BSS RAMs with \ nu-Oracle and the Axiom of Choice Christine Ganer - PowerPoint PPT Presentation

BSS RAM’s with \ nu-Oracle and the Axiom of Choice Christine Gaßner Hamburg 2016

BSS RAM’s with \ nu-Oracle and the Axiom of Choice (History and Outline) ⇓ Stephen C. Kleene Recursion Theory based on recursion and µ -operator ⇓ Yiannis N. Moschovakis Generalized Recursion Theory based on recursion and ν -operator ⇓ Gaßner ν -Operators for BSS RAM’s over arbitrary mathematical structures

BSS RAM’s with \ nu-Oracle and the Axiom of Choice (History and Outline) ⇓ Stephen C. Kleene Recursion Theory based on recursion and µ -operator ⇓ Yiannis N. Moschovakis Generalized Recursion Theory based on recursion and ν -operator ⇓ Gaßner ν -Operators for BSS RAM’s over arbitrary mathematical structures ⇓ Computable multi-valued correspondences ⇓ Question: Are there computable choice functions for these correspondences?

BSS RAM’s with \ nu-Oracle and the Axiom of Choice (History and Outline) ⇓ Stephen C. Kleene Recursion Theory based on recursion and µ -operator ⇓ Yiannis N. Moschovakis Generalized Recursion Theory based on recursion and ν -operator ⇓ Gaßner ν -Operators for BSS RAM’s over arbitrary mathematical structures ⇓ Computable multi-valued correspondences ⇓ Question: Are there computable choice functions for these correspondences? ⇓ Outline: BSS RAM’s A characterization of [non-]deterministic semi-decidability AC n , m (in HPL) and effective AC n , m and AC ∞

Computation by BSS RAM’s over Algebraic Structures (The Machines and the Allowed Instructions) Computation over A = ( U A ; C A ; f 1 , . . . , f n 1 ; R 1 , . . . , R n 2 , = ) . ���� ���� � �� � � �� � constants universe operations relations Z 1 Z 2 Z 3 Z 4 Z 5 . . . Registers for elements in U A . . . I 1 I 2 I 3 I 4 I k M Registers for indices in N

Computation by BSS RAM’s over Algebraic Structures (The Machines and the Allowed Instructions) Computation over A = ( U A ; C A ; f 1 , . . . , f n 1 ; R 1 , . . . , R n 2 , = ) . ���� ���� � �� � � �� � constants universe operations relations Z 1 Z 2 Z 3 Z 4 Z 5 . . . Registers for elements in U A . . . I 1 I 2 I 3 I 4 I k M Registers for indices in N Computation instructions: ℓ : Z j := f k ( Z j 1 , . . . , Z j mk ) ( e.g. ℓ : Z j := Z j 1 + Z j 2 ) ℓ : Z j := d k ( d k ∈ C A ⊆ U A ) Branching instructions: ℓ : if Z i = Z j then goto ℓ 1 else goto ℓ 2 ℓ : if R k ( Z j 1 , . . . , Z j nk ) then goto ℓ 1 else goto ℓ 2 Copy instructions: ℓ : Z I j := Z I k Index instructions: ℓ : I j := 1 ℓ : I j := I j + 1 ℓ : if I j = I k then goto ℓ 1 else goto ℓ 2

Uniform Computation over Algebraic Structures (Input and Output Procedures of Machines in M A ) U A is the universe of A � Input and output space: U ∞ i ≥ 1 U i A = df A x = ( x 1 , . . . , x n ) ∈ U ∞ Input of � A : x 1 x 2 x 3 x 4 x n

Uniform Computation over Algebraic Structures (Input and Output Procedures of Machines in M A ) U A is the universe of A � Input and output space: U ∞ i ≥ 1 U i A = df A x = ( x 1 , . . . , x n ) ∈ U ∞ Input of � A : x 1 x 2 x 3 x 4 x n x n x n ↓ ↓ ↓ ↓ ↓ ↓ ↓ Z 1 Z 2 Z 3 Z 4 . . . Z n Z n + 1 Z n + 2 . . .

