Continuous models of computation: computability, complexity, universality Amaury Pouly Joint work with Olivier Bournez and Daniel Graça 21 january 2019 1 / 21
What is a computer? 2 / 21
What is a computer? 2 / 21
What is a computer? VS 2 / 21
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 / 21
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 / 21
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 / 21
Example of dynamical system ℓ θ m g θ + g ¨ ℓ sin( θ ) = 0 5 / 21
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 ) 5 / 21
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 ) 5 / 21
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 ) Historical remark : the word “analog” The pendulum and the circuit have the same equation. One can study one using the other by analogy. 5 / 21
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 6 / 21
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] 6 / 21
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] 6 / 21
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] 6 / 21
Equivalence with computable analysis Definition (Bournez et al, 2007) f computable by GPAC if ∃ p polynomial such that ∀ x ∈ [ a , b ] y ′ ( t ) = p ( y ( t )) y ( 0 ) = ( x , 0 , . . . , 0 ) satisfies | f ( x ) − y 1 ( t ) | � y 2 ( t ) et y 2 ( t ) − t →∞ 0. − − → y 1 ( t ) − t →∞ f ( x ) − − → y 1 ( t ) f ( x ) y 2 ( t ) = error bound x t 7 / 21
Equivalence with computable analysis Definition (Bournez et al, 2007) f computable by GPAC if ∃ p polynomial such that ∀ x ∈ [ a , b ] y ′ ( t ) = p ( y ( t )) y ( 0 ) = ( x , 0 , . . . , 0 ) satisfies | f ( x ) − y 1 ( t ) | � y 2 ( t ) et y 2 ( t ) − t →∞ 0. − − → y 1 ( t ) − t →∞ f ( x ) − − → y 1 ( t ) f ( x ) y 2 ( t ) = error bound x t Theorem (Bournez et al, 2007) f : [ a , b ] → R computable 1 ⇔ f computable by GPAC 7 / 21
Equivalence with computable analysis Definition (Bournez et al, 2007) f computable by GPAC if ∃ p polynomial such that ∀ x ∈ [ a , b ] y ′ ( t ) = p ( y ( t )) y ( 0 ) = ( x , 0 , . . . , 0 ) satisfies | f ( x ) − y 1 ( t ) | � y 2 ( t ) et y 2 ( t ) − t →∞ 0. − − → y 1 ( t ) − t →∞ f ( x ) − − → y 1 ( t ) f ( x ) y 2 ( t ) = error bound x t Theorem (Bournez et al, 2007) f : [ a , b ] → R computable 1 ⇔ f computable by GPAC 1. In Computable Analysis, a standard model over reals built from Turing machines. 7 / 21
Complexity of analog systems ◮ Turing machines : T ( x ) = number of steps to compute on x 8 / 21
Complexity of analog systems ◮ Turing machines : T ( x ) = number of steps to compute on x ◮ GPAC : Tentative definition T ( x ) = ?? y ′ = p ( y ) y ( 0 ) = ( x , 0 , . . . , 0 ) y 1 ( t ) f ( x ) x t 8 / 21
Complexity of analog systems ◮ Turing machines : T ( x ) = number of steps to compute on x ◮ GPAC : Tentative definition T ( x , µ ) = y ′ = p ( y ) y ( 0 ) = ( x , 0 , . . . , 0 ) y 1 ( t ) f ( x ) x t 8 / 21
Complexity of analog systems ◮ Turing machines : T ( x ) = number of steps to compute on x ◮ GPAC : Tentative definition T ( x , µ ) = first time t so that | y 1 ( t ) − f ( x ) | � e − µ y ′ = p ( y ) y ( 0 ) = ( x , 0 , . . . , 0 ) y 1 ( t ) f ( x ) x t 8 / 21
Complexity of analog systems ◮ Turing machines : T ( x ) = number of steps to compute on x ◮ GPAC : Tentative 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 ) = ( x , 0 , . . . , 0 ) y 1 ( t ) z 1 ( t ) f ( x ) f ( x ) � x x t t 8 / 21
Complexity of analog systems ◮ Turing machines : T ( x ) = number of steps to compute on x ◮ GPAC : Tentative 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 ) = ( x , 0 , . . . , 0 ) y 1 ( t ) z 1 ( t ) f ( x ) f ( x ) � x x t t w ( t ) = y ( e e t ) w 1 ( t ) f ( x ) x t 8 / 21
Complexity of analog systems ◮ Turing machines : T ( x ) = number of steps to compute on x ◮ GPAC : time contraction problem → open problem Tentative 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 ) = ( x , 0 , . . . , 0 ) y 1 ( t ) z 1 ( t ) f ( x ) f ( x ) � x x t t w ( t ) = y ( e e t ) Something is wrong... w 1 ( t ) f ( x ) x All functions have constant t time complexity. 8 / 21
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 9 / 21
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 w ( t ) t 9 / 21
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 w ( t ) must involve time and space! t 9 / 21
Characterization of polynomial time Definition : L ∈ ANALOG-PTIME ⇔ ∃ p polynomial, ∀ word w | w | y ′ = p ( y ) � w i 2 − i y ( 0 ) = ( ψ ( w ) , | w | , 0 , . . . , 0 ) ψ ( w ) = i = 1 y 1 ( t ) 1 ψ ( w ) ℓ ( t ) = length of y − 1 10 / 21
Characterization of polynomial time Definition : L ∈ ANALOG-PTIME ⇔ ∃ p polynomial, ∀ word w | w | y ′ = p ( y ) � w i 2 − i y ( 0 ) = ( ψ ( w ) , | w | , 0 , . . . , 0 ) ψ ( w ) = i = 1 accept : w ∈ L y 1 ( t ) 1 ψ ( w ) ℓ ( t ) = length of y computing − 1 satisfies 1. if y 1 ( t ) � 1 then w ∈ L 10 / 21
Characterization of polynomial time Definition : L ∈ ANALOG-PTIME ⇔ ∃ p polynomial, ∀ word w | w | y ′ = p ( y ) � w i 2 − i y ( 0 ) = ( ψ ( w ) , | w | , 0 , . . . , 0 ) ψ ( w ) = i = 1 accept : w ∈ L 1 ψ ( w ) ℓ ( t ) = length of y computing − 1 y 1 ( t ) reject : w / ∈ L satisfies 2. if y 1 ( t ) � − 1 then w / ∈ L 10 / 21
Characterization of polynomial time Definition : L ∈ ANALOG-PTIME ⇔ ∃ p polynomial, ∀ word w | w | y ′ = p ( y ) � w i 2 − i y ( 0 ) = ( ψ ( w ) , | w | , 0 , . . . , 0 ) ψ ( w ) = i = 1 accept : w ∈ L 1 forbidden y 1 ( t ) ψ ( w ) ℓ ( t ) = length of y poly( | w | ) computing − 1 reject : w / ∈ L satisfies 3. if ℓ ( t ) � poly( | w | ) then | y 1 ( t ) | � 1 10 / 21
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