Analytic Functions and Small Complexity Classes Akitoshi Kawamura, - PowerPoint PPT Presentation

Analytic Functions and Small Complexity Classes Akitoshi Kawamura, Florian Steinberg, Holger Thies Workshop on Mathematical Logic and its Applications Kyoto University, Kyoto, Japan September 16, 2016 1

TTE: Computable Real Numbers Computable Real Number A real number is called computable if it can be approximated up to any desired precision. 2

TTE: Computable Real Numbers Computable Real Number A real number is called computable if it can be approximated up to any desired precision. d n rational approximations x | d n − x | ≤ 2 − n . 0 n d n 2

TTE: Computable Real Functions Computable Real Function A function f : [0 , 1] → R is called computable if the values f ( x ) can be approximated up to any desired precision. 3

TTE: Computable Real Functions Computable Real Function A function f : [0 , 1] → R is called computable if the values f ( x ) can be approximated up to any desired precision. f 0 m d m 3

TTE: Computable Real Functions Computable Real Function A function f : [0 , 1] → R is called computable if the values f ( x ) can be approximated up to any desired precision. x 0 n d n f 0 m d m 3

TTE: Computable Real Functions Computable Real Function A function f : [0 , 1] → R is called computable if the values f ( x ) can be approximated up to any desired precision. Notes x closed under composition 0 n d n f 0 m d m 3

TTE: Computable Real Functions Computable Real Function A function f : [0 , 1] → R is called computable if the values f ( x ) can be approximated up to any desired precision. Notes x closed under composition multidimensional functions: 0 n d n several input oracles and outputs. f 0 m d m 3

TTE: Computable Real Functions Computable Real Function A function f : [0 , 1] → R is called computable if the values f ( x ) can be approximated up to any desired precision. Notes x closed under composition multidimensional functions: 0 n d n several input oracles and outputs. f Computable functions are continuous. 0 m d m 3

Generalization Representation A representation for a set X is a partial surjective function ρ : B → X , that is, objects are encoded by string functions. 4

� � � Generalization Representation A representation for a set X is a partial surjective function ρ : B → X , that is, objects are encoded by string functions. F B B α β � Y X f 4

� � � Generalization Representation A representation for a set X is a partial surjective function ρ : B → X , that is, objects are encoded by string functions. Oracle ϕ F B B r ϕ ( r ) α β � Y X f F q F ( ϕ )( q ) 4

Representation for Real Numbers Example Define a ρ R -name of x ∈ R by letting ρ R (0 n ) encode some d ∈ D such that | d − x | ≤ 2 − n . x ∈ R 2 − n approx. of x 0 n 5

Representation for Real Numbers ( ρ | [0 , 1] , ρ R )-computable R Example Oracle ϕ Define a ρ R -name of x ∈ R by letting ρ R (0 n ) encode some d ∈ D such that | d − x | ≤ 2 − n . r ϕ ( r ) F x ∈ R q F ( ϕ )( q ) 2 − n approx. of x 0 n 5

Representation for Real Numbers Computable Function ( ρ | [0 , 1] , ρ R )-computable R Example x Oracle ϕ Define a ρ R -name of x ∈ R by letting ρ R (0 n ) encode some d ∈ D such that | d − x | ≤ 2 − n . 0 n d n r ϕ ( r ) f F 0 m x ∈ R d m q F ( ϕ )( q ) 2 − n approx. of x 0 n 5

Representation for Functions Example Define a δ � -name of a function f ∈ C [0 , 1] as a pair < µ, φ > where φ is such that | φ ( d , 0 n ) − f ( d ) | ≤ 2 − n for all d ∈ D ∩ [0 , 1] µ encodes the modulus of continuity (in unary) 6

Representation for Functions Modulus of Continuity | x − y | ≤ 2 − µ ( n ) ⇒ | f ( x ) − f ( y ) | ≤ 2 − n Example Define a δ � -name of a function f ∈ C [0 , 1] as a pair < µ, φ > where φ is such that | φ ( d , 0 n ) − f ( d ) | ≤ 2 − n for all d ∈ D ∩ [0 , 1] µ encodes the modulus of continuity (in unary) 6

Representation for Functions Modulus of Continuity | x − y | ≤ 2 − µ ( n ) ⇒ | f ( x ) − f ( y ) | ≤ 2 − n Example Define a δ � -name of a function f ∈ C [0 , 1] as a pair < µ, φ > where φ is such that | φ ( d , 0 n ) − f ( d ) | ≤ 2 − n for all d ∈ D ∩ [0 , 1] µ encodes the modulus of continuity (in unary) f ∈ C [0 , 1] 6

Representation for Functions Modulus of Continuity | x − y | ≤ 2 − µ ( n ) ⇒ | f ( x ) − f ( y ) | ≤ 2 − n Example Define a δ � -name of a function f ∈ C [0 , 1] as a pair < µ, φ > where φ is such that | φ ( d , 0 n ) − f ( d ) | ≤ 2 − n for all d ∈ D ∩ [0 , 1] µ encodes the modulus of continuity (in unary) f ∈ C [0 , 1] 0 m 0 µ ( m ) 6

Representation for Functions Modulus of Continuity | x − y | ≤ 2 − µ ( n ) ⇒ | f ( x ) − f ( y ) | ≤ 2 − n Example Define a δ � -name of a function f ∈ C [0 , 1] as a pair < µ, φ > where φ is such that | φ ( d , 0 n ) − f ( d ) | ≤ 2 − n for all d ∈ D ∩ [0 , 1] µ encodes the modulus of continuity (in unary) f ∈ C [0 , 1] d 0 n q 6

