Continuous models of computation: computability, complexity, - PowerPoint PPT Presentation

Continuous models of computation: computability, complexity, universality Amaury Pouly 21 mars 2017 1 / 19

Digital vs analog computers 2 / 19

Digital vs analog computers VS 2 / 19

Church Thesis Computability logic boolean circuits discrete recursive Turing lambda functions machine calculus continuous quantum analog Church Thesis All reasonable models of computation are equivalent. 3 / 19

Church Thesis Complexity logic boolean circuits discrete recursive Turing lambda functions machine calculus � ? ? continuous quantum analog Effective Church Thesis All reasonable models of computation are equivalent for complexity. 3 / 19

Polynomial Differential Equations u × k uv k v u � + � u + v u u v General Purpose Analog Computer Differential Analyzer polynomial differential Newton mechanics equations : � y ( 0 )= y 0 y ′ ( t )= p ( y ( t )) Reaction networks : chemical Rich class enzymatic Stable (+, × , ◦ ,/,ED) No closed-form solution 4 / 19

Example of dynamical system y 2 � � y 1 × ℓ y 3 y 4 − g � × ℓ θ m � × × − 1 θ + g ¨ ℓ sin( θ ) = 0 y ′  1 = y 2  y 1 = θ   2 = − g y 2 = ˙ y ′   l y 3 θ   ⇔ y ′ 3 = y 2 y 4 y 3 = sin( θ )    y ′  4 = − y 2 y 3 y 4 = cos( θ )   5 / 19

Computing with the GPAC Generable functions � y ( 0 )= y 0 x ∈ R y ′ ( x )= p ( y ( x )) f ( x ) = y 1 ( x ) y 1 ( x ) x Shannon’s notion 6 / 19

Computing with the GPAC Generable functions � y ( 0 )= y 0 x ∈ R y ′ ( x )= p ( y ( x )) f ( x ) = y 1 ( x ) y 1 ( x ) x Shannon’s notion sin , cos , exp , log , ... Strictly weaker than Turing machines [Shannon, 1941] 6 / 19

Computing with the GPAC Generable functions Computable � y ( 0 )= y 0 � y ( 0 )= q ( x ) x ∈ R x ∈ R y ′ ( x )= p ( y ( x )) y ′ ( t )= p ( y ( t )) t ∈ R + f ( x ) = y 1 ( x ) f ( x ) = lim t →∞ y 1 ( t ) y 1 ( x ) y 1 ( t ) x f ( x ) x t Shannon’s notion Modern notion sin , cos , exp , log , ... Strictly weaker than Turing machines [Shannon, 1941] 6 / 19

Computing with the GPAC Generable functions Computable � y ( 0 )= y 0 � y ( 0 )= q ( x ) x ∈ R x ∈ R y ′ ( x )= p ( y ( x )) y ′ ( t )= p ( y ( t )) t ∈ R + f ( x ) = y 1 ( x ) f ( x ) = lim t →∞ y 1 ( t ) y 1 ( x ) y 1 ( t ) x f ( x ) x t Shannon’s notion Modern notion sin , cos , exp , log , ... sin , cos , exp , log , Γ , ζ, ... Strictly weaker than Turing Turing powerful machines [Shannon, 1941] [Bournez et al., 2007] 6 / 19

From discrete to real computability Computable Analysis : lift Turing computability to real numbers [Ko, 1991; Weihrauch, 2000] 7 / 19

From discrete to real computability Computable Analysis : lift Turing computability to real numbers [Ko, 1991; Weihrauch, 2000] Definition x ∈ R is computable iff ∃ a computable f : N → Q such that : | x − f ( n ) | � 10 − n n ∈ N Examples : rational numbers, π , e , ... n f ( n ) | π − f ( n ) | 0 . 14 � 10 − 0 0 3 0 . 04 � 10 − 1 1 3.1 0 . 001 � 10 − 2 2 3.14 0 . 9 · 10 − 10 � 10 − 10 10 3.1415926535 7 / 19

From discrete to real computability Computable Analysis : lift Turing computability to real numbers [Ko, 1991; Weihrauch, 2000] Definition x ∈ R is computable iff ∃ a computable f : N → Q such that : | x − f ( n ) | � 10 − n n ∈ N Examples : rational numbers, π , e , ... n f ( n ) | π − f ( n ) | 0 . 14 � 10 − 0 0 3 0 . 04 � 10 − 1 1 3.1 0 . 001 � 10 − 2 2 3.14 0 . 9 · 10 − 10 � 10 − 10 10 3.1415926535 Beware : there exists uncomputable real numbers! 7 / 19

From discrete to real computability f ( x ) f ( x ) x x 7 / 19

From discrete to real computability f ( x ) f ( y ) � 10 − 0 f ( x ) � 10 − m ( 0 ) x x y Definition (Computable function) f : [ a , b ] → R is computable iff ∃ m : N → N , computable functions such that : | x − y | � 10 − m ( n ) ⇒ | f ( x ) − f ( y ) | � 10 − n x , y ∈ R , n ∈ N m : modulus of continuity 7 / 19

From discrete to real computability f ( x ) � 10 − 1 f ( y ) f ( x ) � 10 − m ( 1 ) x x y Definition (Computable function) f : [ a , b ] → R is computable iff ∃ m : N → N , ψ : Q × N → Q computable functions such that : | x − y | � 10 − m ( n ) ⇒ | f ( x ) − f ( y ) | � 10 − n x , y ∈ R , n ∈ N m : modulus of continuity 7 / 19

