Diapositive 1

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Treasure map
Turing City
Republic of Dynamics
Country of Computers
Poincaréville
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Turing universality in dynamical systems

• Jean-Charles Delvenne
• Caltech 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 ?
 Is 97 prime? 
Its computing
Im computing... 
Aha! 97 is prime.
 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
• Questionis 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
• Instancetwo 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
• Instancea clopen partition, a finite automaton
• QuestionIs there a trace accepted by the finite
  automaton ?
• Universality r.e.-completeness of Generalized
  Halting problem
• Interpretation (cf. Turings argument)
• finite automatonobservers brain
• initial state of the automaton start
  computation 
• final state of the automaton  I have the
  answer 

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Is 97 prime ?
 Is 97 prime? 
Its computing
Im computing... 
Aha! 97 is prime.
 97 is prime. 
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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