Polynomial Time corresponds to Solutions of Polynomial Ordinary Differential Equations of Polynomial Length Olivier Bournez, Daniel Graça and Amaury Pouly July 13, 2015
Main result and consequences Theorem (Informal) PTIME = PIVP of polynomial length PIVP: Ordinary Differential Equations (ODE) with polynomial right-hand side. ◮ Implicit complexity: purely continuous (time and space) characterization of PTIME ◮ Continuous-time models of computations: Turing machines and the GPAC are equivalent at the complexity level
Digital vs analog computers
Digital vs analog computers VS
Let’s model! Physical Computer Model Laptop, ... Turing machines λ -calculus Recursive functions Circuits Discrete dynamical systems Differential Analyzer, ... GPAC Continuous dynamical systems
Let’s model! Physical Computer Model Laptop, ... Turing machines λ -calculus Recursive functions Circuits Discrete dynamical systems Differential Analyzer, ... GPAC Continuous dynamical systems Church Thesis All reasonable models of computation are equivalent.
Let’s model! Physical Computer Model Laptop, ... Turing machines λ -calculus Recursive functions Circuits Discrete dynamical systems Differential Analyzer, ... GPAC Continuous dynamical systems Church Thesis All reasonable models of computation are equivalent. Implicit corollary Some models are too general/unreasonable.
Let’s model! Physical Computer Model Laptop, ... Turing machines λ -calculus Recursive functions Circuits Discrete dynamical systems Differential Analyzer, ... GPAC → reasonable ? Continuous dynamical systems Church Thesis All reasonable models of computation are equivalent. Implicit corollary Some models are too general/unreasonable.
General Purpose Analog Computer (GPAC) ◮ invented by Shannon (1941) ◮ idealization of the Differential Analyzer: ◮ circuits made of: u × uv Multiplier k k Constant v u � � + u + v u u Integrator Adder v
Examples of GPAC Exponential: � � y ( t ) y = y y ( t ) = exp ( t ) � �
Examples of GPAC Exponential: y ′ = y � y ( t ) y ( t ) = exp ( t ) � �
Examples of GPAC Exponential: y ′ = y � y ( t ) y ( t ) = exp ( t ) � � (Co)sine: � � × y 1 ( t ) − 1 y 2 ( t ) � y ′ � y 1 ( t )= sin ( t ) 1 = y 2 � y ′ 2 = − y 1 y 2 ( t )= cos ( t )
Examples of GPAC Rational function: − 2 × � × y 1 ( t ) × � 1 y 2 ( t ) 1 � y ′ 1 = − 2 y 2 y 2 � y 1 ( t )= 1 1 + t 2 � y ′ 2 = 1 y 2 ( t )= t
Examples of GPAC Rational function: − 2 × � × y 1 ( t ) × � 1 y 2 ( t ) 1 � y ′ 1 = − 2 y 2 y 2 � y 1 ( t )= 1 1 + t 2 � y ′ 2 = 1 y 2 ( t )= t Theorem (Graça and Costa) y = ( y 1 , . . . , y d ) is generated by a GPAC iff it satisfies a Polyno- mial Initial Value Problem (PIVP): � y ′ = p ( y ) y ( t 0 )= y 0 where p is a vector of polynomials.
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
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]
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 ) q ( x ) t Shannon’s notion Modern notion sin , cos , exp , log , ... Strictly weaker than Turing machines [Shannon, 1941]
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 ) q ( 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]
Different kinds of equivalence Theorem (Bournez et al) The GPAC is equivalent to Turing machines for computability. ◮ Computability: compute the same functions
Different kinds of equivalence Theorem (Bournez et al) The GPAC is equivalent to Turing machines for computability. ◮ Computability: compute the same functions ◮ Complexity: same functions with same “complexity” Quantum computers Believed different Boolean circuits Turing machines Equivalent Unknown Recursive functions GPACs
Different kinds of equivalence Theorem (Bournez et al) The GPAC is equivalent to Turing machines for computability. ◮ Computability: compute the same functions ◮ Complexity: same functions with same “complexity” Quantum computers Believed different Boolean circuits Turing machines Equivalent Unknown Recursive functions GPACs Main Result of the paper Turing machines and GPACs are equivalent for complexity.
