Treasure map

2. Treasure map Turing City Republic of Dynamics Country of Computers Poincaréville

3. Turing universality in dynamical systems Jean-Charles Delvenne Caltech and University of Louvain July 1st, 2006

4. 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?

5. This is about… • Turing universality • =computing functions: • =deciding subsets of integers • Dynamical systems • = function: • = state space • Or in continuous time

6. This is not about… • Computing real functions • Deciding sets of reals • Super-Turing power • Simulation universality • Quantum systems • Stochastic systems

7. Summary • Definitions of universality • Point-to-point • Point-to-set • Set-to-set • Properties of universality • Robustness to noise • Chaos

8. Definitions of universality

9. Is 97 prime ? « Is 97 prime? » It’s computing… « I’m computing... » Aha! 97 is prime. « 97 is prime. »

10. 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

11. 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

12. Point-to-point universality • Set X, family • Function • Effectivity: with k total computable • Reflection principle (Sutner): • if • then • Universal iff is r.e.-complete

13. Point-to-set universality • Set X, family of points, family • Function • Effectivity, reflection principle • is decidable • Universality iff is r.e.-complete

14. Examples • Turing machine, with finite configurations • Game of Life, with almost blank configurations (Conway)

15. Examples • Rule 110, with almost periodic configurations (Cook, Wolfram) • Reversible and Billiard Ball cellular automata (Margolus, Toffoli)

16. 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)

17. Universal continuous-time systems • Piecewise-constant derivative system (Asarin, Maler, Pnueli) • Ray of light between mirrors (Moore) • Billiard ball computer (Fredkin, Toffoli)

18. 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

19. 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 »

20. Is 97 prime ? « Is 97 prime? » It’s computing… « I’m computing... » Aha! 97 is prime. « 97 is prime. »

21. Examples • Universal Turing machines • A cellular automaton • A subshift • Game of Life? • Rule 110?

22. Properties of universal systems

23. 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)

24. 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)

25. 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

26. 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)

27. 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.)

28. Thank you