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Computational Complexity of the GPAC Amaury Pouly Joint work with - - PowerPoint PPT Presentation

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Computational Complexity of the GPAC Amaury Pouly Joint work with Olivier Bournez and Daniel Graa April 10, 2014 Pouly, Bournez, Graa Computational Complexity of the GPAC April 10, 2014 / 17 Outline Introduction 1 GPAC

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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

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Outline

1

Introduction GPAC Computable Analysis Analog Church Thesis Complexity

2

Toward a Complexity Theory for the GPAC What is the problem Computational Complexity (Real Number)

3

Conclusion

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 −∞ / 17

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Introduction GPAC

GPAC

General Purpose Analog Computer by Claude Shanon (1941)

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 1 / 17

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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

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Introduction GPAC

GPAC

General Purpose Analog Computer by Claude Shanon (1941) idealization of an analog computer: Differential Analyzer circuit built from: k k A constant unit + u + v An adder unit u v × uv An multiplier unit u v

  • u dv

An integrator unit u v

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 1 / 17

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Introduction GPAC

GPAC: beyond the circuit approach

Theorem y is generated by a GPAC iff it is a component of the solution y = (y1, . . . , yd) of the Polynomial Initial Value Problem (PIVP): y′ = p(y) y(t0)= y0 where p is a vector of polynomials.

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 2 / 17

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Introduction GPAC

GPAC: beyond the circuit approach

Theorem y is generated by a GPAC iff it is a component of the solution y = (y1, . . . , yd) of the Polynomial Initial Value Problem (PIVP): y′ = p(y) y(t0)= y0 where p is a vector of polynomials. Remark continuous dynamical system

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 2 / 17

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Introduction GPAC

GPAC: beyond the circuit approach

Theorem y is generated by a GPAC iff it is a component of the solution y = (y1, . . . , yd) of the Polynomial Initial Value Problem (PIVP): y′ = p(y) y(t0)= y0 where p is a vector of polynomials. Remark continuous dynamical system the GPAC is just one reason to look at thema

aAsk question Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 2 / 17

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Introduction GPAC

GPAC: examples

Example (One variable, linear system)

  • et

y′ = y y(0)= 1 t

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 3 / 17

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Introduction GPAC

GPAC: examples

Example (One variable, linear system)

  • et

y′ = y y(0)= 1 t Example (One variable, nonlinear system) × × −2 ×

  • 1

1+t2

y′ = −2ty2 y(0)= 1 t

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 3 / 17

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Introduction GPAC

GPAC: examples

Example (One variable, linear system)

  • et

y′ = y y(0)= 1 t Example (Two variable, nonlinear system) × × −2 ×

  • 1

1+t2

       y′ = −2ty2 y(0)= 1 t′ = 1 t(0)= 0 t

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 3 / 17

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Introduction GPAC

GPAC: examples

Example (Two variables, linear system) −1 ×

  • sin(t)

       y′ = z z′ = −y y(0)= 0 z(0)= 1 t

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 4 / 17

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Introduction GPAC

GPAC: examples

Example (Two variables, linear system) −1 ×

  • sin(t)

       y′ = z z′ = −y y(0)= 0 z(0)= 1 t Example (Not so nice example)

  • . . .
  • t

yn(t)          y′

1= y1

y′

2= y2y′ 1

. . . y′

n= yny′ n−1

n integrators

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 4 / 17

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Introduction GPAC

GPAC: examples

Example (Two variables, linear system) −1 ×

  • sin(t)

       y′ = z z′ = −y y(0)= 0 z(0)= 1 t Example (Not so nice example)

  • . . .
  • t

yn(t)          y′

1= y1

y′

2= y2y1

. . . y′

n= ynyn−1 · · · y2y1

n integrators

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 4 / 17

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Introduction GPAC

GPAC: examples

Example (Two variables, linear system) −1 ×

  • sin(t)

       y′ = z z′ = −y y(0)= 0 z(0)= 1 t Example (Not so nice example)

