Treasure map

Poincaréville

Turing City

Turing universality in dynamical systems

Jean-Charles DelvenneCaltech and University of Louvain

July 1st, 2006

Questions

There is a universal Turing machine (Turing) Game of Life is universal (Conway) Is the solar system universal? (Moore) A neural network is universal (Siegelmann) What is a universal dynamical system? What is a computer? Is universality robust to noise? Is a chaotic system universal?

This is about…

Turing universality =computing functions: =deciding subsets of integers

Dynamical systems = function: = state space Or in continuous time

This is not about…

Computing real functions Deciding sets of reals Super-Turing power Simulation universality

Quantum systems Stochastic systems

Summary

Definitions of universality Point-to-point Point-to-set Set-to-set

Properties of universality Robustness to noise Chaos

Definitions of universality

« Is 97 prime? »

« 97 is prime. »

Is 97 prime ?

« I’m computing... »

It’s computing…

Aha! 97 is prime.

Davis universality

A universal Turing machine has an r.e.-complete halting problem

… and conversely Davis: A Turing machine is said universal iff

its halting problem is r.e.-complete No explicit coding/decoding Universal dynamical system= system with

r.e.-complete halting problem

Halting problem for dynamical systems

Dynamical system

Instance= a point , a subset Question= Is there an such that ?

Instance= two points Question=is there an such that ?

Need to specify a family of points/family of sets Function must be effective

Point-to-point universality

Set X, family Function Effectivity: with k total computable Reflection principle (Sutner):

if then

Universal iff is r.e.-complete

Point-to-set universality

Set X, family of points,

family Function Effectivity, reflection principle is decidable Universality iff is r.e.-

complete

Examples

Turing machine, with finite configurations Game of Life, with almost blank

configurations (Conway)

Examples Rule 110, with almost periodic configurations (Cook,

Wolfram)

Reversible and Billiard Ball cellular automata(Margolus, Toffoli)

Examples

Piecewise-affine continuous map in dimension 2, with rational points and rational polyhedra (Koiran, Cosnard, Garzon)

Artificial neural networks (Siegelmann, Kilian, Sontag)

An one-dimensional analytic map with closed-form formula, with integers (Koiran, Moore)

Universal continuous-time systems

Piecewise-constant derivative system (Asarin, Maler, Pnueli)

Ray of light between mirrors (Moore)

Billiard ball computer (Fredkin, Toffoli)

Set-to-set universality (D., Kurka, Blondel)

Symbolic systems= cellular automata, Turing machines, subshifts, any continuous

Clopen sets= sets ( finite word) or boolean combinations

Halting problem: Instance=two clopen sets A and B Question= Is there a trajectory from A to B ?

At the cost of topology, no need for family of points

Set-to-set universality

Generalized Halting problem: Instance=a clopen partition, a finite automaton Question=Is there a trace accepted by the finite

automaton ? Universality= r.e.-completeness of

Generalized Halting problem Interpretation (cf. Turing’s argument):

finite automaton=observer’s brain initial state of the automaton=« start computation » final state of the automaton= « I have the answer »

« Is 97 prime? »

« 97 is prime. »

Is 97 prime ?

« I’m computing... »

It’s computing…

Aha! 97 is prime.

Examples

Universal Turing machines

A cellular automaton

A subshift

Game of Life?

Rule 110?

Properties of universal systems

Robustness

What if small perturbation on the state? A set-to-set universal symbolic system is

robust to perturbation on initial state What if perturbation at every time? Many systems become non universal (Asarin,

Boujjani, Orponen, Maass) There exists a (point-to-set) universal cellular

automaton with noise (Gacs)

Chaos

Are universal systems at the edge of chaos?(Langton) Neither too predictible (one globally attracting fixed point) Not too unpredictible (chaotic)

Intuition: chaos ~ noise Devaney-chaotic

There is a trajectory from any open set to any open set Periodic trajectories are dense Sensitivity to initial conditions (butterfly effect)

Universal cellular automata are in « class four » (Wolfram)

Results

Point-to-set, point-to-point definitions: little to be said in general

Set-to-set definition: there exists a Devaney-chaotic universal cellular

automaton In a universal system, at least one point must be

sensitive (butterfly effect) An attracting fixed point is not universal «  Edge of chaos » statement is half-true

Decidability vs universality

Universality: one system, a property of points/subsets is undecidable

Compare with: a family of systems, a property of the system is undecidable

Examples Stability of piecewise affine systems (Blondel, Bournez,

Koiran, Tsitsiklis) Reversibility of cellular automata (Kari)

Conclusion

What is a computer? Kaleidoscopic answer Many examples Little known about links

computation/dynamics Motivating open problems (Moore):

Is a solar system universal? Is there a liquid computer? (Navier-Stokes equ.)

Thank you