Dynamical Systems: Computability, Verification, Analysis Amaury Pouly Journée des nouveaux arrivants, IRIF 15 october 2018 1 / 22
Trajectory 2011 Master : ENS Lyon 2015 PhD : LIX, Polytechnique and University of Algarve, Portugal Olivier Bournez and Daniel S. Graça 2016 Postdoc : Oxford Joel Ouaknine and James Worrell 2017 Postdoc : Max Planck Institute for Software Systems Joel Ouaknine 3 / 22
Trajectory 2011 Master : ENS Lyon 2015 PhD : LIX, Polytechnique and University of Algarve, Portugal Olivier Bournez and Daniel S. Graça 2016 Postdoc : Oxford Joel Ouaknine and James Worrell 2017 Postdoc : Max Planck Institute for Software Systems Joel Ouaknine 2018 Attracted to IRIF : starting 1st January 3 / 22
Research Interests Verification � Decidability � Dynamical questions in Complexity Systems with continuous space Models of Computable Computation Analysis 4 / 22
Research Interests Verification � Decidability � Dynamical questions in Complexity Systems with continuous space Models of Computable Computation Analysis 4 / 22
What is a computer? 5 / 22
What is a computer? 5 / 22
What is a computer? VS Differential Analyser Admiralty Fire Control Table British Navy ships (WW2) “Mathematica of the 1920s” 5 / 22
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. 6 / 22
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. 6 / 22
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 7 / 22
Example of dynamical system ℓ θ m g θ + g ¨ ℓ sin( θ ) = 0 8 / 22
Example of dynamical system ℓ θ m y ′ 1 = y 2 y 1 = θ g 2 = − g y 2 = ˙ y ′ l y 3 θ ⇔ y ′ 3 = y 2 y 4 y 3 = sin( θ ) θ + g ¨ ℓ sin( θ ) = 0 y ′ 4 = − y 2 y 3 y 4 = cos( θ ) 8 / 22
Example of dynamical system y 2 � � y 1 × ℓ y 3 y 4 − g � × ℓ θ m � × × − 1 y ′ 1 = y 2 y 1 = θ g 2 = − g y 2 = ˙ y ′ l y 3 θ ⇔ y ′ 3 = y 2 y 4 y 3 = sin( θ ) θ + g ¨ ℓ sin( θ ) = 0 y ′ 4 = − y 2 y 3 y 4 = cos( θ ) 8 / 22
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 9 / 22
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] 9 / 22
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] 9 / 22
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] 9 / 22
Highlights of some results ANALOG-P R ANALOG-PTIME w ∈L y 1 ( t ) 1 y 1 ( t ) f ( x ) y 1 ( t ) ψ ( w ) ℓ ( t ) poly( | w | ) x − 1 ℓ ( t ) y 1 ( t ) w / ∈L Theorem ◮ PTIME = ANALOG-PTIME ◮ f : [ a , b ] → R computable in polynomial time ⇔ f ∈ ANALOG-P R ◮ Analog complexity theory based on length ◮ Time of Turing machine ⇔ length of the GPAC ◮ Purely continuous characterization of PTIME 10 / 22
Research Interests Verification � Decidability � Dynamical questions in Complexity Systems with continuous space Models of Computable Computation Analysis 11 / 22
A word on computability for real functions Classical computability (Turing machine) : compute on words, integers, rationals, ... 12 / 22
A word on computability for real functions Classical computability (Turing machine) : compute on words, integers, rationals, ... Real computability : 12 / 22
A word on computability for real functions Classical computability (Turing machine) : compute on words, integers, rationals, ... Real computability : at least two different notions ◮ BSS (Blum-Shub-Smale) machine : register machine that can store arbitrary real numbers and that can compute rational functions over reals at unit cost. Comparisons between reals are allowed. 12 / 22