  • . . .
  • t

yn(t)            y1(t)= et y2(t)= eet . . . yn(t)= ee...t n integrators

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 4 / 17

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Introduction GPAC

Motivation

1

Study the computational power of such systems:

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 5 / 17

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Introduction GPAC

Motivation

1

Study the computational power of such systems:

(asymptotical) (properties of) solutions

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 5 / 17

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Introduction GPAC

Motivation

1

Study the computational power of such systems:

(asymptotical) (properties of) solutions reachability properties

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 5 / 17

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Introduction GPAC

Motivation

1

Study the computational power of such systems:

(asymptotical) (properties of) solutions reachability properties attractors

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 5 / 17

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Introduction GPAC

Motivation

1

Study the computational power of such systems:

(asymptotical) (properties of) solutions reachability properties attractors

2

Use these systems as a model of computation

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 5 / 17

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Introduction GPAC

Motivation

1

Study the computational power of such systems:

(asymptotical) (properties of) solutions reachability properties attractors

2

Use these systems as a model of computation

  • n words

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 5 / 17

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Introduction GPAC

Motivation

1

Study the computational power of such systems:

(asymptotical) (properties of) solutions reachability properties attractors

2

Use these systems as a model of computation

  • n words
  • n real numbers

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 5 / 17

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Introduction Computable Analysis

Computable real

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 6 / 17

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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

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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 rp s.t. |r − rp| 2−p

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 6 / 17

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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 rp s.t. |r − rp| 2−p Example Rational numbers, π, e, . . .

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 6 / 17

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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 rp s.t. |r − rp| 2−p Example Rational numbers, π, e, . . . Example (Counter-Example) r =

∞

  • n=0

dn2−n where dn = 1 ⇔ the nth Turing Machine halts on input n

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 6 / 17

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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, MO computes f(x).

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 7 / 17

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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, MO 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

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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, MO 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). Example Polynomials, trigonometric functions, e·, √·, . . .

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 7 / 17

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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, MO 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). Example Polynomials, trigonometric functions, e·, √·, . . . Example (Counter-Example) f(x) = ⌈x⌉

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 7 / 17

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Introduction Analog Church Thesis

Computable Analysis = GPAC ?

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 8 / 17

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Introduction Analog Church Thesis

Computable Analysis = GPAC ?

Seems not:

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 8 / 17

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Introduction Analog Church Thesis

Computable Analysis = GPAC ?

Seems not: Solutions of a GPAC are analytic

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 8 / 17

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Introduction Analog Church Thesis

Computable Analysis = GPAC ?

Seems not: Solutions of a GPAC are analytic x → |x| is computable but not analytic Theorem ( ) Computable Analysis = General Purpose Analog Computer

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 8 / 17

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Introduction Analog Church Thesis

Computable Analysis = GPAC ?

Seems not: Solutions of a GPAC are analytic x → |x| is computable but not analytic Theorem ( ) Computable Analysis = General Purpose Analog Computer Can we fix this ?

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 8 / 17

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Introduction Analog Church Thesis

GPAC: back to the basics

Definition y is generated by a GPAC iff it is a component of the solution y = (y1, . . . , yd) of the ordinary differential equation (ODE): y′ = p(y) y(t0)= y0 where p is a vector of polynomials

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 9 / 17

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Introduction Analog Church Thesis

GPAC: back to the basics

Definition y is generated by a GPAC iff it is a component of the solution y = (y1, . . . , yd) of the ordinary differential equation (ODE): y′ = p(y) y(t0)= y0 where p is a vector of polynomials Definition f is computable by a GPAC iff for all x ∈ R the solution y = (y1, . . . , yd)

  • f the ordinary differential equation (ODE):

y′ = p(y) y(t0)= q(x) where p,q is a vector of polynomials satisfies for all f(x) = limt→∞ y1(t).

