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Solving Analytic Differential Equations in Polynomial Time over Unbounded Domains Olivier Bournez Daniel S. Graa Amaury Pouly ENS Lyon May 24, 2011 Olivier Bournez, Daniel S. Graa, Amaury Pouly (ENS Lyon) Solving Analytic Differential

  1. Solving Analytic Differential Equations in Polynomial Time over Unbounded Domains Olivier Bournez Daniel S. Graça Amaury Pouly ENS Lyon May 24, 2011 Olivier Bournez, Daniel S. Graça, Amaury Pouly (ENS Lyon) Solving Analytic Differential Equations in Polynomial Time over Unbounded Domains May 24, 2011 −∞ / 17

  2. Outline Computing with reals 1 Introduction GPAC Computable analysis Church Thesis Solving differential equations 2 Preliminary remarks Solving differential equations over C Back to R Olivier Bournez, Daniel S. Graça, Amaury Pouly (ENS Lyon) Solving Analytic Differential Equations in Polynomial Time over Unbounded Domains May 24, 2011 −∞ / 17

  3. Computing with reals Introduction The case of integers Many models: Recursive functions Turing machines λ -calculus circuits . . . Olivier Bournez, Daniel S. Graça, Amaury Pouly (ENS Lyon) Solving Analytic Differential Equations in Polynomial Time over Unbounded Domains May 24, 2011 1 / 17

  4. Computing with reals Introduction The case of integers Many models: Recursive functions Turing machines λ -calculus circuits . . . And Church Thesis All reasonable discrete models of computation are equivalent. Olivier Bournez, Daniel S. Graça, Amaury Pouly (ENS Lyon) Solving Analytic Differential Equations in Polynomial Time over Unbounded Domains May 24, 2011 1 / 17

  5. Computing with reals Introduction The case of analog computations Several models: BSS model (Blum Shub Smale) Computable analysis GPAC (General Purpose Analog Computer) . . . Olivier Bournez, Daniel S. Graça, Amaury Pouly (ENS Lyon) Solving Analytic Differential Equations in Polynomial Time over Unbounded Domains May 24, 2011 2 / 17

  6. Computing with reals Introduction The case of analog computations Several models: BSS model (Blum Shub Smale) Computable analysis GPAC (General Purpose Analog Computer) . . . Questions: Church Thesis for analog computers ? Olivier Bournez, Daniel S. Graça, Amaury Pouly (ENS Lyon) Solving Analytic Differential Equations in Polynomial Time over Unbounded Domains May 24, 2011 2 / 17

  7. Computing with reals Introduction The case of analog computations Several models: BSS model (Blum Shub Smale) Computable analysis GPAC (General Purpose Analog Computer) . . . Questions: Church Thesis for analog computers ? ⇒ No (GPAC � BSS) Olivier Bournez, Daniel S. Graça, Amaury Pouly (ENS Lyon) Solving Analytic Differential Equations in Polynomial Time over Unbounded Domains May 24, 2011 2 / 17

  8. Computing with reals Introduction The case of analog computations Several models: BSS model (Blum Shub Smale) Computable analysis GPAC (General Purpose Analog Computer) . . . Questions: Church Thesis for analog computers ? ⇒ No (GPAC � BSS) Comparison with digital models of computation ? Olivier Bournez, Daniel S. Graça, Amaury Pouly (ENS Lyon) Solving Analytic Differential Equations in Polynomial Time over Unbounded Domains May 24, 2011 2 / 17

  9. Computing with reals Introduction The case of analog computations Several models: BSS model (Blum Shub Smale) Computable analysis GPAC (General Purpose Analog Computer) . . . Questions: Church Thesis for analog computers ? ⇒ No (GPAC � BSS) Comparison with digital models of computation ? ⇒ How ? Olivier Bournez, Daniel S. Graça, Amaury Pouly (ENS Lyon) Solving Analytic Differential Equations in Polynomial Time over Unbounded Domains May 24, 2011 2 / 17

  10. Computing with reals Introduction The case of analog computations Several models: BSS model (Blum Shub Smale) Computable analysis GPAC (General Purpose Analog Computer) . . . Questions: Church Thesis for analog computers ? ⇒ No (GPAC � BSS) Comparison with digital models of computation ? ⇒ How ? What is a reasonable model ? Olivier Bournez, Daniel S. Graça, Amaury Pouly (ENS Lyon) Solving Analytic Differential Equations in Polynomial Time over Unbounded Domains May 24, 2011 2 / 17

  11. Computing with reals GPAC GPAC General Purpose Analog Computer by Claude Shanon (1941) idealization of an analog computer: Differential Analyzer circuit by from: u k k + u + v v A constant unit An adder unit u u � × � w ′ ( t ) = u ( t ) v ′ ( t ) uv w v v w ( t 0 ) = α An integrator unit A multiplier unit Olivier Bournez, Daniel S. Graça, Amaury Pouly (ENS Lyon) Solving Analytic Differential Equations in Polynomial Time over Unbounded Domains May 24, 2011 3 / 17

