A universal differential equation Olivier Bournez, Amaury Pouly 21 - PowerPoint PPT Presentation

A universal differential equation Olivier Bournez, Amaury Pouly 21 mars 2017 1 / 14

Digital vs analog computers 2 / 14

Digital vs analog computers VS 2 / 14

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 / 14

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 / 14

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 / 14

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 / 14

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 / 14

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 / 14

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 / 14

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 / 14

Universal differential equations Computable functions Generable functions y 1 ( t ) f ( x ) y 1 ( x ) x x t subclass of analytic functions any computable function 7 / 14

Universal differential equations Computable functions Generable functions y 1 ( t ) f ( x ) y 1 ( x ) x x t subclass of analytic functions any computable function y 1 ( x ) x 7 / 14

Universal differential equation (Rubel) y 1 ( x ) x Theorem (Rubel) There exists a fixed polynomial p and k ∈ N such that for any conti- nuous functions f and ε , there exists a solution y to p ( y , y ′ , . . . , y ( k ) ) = 0 such that | y ( t ) − f ( t ) | � ε ( t ) . 8 / 14

Universal differential equation (Rubel) y 1 ( x ) x Theorem (Rubel) There exists a fixed polynomial p and k ∈ N such that for any conti- nuous functions f and ε , there exists a solution y to ′′′′ + 6 y ′ 3 y ′′′′ + 24 y ′ 2 y ′′′′ 2 ′′′ 2 y ′′ 2 y ′′ 4 y 3 y ′ 4 y − 4 y ′ 4 y ′′ y ′′′ y ′′′′ ′′′ 3 − 29 y ′ 2 y ′′′ 2 + 12 y ′′ 3 y ′′ 7 − 12 y ′ 3 y ′′ y = 0 such that | y ( t ) − f ( t ) | � ε ( t ) . Problem : Rubel is «cheating». 8 / 14

Rubel’s proof in one slide − 1 1 − t 2 for − 1 < t < 1 and f ( t ) = 0 otherwise Take f ( t ) = e ( 1 − t 2 ) 2 f ′′ ( t ) + 2 tf ′ ( t ) = 0 . t 9 / 14

Rubel’s proof in one slide − 1 1 − t 2 for − 1 < t < 1 and f ( t ) = 0 otherwise Take f ( t ) = e ( 1 − t 2 ) 2 f ′′ ( t ) + 2 tf ′ ( t ) = 0 . Can do the same with cf ( at + b ) (translation+scaling) t 9 / 14

Rubel’s proof in one slide − 1 1 − t 2 for − 1 < t < 1 and f ( t ) = 0 otherwise Take f ( t ) = e ( 1 − t 2 ) 2 f ′′ ( t ) + 2 tf ′ ( t ) = 0 . Can do the same with cf ( at + b ) (translation+scaling) Can glue together arbitrary many such pieces t 9 / 14

Rubel’s proof in one slide − 1 1 − t 2 for − 1 < t < 1 and f ( t ) = 0 otherwise Take f ( t ) = e ( 1 − t 2 ) 2 f ′′ ( t ) + 2 tf ′ ( t ) = 0 . Can do the same with cf ( at + b ) (translation+scaling) Can glue together arbitrary many such pieces � Can arrange so that f is solution : piecewise pseudo-linear t 9 / 14

Rubel’s proof in one slide − 1 1 − t 2 for − 1 < t < 1 and f ( t ) = 0 otherwise Take f ( t ) = e ( 1 − t 2 ) 2 f ′′ ( t ) + 2 tf ′ ( t ) = 0 . Can do the same with cf ( at + b ) (translation+scaling) Can glue together arbitrary many such pieces � Can arrange so that f is solution : piecewise pseudo-linear t Conclusion : Rubel’s equation allows any piecewise pseudo-linear functions, and those are dense in C 0 9 / 14

Why I don’t like Rubel’s result the solution y is not unique, even with added initial conditions : p ( y , y ′ , . . . , y ( k ) ) = 0 , y ( 0 ) = α 0 , y ′ ( 0 ) = α 1 , . . . , y ( k ) ( 0 ) = α k 10 / 14

Why I don’t like Rubel’s result the solution y is not unique, even with added initial conditions : p ( y , y ′ , . . . , y ( k ) ) = 0 , y ( 0 ) = α 0 , y ′ ( 0 ) = α 1 , . . . , y ( k ) ( 0 ) = α k ...even with a countable number of extra conditions : p ( y , y ′ , . . . , y ( k ) ) = 0 , y ( d i ) ( a i ) = b i , i ∈ N In fact, this is fundamental for the proof to work! 10 / 14

Why I don’t like Rubel’s result the solution y is not unique, even with added initial conditions : p ( y , y ′ , . . . , y ( k ) ) = 0 , y ( 0 ) = α 0 , y ′ ( 0 ) = α 1 , . . . , y ( k ) ( 0 ) = α k ...even with a countable number of extra conditions : p ( y , y ′ , . . . , y ( k ) ) = 0 , y ( d i ) ( a i ) = b i , i ∈ N In fact, this is fundamental for the proof to work! Rubel’s interpretation : this equation is universal My interpretation : this equation allows almost anything 10 / 14

Universal differential equation (PIVP) y 1 ( x ) x Theorem There exists a fixed polynomial p and d ∈ N such that for any conti- nuous functions f and ε , there exists α ∈ R d such that y ′ ( t ) = p ( y ( t )) y ( 0 ) = α, has a unique solution and this solution satisfies such that | y ( t ) − f ( t ) | � ε ( t ) . 11 / 14

Universal differential equation (DAE) y 1 ( x ) x Theorem There exists a fixed polynomial p and k ∈ N such that for any conti- nuous functions f and ε , there exists α 0 , . . . , α k ∈ R such that p ( y , y ′ , . . . , y ( k ) ) = 0 , y ( 0 ) = α 0 , y ′ ( 0 ) = α 1 , . . . , y ( k ) ( 0 ) = α k has a unique analytic solution and this solution satisfies such that | y ( t ) − f ( t ) | � ε ( t ) . 12 / 14

Vague proof idea Key ingredients : fast-growing function (analog) bit generator → On the white board. 13 / 14

A new notion of computability Almost-Theorem f : [ 0 , 1 ] → R is computable if and only if there exists τ > 1, y 0 ∈ R d and p polynomial such that y ′ ( 0 ) = y 0 , y ′ ( t ) = p ( y ( t )) satisfies | f ( x ) − y ( x + n τ ) | � 2 − n , ∀ x ∈ [ 0 , 1 ] , ∀ n ∈ N y ( t ) f ( t mod τ ) t τ 0 1 τ + 1 2 τ 2 τ + 1 3 τ 14 / 14

Conclusion Rubel’s universal differential is very weak We provide a stronger result Another notion of analog computability 15 / 14