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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