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 9 / 17

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Introduction Analog Church Thesis

GPAC: back to the basics

Definition f is computable by a GPAC iff for all x ∈ R the solution y = (y1, . . . , yd)

  • f the ordinary differential equation (ODE):

y′ = p(y) y(t0)= q(x) where p,q is a vector of polynomials satisfies for all f(x) = limt→∞ y1(t). Example t f(x) q(x) y(t)

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 9 / 17

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Introduction Analog Church Thesis

Computable Analysis = GPAC ? (again)

Theorem ( ) The GPAC-computable functions are exactly the computable functions

  • f the Computable Analysis.

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 10 / 17

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Introduction Analog Church Thesis

Computable Analysis = GPAC ? (again)

Theorem ( ) The GPAC-computable functions are exactly the computable functions

  • f the Computable Analysis.

Proof.

Any solution to a PIVP is computable + convergence

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 10 / 17

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Introduction Analog Church Thesis

Computable Analysis = GPAC ? (again)

Theorem ( ) The GPAC-computable functions are exactly the computable functions

  • f the Computable Analysis.

Proof.

Any solution to a PIVP is computable + convergence Simulate a Turing machine with a GPACa

aDetails on blackboard Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 10 / 17

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Introduction Complexity

What about complexity ?

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 11 / 17

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Introduction Complexity

What about complexity ?

Computable Analysis: nice complexity theory (from Turing Machines)

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 11 / 17

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Introduction Complexity

What about complexity ?

Computable Analysis: nice complexity theory (from Turing Machines) General Purpose Analog Computer: nothing

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 11 / 17

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Introduction Complexity

What about complexity ?

Computable Analysis: nice complexity theory (from Turing Machines) General Purpose Analog Computer: nothing

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 11 / 17

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Introduction Complexity

What about complexity ?

Computable Analysis: nice complexity theory (from Turing Machines) General Purpose Analog Computer: nothing Conjecture ( ) Computable Analysis = General Purpose Analog Computer, at the com- plexity level

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 11 / 17

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Toward a Complexity Theory for the GPAC What is the problem

Time Scaling

System #1 #2 ODE y′(t) = p(y(t)) y(1) = y0        z′(t) = u(t)p(z(t)) u′(t) = u(t) z(t0) = y0 u(1) = 1

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 12 / 17

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Toward a Complexity Theory for the GPAC What is the problem

Time Scaling

System #1 #2 ODE y′(t) = p(y(t)) y(1) = y0        z′(t) = u(t)p(z(t)) u′(t) = u(t) z(t0) = y0 u(1) = 1 Remark Same curve, different speed: u(t) = et and z(t) = y(et) Example t f(x) y0(x) y(t)

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 12 / 17

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Toward a Complexity Theory for the GPAC What is the problem

Time Scaling

System #1 #2 ODE y′(t) = p(y(t)) y(1) = y0        z′(t) = u(t)p(z(t)) u′(t) = u(t) z(t0) = y0 u(1) = 1 Computed Function f(x) = limt→∞ y1(t) = limt→∞ z1(t) Remark Same curve, different speed: u(t) = et and z(t) = y(et) Example t f(x) y0(x) y(t)

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 12 / 17

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Toward a Complexity Theory for the GPAC What is the problem

Time Scaling

System #1 #2 ODE y′(t) = p(y(t)) y(1) = y0        z′(t) = u(t)p(z(t)) u′(t) = u(t) z(t0) = y0 u(1) = 1 Computed Function f(x) = limt→∞ y1(t) = limt→∞ z1(t) Convergence Eventually Exponentially faster Example t f(x) y0(x) y(t)

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 12 / 17

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Toward a Complexity Theory for the GPAC What is the problem

Time Scaling

ODE y′(t) = p(y(t)) y(1) = y0        z′(t) = u(t)p(z(t)) u′(t) = u(t) z(t0) = y0 u(1) = 1 Computed Function f(x) = limt→∞ y1(t) = limt→∞ z1(t) Convergence Eventually Exponentially faster Time for precision µ tm(µ) tm′(µ) = log(tm(µ)) Example t f(x) y0(x) y(t) z(t) tm(µ) tm′(µ) µ y1(tm(µ)) − f(x) µ

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 12 / 17

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Toward a Complexity Theory for the GPAC What is the problem