From discrete to real computability f ( x ) � 10 − 2 f ( y ) f ( x ) � 10 − m ( 2 ) x x y Definition (Computable function) f : [ a , b ] → R is computable iff ∃ m : N → N , ψ : Q × N → Q computable functions such that : | x − y | � 10 − m ( n ) ⇒ | f ( x ) − f ( y ) | � 10 − n x , y ∈ R , n ∈ N m : modulus of continuity 7 / 19

From discrete to real computability f ( x ) ψ ( r , 0 ) � 10 − 0 f ( r ) x r ∈ Q Definition (Computable function) f : [ a , b ] → R is computable iff ∃ m : N → N , ψ : Q × N → Q computable functions such that : | x − y | � 10 − m ( n ) ⇒ | f ( x ) − f ( y ) | � 10 − n x , y ∈ R , n ∈ N | f ( r ) − ψ ( r , n ) | � 10 − n r ∈ Q , n ∈ N 7 / 19

From discrete to real computability f ( x ) ψ ( r , 1 ) � 10 − 1 f ( r ) x r ∈ Q Definition (Computable function) f : [ a , b ] → R is computable iff ∃ m : N → N , ψ : Q × N → Q computable functions such that : | x − y | � 10 − m ( n ) ⇒ | f ( x ) − f ( y ) | � 10 − n x , y ∈ R , n ∈ N | f ( r ) − ψ ( r , n ) | � 10 − n r ∈ Q , n ∈ N 7 / 19

From discrete to real computability f ( x ) ψ ( r , 2 ) � 10 − 2 f ( r ) x r ∈ Q Definition (Computable function) f : [ a , b ] → R is computable iff ∃ m : N → N , ψ : Q × N → Q computable functions such that : | x − y | � 10 − m ( n ) ⇒ | f ( x ) − f ( y ) | � 10 − n x , y ∈ R , n ∈ N | f ( r ) − ψ ( r , n ) | � 10 − n r ∈ Q , n ∈ N 7 / 19

From discrete to real computability Definition (Computable function) f : [ a , b ] → R is computable iff ∃ m : N → N , ψ : Q × N → Q computable functions such that : | x − y | � 10 − m ( n ) ⇒ | f ( x ) − f ( y ) | � 10 − n x , y ∈ R , n ∈ N | f ( r ) − ψ ( r , n ) | � 10 − n r ∈ Q , n ∈ N Examples : polynomials, sin , exp , √· Note : all computable functions are continuous Beware : there exists (continuous) uncomputable real functions! 7 / 19

From discrete to real computability Definition (Computable function) f : [ a , b ] → R is computable iff ∃ m : N → N , ψ : Q × N → Q computable functions such that : | x − y | � 10 − m ( n ) ⇒ | f ( x ) − f ( y ) | � 10 − n x , y ∈ R , n ∈ N | f ( r ) − ψ ( r , n ) | � 10 − n r ∈ Q , n ∈ N Examples : polynomials, sin , exp , √· Note : all computable functions are continuous Beware : there exists (continuous) uncomputable real functions! Polytime complexity Add “polynomial time computable” everywhere. 7 / 19

Equivalence with computable analysis Definition (Bournez et al) f computable by GPAC if ∃ p polynomial such that ∀ x y ′ ( t ) = p ( y ( t )) y ( 0 ) = ( x , 0 , . . . , 0 ) satisfies | f ( x ) − y 1 ( t ) | � y 2 ( t ) et y 2 ( t ) − t →∞ 0. − − → y 1 ( t ) − t →∞ f ( x ) − − → y 1 ( t ) f ( x ) y 2 ( t ) = error bound x t 8 / 19

Equivalence with computable analysis Definition (Bournez et al) f computable by GPAC if ∃ p polynomial such that ∀ x y ′ ( t ) = p ( y ( t )) y ( 0 ) = ( x , 0 , . . . , 0 ) satisfies | f ( x ) − y 1 ( t ) | � y 2 ( t ) et y 2 ( t ) − t →∞ 0. − − → y 1 ( t ) − t →∞ f ( x ) − − → y 1 ( t ) f ( x ) y 2 ( t ) = error bound x t Theorem (Bournez, Campagnolo, Graça, Hainry) f : [ a , b ] → R computable ⇔ f computable by GPAC 8 / 19

Complexity of analog systems Turing machines : T ( x ) = number of steps to compute on x 9 / 19

Complexity of analog systems Turing machines : T ( x ) = number of steps to compute on x GPAC : time contraction problem Tentative definition T ( x , µ ) = first time t so that | y 1 ( t ) − f ( x ) | � e − µ y ′ = p ( y ) y ( 0 ) = ( x , 0 , . . . , 0 ) y 1 ( t ) f ( x ) x t 9 / 19

Complexity of analog systems Turing machines : T ( x ) = number of steps to compute on x GPAC : time contraction problem Tentative definition T ( x , µ ) = first time t so that | y 1 ( t ) − f ( x ) | � e − µ y ′ = p ( y ) z ( t ) = y ( e t ) y ( 0 ) = ( x , 0 , . . . , 0 ) y 1 ( t ) z 1 ( t ) f ( x ) f ( x ) � x x t t 9 / 19

Complexity of analog systems Turing machines : T ( x ) = number of steps to compute on x GPAC : time contraction problem Tentative definition T ( x , µ ) = first time t so that | y 1 ( t ) − f ( x ) | � e − µ y ′ = p ( y ) z ( t ) = y ( e t ) y ( 0 ) = ( x , 0 , . . . , 0 ) y 1 ( t ) z 1 ( t ) f ( x ) f ( x ) � x x t t w ( t ) = y ( e e t ) w 1 ( t ) f ( x ) x t 9 / 19