Time complexity for continuous systems ◮ Turing machines: T ( x ) = number of steps to compute on x
Time complexity for continuous systems ◮ Turing machines: T ( x ) = number of steps to compute on x ◮ GPAC: time contraction problem Intuitive definition T ( x , µ ) = first time t so that | y 1 ( t ) − f ( x ) | � e − µ y ′ = p ( y ) y ( 0 ) = q ( x ) y 1 ( t ) f ( x ) g ( x ) t
Time complexity for continuous systems ◮ Turing machines: T ( x ) = number of steps to compute on x ◮ GPAC: time contraction problem Intuitive definition T ( x , µ ) = first time t so that | y 1 ( t ) − f ( x ) | � e − µ y ′ = p ( y ) z ( t ) = y ( e t ) y ( 0 ) = q ( x ) y 1 ( t ) z 1 ( t ) f ( x ) f ( x ) � ˜ g ( x ) g ( x ) t t
Time complexity for continuous systems ◮ Turing machines: T ( x ) = number of steps to compute on x ◮ GPAC: time contraction problem Intuitive definition T ( x , µ ) = first time t so that | y 1 ( t ) − f ( x ) | � e − µ y ′ = p ( y ) z ( t ) = y ( e t ) y ( 0 ) = q ( x ) y 1 ( t ) z 1 ( t ) f ( x ) f ( x ) � ˜ g ( x ) g ( x ) t t w ( t ) = y ( e e t ) w 1 ( t ) f ( x ) � ˆ g ( x ) t
Time complexity for continuous systems ◮ Turing machines: T ( x ) = number of steps to compute on x ◮ GPAC: time contraction problem → open problem Intuitive definition T ( x , µ ) = first time t so that | y 1 ( t ) − f ( x ) | � e − µ y ′ = p ( y ) z ( t ) = y ( e t ) y ( 0 ) = q ( x ) y 1 ( t ) z 1 ( t ) f ( x ) f ( x ) � ˜ g ( x ) g ( x ) t t Observation w ( t ) = y ( e e t ) This definition is broken: all functions have arbitrar- w 1 ( t ) f ( x ) � ily small complexity. ˆ g ( x ) t
Time-space correlation of the GPAC y ′ = p ( y ) y ( 0 ) = q ( x ) y 1 ( t ) f ( x ) q ( x ) t
Time-space correlation of the GPAC y ′ = p ( y ) z ( t ) = y ( e t ) y ( 0 ) = q ( x ) y 1 ( t ) z 1 ( t ) f ( x ) f ( x ) � ˜ q ( x ) q ( x ) t t
Time-space correlation of the GPAC y ′ = p ( y ) z ( t ) = y ( e t ) y ( 0 ) = q ( x ) y 1 ( t ) z 1 ( t ) f ( x ) f ( x ) � ˜ q ( x ) q ( x ) t t extra component: w ( t ) = e t Observation Time scaling costs “space”. w ( t ) t
Time-space correlation of the GPAC y ′ = p ( y ) z ( t ) = y ( e t ) y ( 0 ) = q ( x ) y 1 ( t ) z 1 ( t ) f ( x ) f ( x ) � ˜ q ( x ) q ( x ) t t extra component: w ( t ) = e t Observation Time scaling costs “space”. � Time complexity for the GPAC must involve time and space ! w ( t ) t
Two equivalent notions of complexity � y ( 0 )= q ( x ) y 1 ( t ) y ′ ( t )= p ( y ( t )) f ( x ) f ( x ) = lim t →∞ y 1 ( t ) q ( x ) t
Two equivalent notions of complexity � y ( 0 )= q ( x ) y 1 ( t ) y ′ ( t )= p ( y ( t )) f ( x ) f ( x ) = lim t →∞ y 1 ( t ) q ( x ) t Length based complexity: L ℓ ( t ) = length of y over [ 0 , t ] � t = � p ( y ( u )) � du 0 L ( x , µ ) = length ℓ ( t ) so that � y 1 ( t ) − f ( x ) � � e − µ
Characterization of Turing polynomial time Definition: L ⊆ { 0 , 1 } ∗ is polytime-recognizable iff for all w :
Characterization of Turing polynomial time Definition: L ⊆ { 0 , 1 } ∗ is polytime-recognizable iff for all w : | w | y ′ = p ( y ) � w i 2 − i y ( 0 ) = q ( ψ ( w )) ψ ( w ) = i = 1 satisfies:
Characterization of Turing polynomial time Definition: L ⊆ { 0 , 1 } ∗ is polytime-recognizable iff for all w : | w | y ′ = p ( y ) � w i 2 − i y ( 0 ) = q ( ψ ( w )) ψ ( w ) = i = 1 satisfies: y 1 ( t ) 1 � t 0 � y ′ � = ℓ ( t )= length of y q ( ψ ( w )) over [ 0 , t ] − 1
Characterization of Turing polynomial time Definition: L ⊆ { 0 , 1 } ∗ is polytime-recognizable iff for all w : | w | y ′ = p ( y ) � w i 2 − i y ( 0 ) = q ( ψ ( w )) ψ ( w ) = i = 1 satisfies: accept: w ∈ L y 1 ( t ) 1 � t 0 � y ′ � = ℓ ( t )= length of y computing q ( ψ ( w )) over [ 0 , t ] − 1 if y 1 ( t ) � 1 then w ∈ L 1
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