Hierarchies of Decision Problems over Algebraic Structures Defined - PowerPoint PPT Presentation

Hierarchies of Decision Problems over Algebraic Structures Defined by Quantifiers Christine Gaßner University of Greifswald A shortened version of the slides that were presented at the CCC 2015

Hierarchies over Algebraic Structures Introduction Subject: BSS RAM model over first order structures a framework for study of the abstract computability by machines over several structures the uniform abstract decidability and the reducibility of decision problems over algebraic structures on a high abstraction level includes several types of register machines, the Turing machine, and the uniform BSS model of computation over the reals hierarchies of undecidable decision problems within this model Meaning: better understanding the structural complexity of decision problems the methods used in the recursion theory the limits of computations over several structures

Outline The BSS RAM’s uniform machines over first order structures Halting problems uniformity and codes for machines Known hierarchies derived from the arithmetical hierarchy Kleene–Mostowski, Cucker, . . . A hierarchy over first order structures defined by quantifiers characterized by halting problems complete problems

Computation over Algebraic Structures The Allowed Instructions (for BSS RAM’s) Computation over A = ( U A ; U A ; f 1 , . . . , f n 1 ; R 1 , . . . , R n 2 , = ) . ���� ���� � �� � � �� � constants universe operations relations Z 1 Z 2 Z 3 Z 4 . . . Registers for elements in U A . . . I 1 I 2 I 3 I 4 Registers for indices / addresses 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 ∈ 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 Inputs and Outputs for BSS RAM’s in M [ ND ] A � U ∞ i ≥ 1 U i A = df A — input and output space (for the universe U A ) x = ( x 1 , . . . , x n ) ∈ U ∞ Input of � A : x 1 x 2 x 3 x n x n x n ↓ ↓ ↓ ↓ ↓ ↓ Z 1 Z 2 Z 3 . . . Z n Z n + 1 Z n + 2 . . . k M index registers I 1 I 2 I 3 I 4 . . . I k M ↑ ↑ ↑ ↑ ↑ n 1 1 1 1 Input and guessing procedures of nondeterministic machines: 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 m ∈ N + is also guessed. I 1 I 2 I 3 I 4 . . . I k M ↑ ↑ ↑ ↑ ↑ n 1 1 1 1 Output of Z 1 , . . . , Z I 1 .

Two Hierarchies Analogous to the Arithmetical Hierarchy A is fixed. The first hierarchy defined semantically by deterministic machines: Σ 0 = DEC A 0 Π 0 { U ∞ A \ P | P ∈ Σ 0 = n } n ∆ 0 Σ 0 n ∩ Π 0 = n n n )( P ∈ SDEC Q { P ⊆ U ∞ Σ 0 A | ( ∃ Q ∈ Π 0 = A ) } n + 1 The second hierarchy defined syntactically by formulas: Σ ND = DEC A 0 { U ∞ Π ND A \ P | P ∈ Σ ND = n } n ∆ ND Σ ND ∩ Π ND = n n n Σ ND { P ⊆ U ∞ A | ( ∃ Q ∈ Π ND = n ) n + 1 y ∈ U ∞ ∀ � x ( � x ∈ P ⇔ ( ∃ � A )(( � y .� x ) ∈ Q )) }

The Arithmetical Hierarchy (Kleene-Mostowski) For Turing Machines For ( { 0 , 1 } ; 0 , 1 ; ; =) , both definitions provide the same hierarchy: { P ⊆ { 0 , 1 } ∞ | ( ∃ Q ∈ Π 0 n )( P ∈ SDEC Q ) } Σ 0 = n + 1 || { P ⊆ { 0 , 1 } ∞ | ( ∃ Q ∈ Π 0 Σ ND = n ) n + 1 y ∈ { 0 , 1 } ∞ )(( � ∀ � x ( � x ∈ P ⇔ ( ∃ � y .� x ) ∈ Q )) } Σ 0 Σ 0 Σ 0 · · · 1 2 3 ր ց ր ց ր ∆ 0 ∆ 0 ∆ 0 1 2 3 ց ր ց ր ց Π 0 Π 0 Π 0 · · · 1 2 3 “ → ” means “strong ⊂ ”.