Uniform Computation over Algebraic Structures (Input and Output Procedures of Machines in M A ) U A is the universe of A � Input and output space: U ∞ i ≥ 1 U i A = df A x = ( x 1 , . . . , x n ) ∈ U ∞ Input of � A : x 1 x 2 x 3 x 4 x n x n x n ↓ ↓ ↓ ↓ ↓ ↓ ↓ Z 1 Z 2 Z 3 Z 4 . . . Z n Z n + 1 Z n + 2 . . . I 1 I 2 I 3 I 4 . . . I k M ↑ ↑ ↑ ↑ ↑ n 1 1 1 1

Uniform Computation over Algebraic Structures (Input and Output Procedures of Machines in M A ) U A is the universe of A � Input and output space: U ∞ i ≥ 1 U i A = df A x = ( x 1 , . . . , x n ) ∈ U ∞ Input of � A : x 1 x 2 x 3 x 4 x n x n x n ↓ ↓ ↓ ↓ ↓ ↓ ↓ Z 1 Z 2 Z 3 Z 4 . . . Z n Z n + 1 Z n + 2 . . . I 1 I 2 I 3 I 4 . . . I k M ↑ ↑ ↑ ↑ ↑ n 1 1 1 1 ր !!!

Uniform Computation over Algebraic Structures (Input and Output Procedures of Machines in M A ) U A is the universe of A � Input and output space: U ∞ i ≥ 1 U i A = df A x = ( x 1 , . . . , x n ) ∈ U ∞ Input of � A : x 1 x 2 x 3 x 4 x n x n x n ↓ ↓ ↓ ↓ ↓ ↓ ↓ Z 1 Z 2 Z 3 Z 4 . . . Z n Z n + 1 Z n + 2 . . . I 1 I 2 I 3 I 4 . . . I k M ↑ ↑ ↑ ↑ ↑ n 1 1 1 1 ր !!! Output of Z 1 , . . . , Z I 1 .

[ ν -]Semi-Decidability (The Definitions) P ⊆ U ∞ A is a decision problem . P ⊆ U ∞ A is semi-decidable x ∈ P ⇔ M ( � x ) halts on � . if there is a BSS RAM M such that � x � �� � M ( � x ) ↓ We will also use: P ⊆ U ∞ A is nondeterministically semi-decidable x ∈ P if there is a nondeterministic BSS RAM M such that � ⇔ M halts on � x for some guesses . � �� � M ( � x ) ↓ P ⊆ U ∞ A is ν -semi-decidable if there is a ν -oracle BSS RAM semi-deciding P . . . . ν -oracle BSS RAM = BSS RAM being able to use operator ν . . .

µ -Oracle BSS RAM’s with µ -Operators for N ⊆ U A (Kleene’s Operator µ ) A fixed, N ⊆ U A effectively enumerable over A , f : U ∞ A → { a , b } partial function, computable over A . � �� � { 1 , 0 } Definition (Kleene’s operator for A ) µ [ f ]( x 1 , . . . , x n ) = df min { k ∈ N | f ( x 1 , . . . , x n , k ) = 1 & f ( x 1 , . . . , x n , l ) ↓ for l < k , l ∈ N }

µ -Oracle BSS RAM’s with µ -Operators for N ⊆ U A (Kleene’s Operator µ ) A fixed, N ⊆ U A effectively enumerable over A , f : U ∞ A → { a , b } partial function, computable over A . � �� � { 1 , 0 } Definition (Kleene’s operator for A ) µ [ f ]( x 1 , . . . , x n ) = df min { k ∈ N | f ( x 1 , . . . , x n , k ) = 1 & f ( x 1 , . . . , x n , l ) ↓ for l < k , l ∈ N } Example A = ( N ; 0 ; + , − ; ≤ , =)  if x n + a n x n − 1 + · · · + a 1 x 0 = 0 , 1   � �� � f 0 ( a 1 , . . . , a n , x ) := p ( x )   0 otherwise . ⇒ µ [ f 0 ]( a 1 , . . . , a n ) = the smallest zero of p