Time Scaling

ODE y′ = p(y) z′ = up(z) u′ = u Computed Function f(x) = limt→∞ y1(t) = limt→∞ z1(t) Time for precision µ tm(µ) tm′(µ) = log(tm(µ)) Bounding box for ODE at time t sp(t) sp′(t) = max(sp(et), et) Example t f(x) y(t) z(t) u(t) sp(t) sp′(t) t sp(t) = sup

ξ∈[1,t]

y(ξ) sp′(t) = sup

ξ∈[1,t]

z(ξ), u(ξ)

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 12 / 17

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Toward a Complexity Theory for the GPAC What is the problem

Time Scaling

ODE y′ = p(y) z′ = up(z) u′ = u Computed Function f(x) = limt→∞ y1(t) = limt→∞ z1(t) Time for precision µ tm(µ) tm′(µ) = log(tm(µ)) Bounding box for ODE at time t sp(t) sp′(t) = max(sp(et), et) Bounding box for ODE at precision µ sp(tm(µ)) max(sp(tm(µ)), tm(µ)) Remark tm(µ) and sp(t) depend on the convergence rate sp(tm(µ)) seems not

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 12 / 17

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Toward a Complexity Theory for the GPAC What is the problem

Proper Measures

Proper measures of “complexity”: time scaling invariant property of the curve

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 13 / 17

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Toward a Complexity Theory for the GPAC What is the problem

Proper Measures

Proper measures of “complexity”: time scaling invariant property of the curve Possible choices: Bounding Box at precision µ ⇒ Ok but geometric interpretation ?

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 13 / 17

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Toward a Complexity Theory for the GPAC What is the problem

Proper Measures

Proper measures of “complexity”: time scaling invariant property of the curve Possible choices: Bounding Box at precision µ ⇒ Ok but geometric interpretation ? Length of the curve until precision µ ⇒ Much more intuitive

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 13 / 17

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Toward a Complexity Theory for the GPAC Computational Complexity (Real Number)

Definition (Polytime GPAC-Computable Function) f is polytime computable by a GPAC iff for all x ∈ R the solution y = (y1, . . . , yd) of the ordinary differential equation (ODE): y′ = p(y) y(t0)= q(x) where p,q are vectors of polynomials satisfies

  • f(x) − y1(ℓ−1(len(x, µ))
  • e−µ where

len is a polynomial [polytime] ℓ(t) is the length of the curve y from t0 to t. ℓ−1(l) is the time to reach a length l on the curve y

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 14 / 17

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Toward a Complexity Theory for the GPAC Computational Complexity (Real Number)

Definition (Polytime GPAC-Computable Function) f is polytime computable by a GPAC iff for all x ∈ R the solution y = (y1, . . . , yd) of the ordinary differential equation (ODE): y′ = p(y) y(t0)= q(x) where p,q are vectors of polynomials satisfies

  • f(x) − y1(ℓ−1(len(x, µ))
  • e−µ where

len is a polynomial [polytime] ℓ(t) is the length of the curve y from t0 to t. ℓ−1(l) is the time to reach a length l on the curve y Remark implies f(x) = limt→∞ y1(t) length of a curve: ℓ(t) = t

t0 p(y(u)) du

y1(ℓ−1(l)) = position after travelling a length l on the curve y

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 14 / 17

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Toward a Complexity Theory for the GPAC Computational Complexity (Real Number)

Computable Analysis = GPAC ?

Theorem (Almost ) The polytime GPAC-computable functions are exactly the polytime com- putable functions of the Computable Analysis.

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 15 / 17

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Toward a Complexity Theory for the GPAC Computational Complexity (Real Number)

Computable Analysis = GPAC ?