Complete Problems in the Arithmetical Hierarchy For Turing Machines . . . . . . Σ 0 Π 0 CoFIN ∈ 3 3 տ ր ∆ 0 3 ր տ ∈ Σ 0 Π 0 ∋ FIN TOTAL 2 2 տ ր ∆ 0 2 ր տ H spec ∈ Σ 0 Π 0 ∋ EMPTY 1 1 տ ր ∆ 0 1 x ∈ { 0 , 1 } ( ≥ n ) )( M ( � { code ( M ) | ( ∃ n ∈ N )( ∀ � x ) ↓ ) } CoFIN = x ∈ { 0 , 1 } ( ≥ n ) )( M ( � FIN = { code ( M ) | ( ∃ n ∈ N )( ∀ � x ) ↑ ) } x ∈ { 0 , 1 } ∞ )( M ( � { code ( M ) | ( ∀ � x ) ↓ ) } TOTAL = H spec = { code ( M ) | M ( code ( M )) ↓} x ∈ { 0 , 1 } ∞ )( M ( � EMPTY = { code ( M ) | ( ∀ � x ) ↑ ) } (cp. Soare, Kozen)

Complete Problems in the BSS model (Cucker) For Computation over the Ring of Reals R = ( R ; R ; · , + , − ; ≤ ) . . . ր տ Σ ND Π ND 3 տ ր 3 ∆ ND 3 ր տ TOTAL R , TOTAL ND Σ ND Π ND ∋ Suslin’s proj. hier. = 2 տ ր 2 R ∆ ND = set of Borel sets 2 ( ⊆ R ∞ ) ↑ . . . . . . տ ր . . . ր տ Σ 0 Π 0 ∈ FIN R 2 տ ր 2 ∆ 0 2 ր տ Σ ND Σ 0 Π 0 Π ND = = ∋ INJ R 1 1 տ ր 1 1 ∆ 0 1 x ∈ R ( ≥ n ) )( M ( � = { code ( M ) | ( ∃ n ∈ N )( ∀ � x ) ↑ ) } FIN R x 2 ∈ R ∞ )( M ( � = { code ( M ) | ( ∀ � x 1 ,� x 1 ) ↓ = M ( � x 2 ) ↓ ⇒ � x 1 = � x 2 ) } INJ R TOTAL [ ND ] { code ( M ) | M ∈ M [ ND ] x ∈ R ∞ )( M ( � = & ( ∀ � x ) ↓ ) } R R (cp. Cucker)

A Characterization of the First Hierarchy For BSS RAM’s — Computation over Several Structures For A : a finite number of operations & relations, all elements are constants, contains an infinite set effectively enumerable over A : N ⊆ U A . Recall (the definition): Σ 0 = DEC A 0 Π 0 { U ∞ A \ P | P ∈ Σ 0 n } = n ∆ 0 Σ 0 n ∩ Π 0 = n n n )( P ∈ SDEC Q { P ⊆ U ∞ Σ 0 A | ( ∃ Q ∈ Π 0 A ) } = n + 1 H ( n ) A | P � 1 H ( n + 1 ) ⇒ Σ 0 { P ⊆ U ∞ = A = } SDEC n + 1 A A Proposition (G. 2014) n + 1 = { P ⊆ U ∞ Σ 0 A | ( ∃ Q ∈ Π 0 n ) ∀ � x ∈ P ⇔ ( ∃ k ∈ N )(( � x . k ) ∈ Q )) } x ( � Note: H ( 0 ) = ∅ A H ( n + 1 ) = Halting problem for BSS RAM’s using H ( n ) A as oracle A