µ -Oracle BSS RAM’s with µ -Operators for N ⊆ U A (Kleene’s Operator µ ) A fixed, N ⊆ U A effectively enumerable over A , f : U ∞ A → { a , b } partial function, computable over A . � �� � { 1 , 0 } Definition (Kleene’s operator for A ) µ [ f ]( x 1 , . . . , x n ) = df min { k ∈ N | f ( x 1 , . . . , x n , k ) = 1 & f ( x 1 , . . . , x n , l ) ↓ for l < k , l ∈ N }

µ -Oracle BSS RAM’s with µ -Operators for N ⊆ U A (Kleene’s Operator µ ) A fixed, N ⊆ U A effectively enumerable over A , f : U ∞ A → { a , b } partial function, computable over A . � �� � { 1 , 0 } Definition (Kleene’s operator for A ) µ [ f ]( x 1 , . . . , x n ) = df min { k ∈ N | f ( x 1 , . . . , x n , k ) = 1 & f ( x 1 , . . . , x n , l ) ↓ for l < k , l ∈ N } Definition (Oracle Instruction with Kleene’s operator) z 1 · · · z n ↓ ↓ ℓ : Z j := µ [ f ]( Z 1 , . . . , Z I 1 ) , if I 1 = n no minimum ⇒ the machine loops forever Properties Any µ -semi-decidable problem is semi-decidable over A .

ν -Oracle BSS RAM’s for Structures with a and b (Moschovakis’ Operator ν ) A is fixed. a , b are constants of A . f : U ∞ A → { a , b } partial function, computable over A . Definition (Moschovakis’ operator for A ) ν [ f ]( x 1 , . . . , x n ) = df { y 1 ∈ U A | ( ∃ ( y 2 , . . . , y m ) ∈ U ∞ A )( f ( x 1 , . . . , x n , y 1 , y 2 , . . . , y m ) = a ) } � �� � y ∈ U ∞ � A Definition (Oracle instruction with Moschovakis’ operator) z 1 · · · z n ↓ ↓ NONDETERMINISTIC ! ℓ : Z j := ν [ f ]( Z 1 , . . . , Z I 1 ) ν [ f ]( z 1 , . . . , z n ) � = ∅ ⇒ Z j contains some z ∈ ν [ f ]( z 1 , . . . , z n ) . ν [ f ]( z 1 , . . . , z n ) = ∅ ⇒ no stop (the machine loops forever).

Nondeterministic Machines versus ν -oracle Machines (Guessing Solutions and Nondeterministic Semi-Decidability) f : U ∞ A → { a , b } partial function, computable by M f over A . Properties (By ν -operator of a ν -oracle machine) x 1 · · · x n x 1 · · · x n y 1 ↓ ↓ ↓ ↓ ↓ Z j := ν [ f ]( Z 1 , . . . , Z I 1 ); . . . Z j := ν [ f ]( Z 1 , . . . , Z I 1 − 1 , Z I 1 ) . . . ↓ ↓ ⇒ f ( x 1 , . . . , x n , y 1 , . . . , y m ) = a y 1 y 2 Properties (By input-guessing procedure of nondeterm. machine) x 1 x 2 x n y 1 y 2 y m x n ↓ ↓ ↓ ↓ ↓ ↓ ↓ Z 1 Z 2 . . . Z n Z n + 1 Z n + 2 . . . Z n + m Z n + m + 1 . . . Then, simulate M f . Proposition A ⊆ U ∞ A is ν -semi-decidable iff A is nondeterm. semi-decidable.

ν n -Oracle BSS RAM’s versus ν -Oracle BSS RAM’s (For Motivation: Computable Choice Functions?) A = ( N ; N ; ; =) . � if x i � = x j for all i , j with i � = j , 1 R ( x 1 , . . . , x n ) := 0 otherwise . Example ( ν n for fixed arity n ) Example ( ν for any arity) ( z 1 , . . . , z n ) ∈ N n ∈ N ∞ ( z 1 , . . . , z n ) ↓ ↓ ↓ ↓ ℓ : Z j := ν n [ R ]( Z 1 , . . . , Z n ) ℓ : Z j := ν [ R ]( Z 1 , . . . , Z I 1 )