Theorem (Almost ) The polytime GPAC-computable functions are exactly the polytime com- putable functions of the Computable Analysis. Remark (Polytime computable in CA) f polytime computable: polynomial modulus of continuity mc: x − y 2−mc(µ) ⇒ f(x) − f(y) 2−µ polynomial time computable over Q

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 15 / 17

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Conclusion

Conclusion

Complexity theory for the GPAC

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 16 / 17

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Conclusion

Conclusion

Complexity theory for the GPAC Equivalence with Computable Analysis for polynomial time

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 16 / 17

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Conclusion

Conclusion

Complexity theory for the GPAC Equivalence with Computable Analysis for polynomial time Not mentioned in this talk: The GPAC as a language recogniser

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 16 / 17

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Conclusion

Conclusion

Complexity theory for the GPAC Equivalence with Computable Analysis for polynomial time Not mentioned in this talk: The GPAC as a language recogniser Equivalence with P and NP

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 16 / 17

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Conclusion

Future Work

Notion of reduction ?

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 17 / 17

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Conclusion

Future Work

Notion of reduction ? Space complexity ?

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 17 / 17

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Questions ?

Do you have any questions ?

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 ∞ / 17

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Classical Computational Complexity

GPAC as Language Recogniser

GPAC as computable real function → Computable Analysis

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 ω + 1 / 17

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Classical Computational Complexity

GPAC as Language Recogniser

GPAC as computable real function → Computable Analysis GPAC as language recogniser → classical computability ?

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 ω + 1 / 17

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Classical Computational Complexity

GPAC as Language Recogniser

GPAC as computable real function → Computable Analysis GPAC as language recogniser → classical computability ? Remark words ≈ integers ⊆ real numbers

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 ω + 1 / 17

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Classical Computational Complexity

GPAC as Language Recogniser

GPAC as computable real function → Computable Analysis GPAC as language recogniser → classical computability ? Remark words ≈ integers ⊆ real numbers decide ≈ {Yes, No} ≈ {0, 1} ⊆ real numbers

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 ω + 1 / 17

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Classical Computational Complexity

GPAC as Language Recogniser

GPAC as computable real function → Computable Analysis GPAC as language recogniser → classical computability ? Remark words ≈ integers ⊆ real numbers decide ≈ {Yes, No} ≈ {0, 1} ⊆ real numbers language recogniser: special case of real function ? f : N ⊆ R → {0, 1} ⊆ R

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 ω + 1 / 17

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Classical Computational Complexity

GPAC as Language Recogniser

GPAC as computable real function → Computable Analysis GPAC as language recogniser → classical computability ? Remark words ≈ integers ⊆ real numbers decide ≈ {Yes, No} ≈ {0, 1} ⊆ real numbers language recogniser: special case of real function ? f : N ⊆ R → {0, 1} ⊆ R Yes but there is more !

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 ω + 1 / 17

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Classical Computational Complexity

Definition (GPAC-Recognisable Language) L ⊆ N GPAC-recognisable if for any x ∈ N, the solution y to y′ = p(y) y(t0)= q(x) where p,q are vectors of polynomials satisfies for t t1(x): if x ∈ L then y1(t) 1 (accept) if x / ∈ L then y1(t) −1 (reject)

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 ω + 2 / 17

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Classical Computational Complexity

Definition (GPAC-Recognisable Language) L ⊆ N GPAC-recognisable if for any x ∈ N, the solution y to y′ = p(y) y(t0)= q(x) where p,q are vectors of polynomials satisfies for t t1(x): if x ∈ L then y1(t) 1 (accept) if x / ∈ L then y1(t) −1 (reject) Theorem The GPAC-recognisable languages are exactly the recursive lan- guages.

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 ω + 2 / 17

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Classical Computational Complexity

Definition (GPAC-Recognisable Language) L ⊆ N GPAC-recognisable if for any x ∈ N, the solution y to y′ = p(y) y(t0)= q(x) where p,q are vectors of polynomials satisfies for t t1(x): if x ∈ L then y1(t) 1 (accept) if x / ∈ L then y1(t) −1 (reject) Theorem The GPAC-recognisable languages are exactly the recursive lan- guages. Remark What about complexity ?