A Characterization of the Second Hierarchy For BSS RAM’s — Computation over Several Structures A : a finite number of operations & relations, all elements ˆ = constants. Recall (the definition): Σ ND = DEC A 0 Π ND { U ∞ A \ P | P ∈ Σ ND = n } n ∆ ND Σ ND ∩ Π ND = n n n Σ ND { P ⊆ U ∞ A | ( ∃ Q ∈ Π ND = n ) n + 1 y ∈ U ∞ ∀ � x ∈ P ⇔ ( ∃ � x ) ∈ Q )) } x ( � A )(( � y .� Proposition (G. 2015) n + 1 = { P ⊆ U ∞ n )( P ∈ ( SDEC ND Σ ND A | ( ∃ Q ∈ Π ND A ) Q ) } A ) ( n ) = { P ⊆ U ∞ A ) ( H ND n + 1 = ( SDEC ND A ) ( n + 1 ) } Σ ND A | P � 1 ( H ND ⇒ Note: ( H ND A ) ( 0 ) = ∅ A ) ( n + 1 ) = Halting p. for ND-machines using ( H ND A ) ( n ) as oracle ( H ND

Complete Problems in the First Hierarchy For BSS RAM’s — Computation over Several Structures For A : a finite number of operations & relations, all elements are constants, contains an infinite set effectively enumerable over A : N ⊆ U A . . . . ր տ ∈ Σ 0 Π 0 ∋ FIN N TOTAL N , INCL N 2 2 տ ր ∆ 0 2 ր տ H spec ∈ Σ 0 Π 0 A , H A 1 1 տ ր ∆ 0 1 { code ( M ) ∈ U ∞ A | | H M ∩ N ∞ | < ∞} FIN N = ( H M = halting set) { code ( M ) ∈ U ∞ x ∈ N ∞ )( M ( � TOTAL N = A | ( ∀ � x ) ↓ ) } { ( code ( M ) . code ( N )) ∈ U ∞ A | ( H M ∩ N ∞ ) ⊆ ( H N ∩ N ∞ ) } = INCL N H [ spec ] = ˆ Halting problems for BSS RAM’s over A (cp. Gaßner) A

Complete Problems in the Second Hierarchy (Blue) For BSS RAM’s — Computation over Several Structures A : a finite number of operations & relations, all elements ˆ = constants. . . . ր տ TOTFIN ND ∈ Σ ND Π ND A 3 տ ր 3 ∆ ND 3 ր տ FIN ND ∈ Σ ND Π ND TOTAL ND A , INCL ND A , i , CONST ND ∋ A A 2 տ ր 2 ∆ ND 2 ր տ INJ ND H ND ∈ Σ ND Π ND ∋ A A 1 տ ր 1 ∆ ND 1 TOTAL ND x ∈ R ∞ )( M ( � ( M ∈ M ND = { code ( M ) | ( ∀ � x ) ↓ ) } A ) A INJ ND = { code ( M ) | M computes a/an [super] injective function } A CONST ND = { code ( M ) | M computes a total constant function } A FIN ND { code ( M ) | ( ∀ i ∈ N \ I )( H M ∩ U i A = ∅ ) for some | I | < ω } = A TOTFIN ND { code ( M ) | ( ∀ i ∈ N \ I )( H M ∩ U i A � = U i = A ) for some | I | < ω } A H ND = Halting problem for ND-machines over A (in M ND A ) A INCL ND { ( code ( M ) . code ( N )) | ( M , N ) ∈ M A , i × M ND = A & H M ⊆ H N } A , i M A , 3 = M A ( H ND M A , 4 = M ND A ( H ND M A , 1 = M A , M A , 2 = M A ( H A ) , A ) , A )

Summary For BSS RAM’s — Computation over Several Structures 1st hierarchy: . . . ր տ ∈ Σ 0 Π 0 2 ∋ FIN N TOTAL N , INCL N 2 տ ր ∆ 0 2 ր տ ∈ Σ 0 Π 0 H A 1 տ ր 1 ∆ 0 1 2nd hierarchy: . . . TOTFIN ND ր տ ∈ Σ ND Π ND A 3 տ ր 3 ∆ ND 3 ր տ FIN ND ∈ Σ ND Π ND TOTAL ND A , INCL ND A , i , CONST ND ∋ A A 2 տ ր 2 ∆ ND 2 ր տ H ND ∈ Σ ND Π ND INJ ND ∋ A A 1 տ ր 1 ∆ ND 1 Thank you very much for your attention!