  12. Computing with reals GPAC GPAC General Purpose Analog Computer by Claude Shanon (1941) idealization of an analog computer: Differential Analyzer circuit by from: u k k + u + v v A constant unit An adder unit u u � × � w ′ ( t ) = u ( t ) v ′ ( t ) uv w v v w ( t 0 ) = α An integrator unit A multiplier unit Olivier Bournez, Daniel S. Graça, Amaury Pouly (ENS Lyon) Solving Analytic Differential Equations in Polynomial Time over Unbounded Domains May 24, 2011 3 / 17

  13. Computing with reals GPAC GPAC General Purpose Analog Computer by Claude Shanon (1941) idealization of an analog computer: Differential Analyzer circuit by from: u k k + u + v v A constant unit An adder unit u u � × � w ′ ( t ) = u ( t ) v ′ ( t ) uv w v v w ( t 0 ) = α An integrator unit A multiplier unit Olivier Bournez, Daniel S. Graça, Amaury Pouly (ENS Lyon) Solving Analytic Differential Equations in Polynomial Time over Unbounded Domains May 24, 2011 3 / 17

  14. Computing with reals GPAC GPAC: examples Example (Exponential) � y ′ = y � e t • y ( 0 )= 1 t Olivier Bournez, Daniel S. Graça, Amaury Pouly (ENS Lyon) Solving Analytic Differential Equations in Polynomial Time over Unbounded Domains May 24, 2011 4 / 17

  15. Computing with reals GPAC GPAC: examples Example (Exponential) � y ′ = y � e t • y ( 0 )= 1 t Example (Nonlinear) • × � y ′ = − 2 ty 2 × � 1 t • 1+ t 2 y ( 0 )= 1 − 2 × • Olivier Bournez, Daniel S. Graça, Amaury Pouly (ENS Lyon) Solving Analytic Differential Equations in Polynomial Time over Unbounded Domains May 24, 2011 4 / 17

  16. Computing with reals GPAC GPAC: beyond the circuit approach Theorem y is generated by a GPAC iff it is a component of the solution y = ( y 1 , . . . , y d ) of the ordinary differential equation (ODE): � ˙ y = p ( y ) y ( t 0 )= y 0 where p is a vector or polynomials. Example (Counter-example) y = 1 � ˙ y y ( 0 )= 1 Olivier Bournez, Daniel S. Graça, Amaury Pouly (ENS Lyon) Solving Analytic Differential Equations in Polynomial Time over Unbounded Domains May 24, 2011 5 / 17

  17. Computing with reals GPAC GPAC: beyond the circuit approach Theorem y is generated by a GPAC iff it is a component of the solution y = ( y 1 , . . . , y d ) of the ordinary differential equation (ODE): � ˙ y = p ( y ) y ( t 0 )= y 0 where p is a vector or polynomials. Example (Counter-example) y = 1 � ˙ y y ( 0 )= 1 Olivier Bournez, Daniel S. Graça, Amaury Pouly (ENS Lyon) Solving Analytic Differential Equations in Polynomial Time over Unbounded Domains May 24, 2011 5 / 17

  18. Computing with reals Computable analysis Computable real Definition A real r ∈ R is computable is one can compute an arbitrary close approximation for a given precision: Given p ∈ N , compute r p s.t. | r − r p | � 2 − p Olivier Bournez, Daniel S. Graça, Amaury Pouly (ENS Lyon) Solving Analytic Differential Equations in Polynomial Time over Unbounded Domains May 24, 2011 6 / 17

  19. Computing with reals Computable analysis Computable real Definition A real r ∈ R is computable is one can compute an arbitrary close approximation for a given precision: Given p ∈ N , compute r p s.t. | r − r p | � 2 − p Olivier Bournez, Daniel S. Graça, Amaury Pouly (ENS Lyon) Solving Analytic Differential Equations in Polynomial Time over Unbounded Domains May 24, 2011 6 / 17

  20. Computing with reals Computable analysis Computable real Definition A real r ∈ R is computable is one can compute an arbitrary close approximation for a given precision: Given p ∈ N , compute r p s.t. | r − r p | � 2 − p Example Rationals, π , e , . . . Olivier Bournez, Daniel S. Graça, Amaury Pouly (ENS Lyon) Solving Analytic Differential Equations in Polynomial Time over Unbounded Domains May 24, 2011 6 / 17

  21. Computing with reals Computable analysis Computable real Definition A real r ∈ R is computable is one can compute an arbitrary close approximation for a given precision: Given p ∈ N , compute r p s.t. | r − r p | � 2 − p Example Rationals, π , e , . . . Counter-Example ∞ � d n 2 − n r = n = 0 where d n = 1 ⇔ the n th Turing Machine halts on input n Olivier Bournez, Daniel S. Graça, Amaury Pouly (ENS Lyon) Solving Analytic Differential Equations in Polynomial Time over Unbounded Domains May 24, 2011 6 / 17

  22. Computing with reals Computable analysis Computable function Definition A function f : R → R is computable if there exist a Turing Machine M s.t. for any x ∈ R and oracle O computing x , M O computes f ( x ) . Definition (Simplified) A function f : R → R is computable if f is continuous and for a any rational r one can compute f ( r ) . Olivier Bournez, Daniel S. Graça, Amaury Pouly (ENS Lyon) Solving Analytic Differential Equations in Polynomial Time over Unbounded Domains May 24, 2011 7 / 17

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