treasure map poincaréville turing city turing universality in dynamical systems jean-charles...
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Treasure map
Poincaréville
Turing City
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Turing universality in dynamical systems
Jean-Charles DelvenneCaltech and University of Louvain
July 1st, 2006
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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?
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This is about…
Turing universality =computing functions: =deciding subsets of integers
Dynamical systems = function: = state space Or in continuous time
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This is not about…
Computing real functions Deciding sets of reals Super-Turing power Simulation universality
Quantum systems Stochastic systems
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Summary
Definitions of universality Point-to-point Point-to-set Set-to-set
Properties of universality Robustness to noise Chaos
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Definitions of universality
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« Is 97 prime? »
« 97 is prime. »
Is 97 prime ?
« I’m computing... »
It’s computing…
Aha! 97 is prime.
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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
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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
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Point-to-point universality
Set X, family Function Effectivity: with k total computable Reflection principle (Sutner):
if then
Universal iff is r.e.-complete
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Point-to-set universality
Set X, family of points,
family Function Effectivity, reflection principle is decidable Universality iff is r.e.-
complete
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Examples
Turing machine, with finite configurations Game of Life, with almost blank
configurations (Conway)
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Examples Rule 110, with almost periodic configurations (Cook,
Wolfram)
Reversible and Billiard Ball cellular automata(Margolus, Toffoli)
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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)
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Universal continuous-time systems
Piecewise-constant derivative system (Asarin, Maler, Pnueli)
Ray of light between mirrors (Moore)
Billiard ball computer (Fredkin, Toffoli)
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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
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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 »
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« Is 97 prime? »
« 97 is prime. »
Is 97 prime ?
« I’m computing... »
It’s computing…
Aha! 97 is prime.
![Page 21: Treasure map Poincaréville Turing City Turing universality in dynamical systems Jean-Charles Delvenne Caltech and University of Louvain July 1st, 2006](https://reader035.vdocuments.site/reader035/viewer/2022062421/56649e625503460f94b5d62b/html5/thumbnails/21.jpg)
Examples
Universal Turing machines
A cellular automaton
A subshift
Game of Life?
Rule 110?
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Properties of universal systems
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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)
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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)
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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
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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)
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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.)
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Thank you