Continuous models of computation: computability, complexity, universality Amaury Pouly Joint work with Olivier Bournez and Daniel Graça CNRS, IRIF, Université Paris Diderot 26 march 2019 1 / 23
What is a computer? 2 / 23
What is a computer? 2 / 23
What is a computer? VS 2 / 23
Analog Computers Admiralty Fire Control Table Differential Analyser British Navy ships (WW2) “Mathematica of the 1920s” 3 / 23
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. 4 / 23
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. 4 / 23
Polynomial Differential Equations u × k uv k v u � + � u + v u u v General Purpose Analog Computer Differential Analyzer 5 / 23
Polynomial Differential Equations u × k uv k v u � + � u + v u u v General Purpose Analog Computer Differential Analyzer polynomial differential equations : � y ( 0 )= y 0 y ′ ( t )= p ( y ( t )) 5 / 23
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 ◮ enzymatic 5 / 23
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 5 / 23
Example of dynamical system ℓ θ m g θ + g ¨ ℓ sin( θ ) = 0 6 / 23
Example of dynamical system ℓ θ m y ′ y 1 = θ 1 = y 2 g y 2 = ˙ 2 = − g y ′ θ l y 3 ⇔ y ′ y 3 = sin( θ ) 3 = y 2 y 4 θ + g ¨ ℓ sin( θ ) = 0 y ′ y 4 = cos( θ ) 4 = − y 2 y 3 6 / 23
Example of dynamical system y 2 � � y 1 × ℓ y 3 y 4 − g � × ℓ θ m � × × − 1 y ′ y 1 = θ 1 = y 2 g y 2 = ˙ 2 = − g y ′ θ l y 3 ⇔ y ′ y 3 = sin( θ ) 3 = y 2 y 4 θ + g ¨ ℓ sin( θ ) = 0 y ′ y 4 = cos( θ ) 4 = − y 2 y 3 6 / 23
Example of dynamical system y 2 � � y 1 × ℓ y 3 y 4 − g � × ℓ θ m � × × − 1 y ′ y 1 = θ 1 = y 2 g y 2 = ˙ 2 = − g y ′ θ l y 3 ⇔ y ′ y 3 = sin( θ ) 3 = y 2 y 4 θ + g ¨ ℓ sin( θ ) = 0 y ′ y 4 = cos( θ ) 4 = − y 2 y 3 Historical remark : the word “analog” The pendulum and the circuit have the same equation. One can study one using the other by analogy. 6 / 23
Computing with differential equations 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 7 / 23
Computing with differential equations 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] 7 / 23
Computing with differential equations 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] 7 / 23
Computing with differential equations 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] 7 / 23
From discrete to real computability Computable Analysis : “Turing” computability over real numbers 8 / 23
From discrete to real computability Computable Analysis : “Turing” computability over real numbers Definition (Ko, 1991; Weihrauch, 2000) 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 8 / 23
From discrete to real computability Computable Analysis : “Turing” computability over real numbers Definition (Ko, 1991; Weihrauch, 2000) 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! Γ = { n : the n th Turing machine halts } � 2 − n , x = n ∈ Γ 8 / 23
From discrete to real computability f ( x ) f ( r ) x r ∈ Q 8 / 23
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 , computable functions such that : ◮ effective approx over Q : | f ( r ) − ψ ( r , n ) | � 10 − n 8 / 23
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 , computable functions such that : ◮ effective approx over Q : | f ( r ) − ψ ( r , n ) | � 10 − n 8 / 23
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 , computable functions such that : ◮ effective approx over Q : | f ( r ) − ψ ( r , n ) | � 10 − n 8 / 23
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 , ψ : Q × N → Q computable functions such that : ◮ effective approx over Q : | f ( r ) − ψ ( r , n ) | � 10 − n ◮ effective continuity : | x − y | � 10 − m ( n ) ⇒ | f ( x ) − f ( y ) | � 10 − n m : modulus of continuity 8 / 23
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 : ◮ effective approx over Q : | f ( r ) − ψ ( r , n ) | � 10 − n ◮ effective continuity : | x − y | � 10 − m ( n ) ⇒ | f ( x ) − f ( y ) | � 10 − n m : modulus of continuity 8 / 23
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 : ◮ effective approx over Q : | f ( r ) − ψ ( r , n ) | � 10 − n ◮ effective continuity : | x − y | � 10 − m ( n ) ⇒ | f ( x ) − f ( y ) | � 10 − n m : modulus of continuity 8 / 23
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 : ◮ effective approx over Q : | f ( r ) − ψ ( r , n ) | � 10 − n ◮ effective continuity : | x − y | � 10 − m ( n ) ⇒ | f ( x ) − f ( y ) | � 10 − n m : modulus of continuity All computable functions are continuous! Examples : polynomials, sin , exp , √· Beware : there exists (continuous) uncomputable real functions! 8 / 23
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 : ◮ effective approx over Q : | f ( r ) − ψ ( r , n ) | � 10 − n ◮ effective continuity : | x − y | � 10 − m ( n ) ⇒ | f ( x ) − f ( y ) | � 10 − n m : modulus of continuity All computable functions are continuous! Examples : polynomials, sin , exp , √· Beware : there exists (continuous) uncomputable real functions! Polytime complexity Add “polynomial time computable” everywhere. 8 / 23
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