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Computational Complexity of the GPAC Amaury Pouly Joint work with Olivier Bournez and Daniel Graça April 10, 2014 Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 −∞ / 17

Outline Introduction 1 GPAC Computable Analysis Analog Church Thesis Complexity Toward a Complexity Theory for the GPAC 2 What is the problem Computational Complexity (Real Number) 3 Conclusion Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 −∞ / 17

Introduction GPAC GPAC General Purpose Analog Computer by Claude Shanon (1941) Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 1 / 17

Introduction GPAC GPAC General Purpose Analog Computer by Claude Shanon (1941) idealization of an analog computer: Differential Analyzer Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 1 / 17

Introduction GPAC GPAC General Purpose Analog Computer by Claude Shanon (1941) idealization of an analog computer: Differential Analyzer circuit built from: u + u + v k k v A constant unit An adder unit u u � � × uv u dv v v An integrator unit An multiplier unit Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 1 / 17

Introduction 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 Polynomial Initial Value Problem (PIVP): � y ′ = p ( y ) y ( t 0 )= y 0 where p is a vector of polynomials. Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 2 / 17

Introduction 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 Polynomial Initial Value Problem (PIVP): � y ′ = p ( y ) y ( t 0 )= y 0 where p is a vector of polynomials. Remark continuous dynamical system Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 2 / 17

Introduction 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 Polynomial Initial Value Problem (PIVP): � y ′ = p ( y ) y ( t 0 )= y 0 where p is a vector of polynomials. Remark continuous dynamical system the GPAC is just one reason to look at them a a Ask question Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 2 / 17

Introduction GPAC GPAC: examples Example (One variable, linear system) � y ′ = y e t � t y ( 0 )= 1 Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 3 / 17

Introduction GPAC GPAC: examples Example (One variable, linear system) � y ′ = y � e t t y ( 0 )= 1 Example (One variable, nonlinear system) × � y ′ = − 2 ty 2 × 1 � − 2 1 + t 2 y ( 0 )= 1 × t Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 3 / 17

Introduction GPAC GPAC: examples Example (One variable, linear system) � y ′ = y � e t t y ( 0 )= 1 Example (Two variable, nonlinear system) y ′ = − 2 ty 2  ×   y ( 0 )= 1 ×  1 � t ′ = 1 − 2 1 + t 2 ×   t ( 0 )= 0  t Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 3 / 17

Introduction GPAC GPAC: examples Example (Two variables, linear system) y ′ = z   z ′ = − y   � � − 1 × sin ( t ) t y ( 0 )= 0   z ( 0 )= 1  Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 4 / 17

Introduction GPAC GPAC: examples Example (Two variables, linear system) y ′ = z   z ′ = − y   � � × sin ( t ) t − 1 y ( 0 )= 0   z ( 0 )= 1  Example (Not so nice example) y ′  1 = y 1   y ′ 2 = y 2 y ′   1 . . . � � � y n ( t ) t . . .    y ′ n = y n y ′  n integrators n − 1 Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 4 / 17

Introduction GPAC GPAC: examples Example (Two variables, linear system) y ′ = z   z ′ = − y   � � − 1 × sin ( t ) t y ( 0 )= 0   z ( 0 )= 1  Example (Not so nice example) y ′  1 = y 1  y ′  2 = y 2 y 1   . . . � � � y n ( t ) t . . .    y ′ n = y n y n − 1 · · · y 2 y 1  n integrators Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 4 / 17

Introduction GPAC GPAC: examples Example (Two variables, linear system) y ′ = z   z ′ = − y   � � × sin ( t ) t − 1 y ( 0 )= 0   z ( 0 )= 1  Example (Not so nice example)  y 1 ( t )= e t   y 2 ( t )= e e t    . . . � � � y n ( t ) t . . .  y n ( t )= e e ... t    n integrators  Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 4 / 17

Introduction GPAC Motivation Study the computational power of such systems: 1 Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 5 / 17

Introduction GPAC Motivation Study the computational power of such systems: 1 (asymptotical) (properties of) solutions Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 5 / 17

Introduction GPAC Motivation Study the computational power of such systems: 1 (asymptotical) (properties of) solutions reachability properties Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 5 / 17

Introduction GPAC Motivation Study the computational power of such systems: 1 (asymptotical) (properties of) solutions reachability properties attractors Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 5 / 17

Introduction GPAC Motivation Study the computational power of such systems: 1 (asymptotical) (properties of) solutions reachability properties attractors Use these systems as a model of computation 2 Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 5 / 17

Introduction GPAC Motivation Study the computational power of such systems: 1 (asymptotical) (properties of) solutions reachability properties attractors Use these systems as a model of computation 2 on words Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 5 / 17

Introduction GPAC Motivation Study the computational power of such systems: 1 (asymptotical) (properties of) solutions reachability properties attractors Use these systems as a model of computation 2 on words on real numbers Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 5 / 17

Introduction Computable Analysis Computable real Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 6 / 17

Introduction Computable Analysis Computable real Definition (Computable Real) A real r ∈ R is computable is one can compute an arbitrary close ap- proximation for a given precision: Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 6 / 17

Introduction Computable Analysis Computable real Definition (Computable Real) A real r ∈ R is computable is one can compute an arbitrary close ap- proximation for a given precision: Given p ∈ N , compute r p s.t. | r − r p | � 2 − p Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 6 / 17

Introduction Computable Analysis Computable real Definition (Computable Real) A real r ∈ R is computable is one can compute an arbitrary close ap- proximation for a given precision: Given p ∈ N , compute r p s.t. | r − r p | � 2 − p Example Rational numbers, π , e , . . . Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 6 / 17

Introduction Computable Analysis Computable real Definition (Computable Real) A real r ∈ R is computable is one can compute an arbitrary close ap- proximation for a given precision: Given p ∈ N , compute r p s.t. | r − r p | � 2 − p Example Rational numbers, π , e , . . . Example (Counter-Example) ∞ � d n 2 − n r = n = 0 where d n = 1 ⇔ the n th Turing Machine halts on input n Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 6 / 17

Introduction Computable Analysis Computable function Definition (Computable Function) 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 ) . Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 7 / 17

Introduction Computable Analysis Computable function Definition (Computable Function) 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 (Equivalent) A function f : R → R is computable if f is continuous and for a any rational r one can compute f ( r ) . Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 7 / 17