y 1 , y 2 total and escapes Φ A e , y 1 , y 2 ✫ ✪ kaharris@uchicago.edu Presented @ SEALS, Univ. of Florida 03 / 06

✬ ✩ Kenneth Harris Characterizing low n Degrees 19 low 3 Equivalent to 3-UEP Theorem : For all degrees a TFAE a ✸ ↕ 0 ✸ ✟ � (A) a is low 3 . (B) a has the 3-Uniform Escape Property. ✫ ✪ kaharris@uchicago.edu Presented @ SEALS, Univ. of Florida 03 / 06

✬ ✩ Kenneth Harris Characterizing low n Degrees 20 Strong Normal Form: Π 4 , Σ 4 If A is low 3 then Π A 4 ❸ Π 4 . Strategy of Proof : Pump ( escape ) property ( Π A 2 ) with strong quantifiers to Π A 4 and exploit weakness of A . ♣❉ ✽ y 2 q♣❅ ✽ y 1 q♣❉ ✽ x q♣❉ s q Φ A ✏ e , y 1 , y 2 , s ♣ x qÓ↕ s ✘ & x � W g ♣ v , e , y 1 , y 2 q , s Strong Normal Form ( SNF ): There is a computable g , such that for any V P Π 4 with index e ♣❅ y 2 q♣❅ ✽ y 1 q ✏ ✘ ♣ escape q ùñ W g ♣ v , e , y 1 , y 2 q ✏ ω ♣❅ ✽ y 2 q♣❅ y 1 q ✏ ✘ ✥♣ escape q ùñ W g ♣ v , e , y 1 , y 2 q finite ✫ ✪ kaharris@uchicago.edu Presented @ SEALS, Univ. of Florida 03 / 06