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 ω + 2 / 17

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Classical Computational Complexity

Definition (Polytime GPAC-Recognisable Language) L ⊆ N poyltime GPAC-recognisable if for any x ∈ N, the solution y to y′ = p(y) y(t0)= q(x) where p,q are vectors of polynomials satisfies for t t1(x): if x ∈ L then y1(t) 1 (accept) if x / ∈ L then y1(t) −1 (reject)

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 ω + 3 / 17

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Classical Computational Complexity

Definition (Polytime GPAC-Recognisable Language) L ⊆ N poyltime GPAC-recognisable if for any x ∈ N, the solution y to y′ = p(y) y(t0)= q(x) where p,q are vectors of polynomials satisfies for t t1(x): if x ∈ L then y1(t) 1 (accept) if x / ∈ L then y1(t) −1 (reject) where t1(x) = ℓ−1(len(log(x)) where ℓ(t) is the length of y from t0 to t and len a polynomial.

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 ω + 3 / 17

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Classical Computational Complexity

Definition (Polytime GPAC-Recognisable Language) L ⊆ N poyltime GPAC-recognisable if for any x ∈ N, the solution y to y′ = p(y) y(t0)= q(x) where p,q are vectors of polynomials satisfies for t t1(x): if x ∈ L then y1(t) 1 (accept) if x / ∈ L then y1(t) −1 (reject) where t1(x) = ℓ−1(len(log(x)) where ℓ(t) is the length of y from t0 to t and len a polynomial. Theorem The class of polytime GPAC-recognisable languages is exactly P.

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 ω + 3 / 17

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Classical Computational Complexity

Definition (Polytime GPAC-Recognisable Language) L ⊆ N poyltime GPAC-recognisable if for any x ∈ N, the solution y to y′ = p(y) y(t0)= q(x) where p,q are vectors of polynomials satisfies for t t1(x): if x ∈ L then y1(t) 1 (accept) if x / ∈ L then y1(t) −1 (reject) where t1(x) = ℓ−1(len(log(x)) where ℓ(t) is the length of y from t0 to t and len a polynomial. Theorem The class of polytime GPAC-recognisable languages is exactly P. Remark (Why log(x) ?) Classical complexity measure: length of word ≈ log of value

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 ω + 3 / 17

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Classical Computational Complexity

Definition (Non-deterministic Polytime GPAC-Recognisable Language) L ⊆ N non-deterministic poyltime GPAC-recognisable if for any x ∈ N, the solution y to y′ = p(y, u) y(t0)= q(x) where p,q are vectors of polynomials satisfies for t t1(x): if x ∈ L then y1(t) 1 for at least one digital controller u if x / ∈ L then y1(t) −1 for all digital controller u where t1(x) = ℓ−1(len(log(x)) and len a polynomial.

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 ω + 4 / 17

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Classical Computational Complexity

Definition (Non-deterministic Polytime GPAC-Recognisable Language) L ⊆ N non-deterministic poyltime GPAC-recognisable if for any x ∈ N, the solution y to y′ = p(y, u) y(t0)= q(x) where p,q are vectors of polynomials satisfies for t t1(x): if x ∈ L then y1(t) 1 for at least one digital controller u if x / ∈ L then y1(t) −1 for all digital controller u where t1(x) = ℓ−1(len(log(x)) and len a polynomial. Remark (Digital Controller) Digital Controller ≈ u : R → {0, 1}

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 ω + 4 / 17

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Classical Computational Complexity

Definition (Non-deterministic Polytime GPAC-Recognisable Language) L ⊆ N non-deterministic poyltime GPAC-recognisable if for any x ∈ N, the solution y to y′ = p(y, u) y(t0)= q(x) where p,q are vectors of polynomials satisfies for t t1(x): if x ∈ L then y1(t) 1 for at least one digital controller u if x / ∈ L then y1(t) −1 for all digital controller u where t1(x) = ℓ−1(len(log(x)) and len a polynomial. Remark (Digital Controller) Digital Controller ≈ u : R → {0, 1} Theorem The class of non-deterministic polytime GPAC-recognisable languages is exactly NP.

Pouly, Bournez, Graça Computational Complexity of the GPAC April 10, 2014 ω + 4 / 17