A word on computability for real functions Classical computability (Turing machine) : compute on words, integers, rationals, ... Real computability : at least two different notions ◮ BSS (Blum-Shub-Smale) machine : register machine that can store arbitrary real numbers and that can compute rational functions over reals at unit cost. Comparisons between reals are allowed. ◮ Computable Analysis : reals are represented as converging Cauchy sequences, computations are carried out by rational approximations using Turing machines. Comparisons between reals is not decidable in general. Computable implies continuous. 12 / 22
Computable Analysis : differential equations Let f : R n → R n continuous, consider y ′ = f ( y ) y ( 0 ) = x , (1) Question When is y computable? What about its complexity? y ( t ) y ( t ) x x 13 / 22
Computable Analysis : differential equations Let f : R n → R n continuous, consider y ′ = f ( y ) y ( 0 ) = x , (1) It can be very bad : Theorem (Pour-El and Richards) There exists a computable (hence continuous) f such that none of the solutions to (1) is computable. y ( t ) y ( t ) x x 13 / 22
Computable Analysis : differential equations Let f : R n → R n continuous, consider y ′ = f ( y ) y ( 0 ) = x , (1) Some good news : Theorem (Ruohonen) If f is computable and (1) has a unique solution, then it is computable. But complexity can be unbounded y ( t ) y ( t ) x x 13 / 22
Computable Analysis : differential equations Let f : R n → R n continuous, consider y ′ = f ( y ) y ( 0 ) = x , (1) Still things are bad : Theorem (Buescu, Campagnolo and Graça) Computing the maximum interval of life (or deciding if it is bounded) is undecidable, even if f is a polynomial. y ( t ) y ( t ) x x 13 / 22
Computable Analysis : differential equations Let f : R n → R n continuous, consider y ′ = f ( y ) y ( 0 ) = x , (1) A new hope : Theorem If y ( t ) exists, we can compute r ∈ Q such | r − y ( t ) | � 2 − n in time poly ( size of x and p , n , ℓ ( t )) where ℓ ( t ) ≈ length of the curve y (between x and y ( t ) ) y ( t ) y ( t ) x x 13 / 22
Research Interests Verification � Decidability � Dynamical questions in Complexity Systems with continuous space Models of Computable Computation Analysis 14 / 22
Example : 2D robot ℓ θ Available actions : ◮ rotate arm ◮ change arm length 15 / 22
Example : 2D robot ( x , y ) State : X = ( x θ , y θ , x , y ) ∈ R 4 ℓ ( x θ , y θ ) θ Available actions : ◮ rotate arm ◮ change arm length 15 / 22
Example : 2D robot ( x , y ) State : X = ( x θ , y θ , x , y ) ∈ R 4 ℓ Rotate arm (increase θ ) : ( x θ , y θ ) � ′ � x � 0 � � x � − 1 = θ y 1 0 y � ′ � x θ � 0 � � x θ � − 1 = y θ 1 0 y θ Available actions : ◮ rotate arm ◮ change arm length 15 / 22
Example : 2D robot ( x , y ) State : X = ( x θ , y θ , x , y ) ∈ R 4 ℓ Rotate arm (increase θ ) : ( x θ , y θ ) � ′ � x � 0 � � x � − 1 = θ y 1 0 y � ′ � x θ � 0 � � x θ � − 1 = y θ 1 0 y θ Available actions : ◮ rotate arm Change arm length (increase ℓ ) : ◮ change arm length � ′ � x � x θ � = y y θ 15 / 22
Example : 2D robot ( x , y ) State : X = ( x θ , y θ , x , y ) ∈ R 4 ℓ Rotate arm (increase θ ) : ( x θ , y θ ) � ′ � x � 0 � � x � − 1 = θ y 1 0 y � ′ � x θ � 0 � � x θ � − 1 = y θ 1 0 y θ Available actions : ◮ rotate arm Change arm length (increase ℓ ) : ◮ change arm length � ′ � x � x θ � = → Switched linear system : y y θ X ′ = AX where A ∈ { A rot , A arm } . 15 / 22
Example : mass-spring-damper system State : X = z ∈ R Equation of motion : b k mz ′′ = − kz − bz ′ + mg + u m u ( t ) Model with external input u ( t ) 16 / 22
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