Type-2 Complexity Theory Let Σ ∗∗ be the set of length-monotone string-functions 7

Type-2 Complexity Theory Let Σ ∗∗ be the set of length-monotone string-functions Define | ϕ | : N → N by | ϕ | ( | u | ) = | ϕ ( u ) | 7

Type-2 Complexity Theory Let Σ ∗∗ be the set of length-monotone string-functions Define | ϕ | : N → N by | ϕ | ( | u | ) = | ϕ ( u ) | Bound running time by second order polynomials P ( | ϕ | )( | x | ) 7

Type-2 Complexity Theory Second Order Polynomials Defined inductively by 1 P ( L , n ) := m , for m ∈ N 2 P ( L , n ) := n Let Σ ∗∗ be the set of length-monotone string-functions 3 P , Q second order polynomials ⇒ P + Q , P · Q 4 P a second order polynomial ⇒ L ( P ) Define | ϕ | : N → N by | ϕ | ( | u | ) = | ϕ ( u ) | Example: P ( L , n ) = 2 L ( L ( L ( n ) · L ( n ) + 2 L ( n )) + 2) + n + 10 Bound running time by second order polynomials P ( | ϕ | )( | x | ) 7

Type-2 Complexity Theory Let Σ ∗∗ be the set of length-monotone string-functions Define | ϕ | : N → N by | ϕ | ( | u | ) = | ϕ ( u ) | Bound running time by second order polynomials P ( | ϕ | )( | x | ) Can define complexity classes FP 2 , #P 2 and FPSPACE 2 7

Type-2 Complexity Theory Let Σ ∗∗ be the set of length-monotone string-functions Define | ϕ | : N → N by | ϕ | ( | u | ) = | ϕ ( u ) | Bound running time by second order polynomials P ( | ϕ | )( | x | ) Can define complexity classes FP 2 , #P 2 and FPSPACE 2 Can also define complexity classes P 2 , NP 2 and PSPACE 2 by considering functions ϕ : Σ ∗∗ → (Σ ∗ → { 0 , 1 } ) 7

Small Complexity Classes 8

Small Complexity Classes 8

Small Complexity Classes 8

Small Complexity Classes 9

Small Complexity Classes 9

Small Complexity Classes 9

Small Complexity Classes 9

Small Complexity Classes 9

The stack model (Kawamura and Ota) 10

The stack model (Kawamura and Ota) 10

The stack model (Kawamura and Ota) 10

The stack model (Kawamura and Ota) 10

The stack model (Kawamura and Ota) 10

Example Function Application The function Apply : C [0 , 1] × [0 , 1] → R , ( f , x ) �→ f ( x ) is ([ δ � , ρ | [0 , 1] ] , ρ R ) − FL computable. R 11

Example f ∈ C [0 , 1] 0 m 0 µ ( m ) d 0 n q Function Application The function Apply : C [0 , 1] × [0 , 1] → R , ( f , x ) �→ f ( x ) is ([ δ � , ρ | [0 , 1] ] , ρ R ) − FL computable. R 11

Example f ∈ C [0 , 1] x ∈ R 0 m 0 µ ( m ) d 0 n q Function Application 0 n d The function Apply : C [0 , 1] × [0 , 1] → R , ( f , x ) �→ f ( x ) is ([ δ � , ρ | [0 , 1] ] , ρ R ) − FL computable. R 11

Example f ∈ C [0 , 1] x ∈ R 0 m 0 µ ( m ) d 0 n q Function Application 0 n d The function Apply : C [0 , 1] × [0 , 1] → R , ( f , x ) �→ f ( x ) is ([ δ � , ρ | [0 , 1] ] , ρ R ) − FL computable. R 0 n Input: Stack: - ε Oracle Output: ε Work: ε Output: 11

Example f ∈ C [0 , 1] x ∈ R 0 m 0 µ ( m ) d 0 n q Function Application 0 n d The function Apply : C [0 , 1] × [0 , 1] → R , ( f , x ) �→ f ( x ) is ([ δ � , ρ | [0 , 1] ] , ρ R ) − FL computable. R 0 n Input: ε ; ε ; ε Stack: ε Oracle Output: ε Work: ε Output: 11

Example f ∈ C [0 , 1] x ∈ R 0 m 0 µ ( m ) d 0 n q Function Application 0 n d The function Apply : C [0 , 1] × [0 , 1] → R , ( f , x ) �→ f ( x ) is ([ δ � , ρ | [0 , 1] ] , ρ R ) − FL computable. R 0 n Input: 0 n +1 ; ε ; ε Stack: ε Oracle Output: ε Work: ε Output: 11

Example f ∈ C [0 , 1] x ∈ R 0 m 0 µ ( m ) d 0 n q Function Application 0 n d The function Apply : C [0 , 1] × [0 , 1] → R , ( f , x ) �→ f ( x ) is ([ δ � , ρ | [0 , 1] ] , ρ R ) − FL computable. R 0 n Input: ε ; ε Stack: | x − y | ≤ 2 − m ⇒ | f ( x ) − f ( y ) | ≤ 2 − ( n +1) 0 m Oracle Output: ε Work: ε Output: 11