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Computational Irreducibility & Emergence

Explore the concept of computational irreducibility and its implications in understanding complex systems and emergent phenomena. Examples include cellular automata and Conway's Game of Life.

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Computational Irreducibility & Emergence

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  1. ComputationalIrreducibility&Emergence Hervé ZWIRN CNRS & Paris Diderot University (LIED) ENS Cachan (CMLA) IHPST Histoire et Philosophie de l’Informatique IHPST le 22 juin 2017 Hervé ZWIRN

  2. ComputationalIrreducibility Determinism and Impredictibility A few examples Cellular automata : generalframework (discret) Hervé ZWIRN

  3. Determinism and Impredictibility Langton’s ant Rules: 1. If the antis on a black square, itturns right  and moves forward one unit. 2. If the antis on a white square, itturnsleftand moves forward one unit. 3. When the antleaves a square, itinverts the color. Hervé ZWIRN

  4. Determinism and Impredictibility Langton’s ant Hervé ZWIRN

  5. Determinism and Impredictibility Hervé ZWIRN

  6. Determinism and Impredictibility • This isimpredictible and nobodyknows how to provethat. • The onlyway to know whathappensis to run a simulation of the 10 000 steps of the ant and seewhathappens Hervé ZWIRN

  7. Determinism and Impredictibility Hervé ZWIRN

  8. Determinism and Impredictibility Rule 30 Hervé ZWIRN

  9. Determinism and Impredictibility Rule 110 Hervé ZWIRN

  10. Determinism and Impredictibility Rule 110 If youwantto know whatis the 1 000th line, you have to run the simulation and go through the 999 previoussteps. There is no otherway, no shortcutallowing to compute the nth line withoutgoingthrough the (n-1) previoussteps. Hervé ZWIRN

  11. Determinism and Impredictibility Conway’sgame of life Any live cell with fewer than two or more than three live neighbours dies. Any live cell with two or three live neighbours lives on to the next generation. Any dead cell with exactly three live neighbours becomes a live cell. Hervé ZWIRN

  12. Determinism and Impredictibility Conway’sgame of life Hervé ZWIRN

  13. Determinism and Impredictibility Conway’sgame of life Glider gun Hervé ZWIRN

  14. Determinism and Impredictibility Conway’sgame of life Hervé ZWIRN

  15. Determinism and Impredictibility ComputationalIrreducibility CIR The behavior of the system can be found only by direct simulation or observation: No general predictive procedure is possible. Wolfram S., Undecidability and intractability in theoretical physics, Vol 54, N 8. Phys. Rev. Letters, 1985 Hervé ZWIRN

  16. ComputationalIrreducibility Intuitively a system willbe CIR if there is no other way to reach the nth state than to go successively through the (n-1) previous ones. There is no shortcut. Predicting : Knowing the resultbefore the system computingfasterthan the system findingshortcuts This is not a robustdefinitionsincewewouldlike a process to be CIR even if itis possible to reach the nthstate by goingthrough (n-1) states that are “near” of the (n-1) previous states of the system if it is still impossible to go directly to it. Hervé ZWIRN

  17. ComputationalIrreducibility Giving a rigorous and robustdefinitionis not easybecauseitisnecessary to preciseformallywhichpath of (n-1) stepsis acceptable as beingnear the real pathfollowed by the system. • H. Zwirn & J.P. Delahaye, Irreducibility and Computational Equivalence: Wolfram Science 10 Years After the Publication of A New Kind of Science,H. Zenil (Ed), Springer, 2013 • H. Zwirn, Computational Irredudicibility and Computational Analogy, Complex Systems, Vol 24, Issue 2, 2015 • H. Zwirn"Les systèmes déterministes simples sont-ils toujours prédictibles"  in Complexité et désordre, Grenoble Sciences Ed., 2015 • H. Zwirn “Emergence et irréductibilité computationnelle”, à paraitre dans « Complexité ​et ​ désordre : adaptation, localisation, dynamique », Editions Matériologiques, 2017 Hervé ZWIRN

  18. Computationalirreducibility WhyisComputationalirreducibility (CIR) interesting? From the algorithmic point of view : Is thereanyrobustdefinition? Do anyreally CIR processesexist? Philosophicalreasons: Understanding the behaviour of ComplexSystems Understandingemergentphenomena Hervé ZWIRN

  19. Computationalirreducibility CIR means impossible to speed-up Is everyalgorithm ’’speed able’’ ? The computation model matters Turing machines with k tapes (k ≥ 2) Hervé ZWIRN

  20. Computationalirreducibility Some speed-up theorems: The problem of deciding if a string is a palindrome which is O(n2) in the 1-tape Turing machines model and O(n) in the 2-tape Turing machines model. For any k-tapes Turing machine M operating in time f(n) there exists a k'-tapes Turing machine M' operating in time f'(n)=f(n)+n (where  is an arbitrary small positive constant) which simulates M. Given any k-tape Turing machine M operating within time f(n), it's possible to construct a 1-tape Turing machine M' operating within time O(f(n)2) and such that for any input x, M(x)=M'(x). Hervé ZWIRN

  21. Computationalirreducibility The computation model Turing machines with 3 symbols (0, 1, #) k ≥ 2 tapes and one way write only tape which is used for output. We suppose that when the computation ends, the result is the number written at the right end of the output tape. Given a Turing machine M computing f(n) in time T(M(n)),let's denote by Rn,1, …, Rn,i, …, Rn,T(M(n)) the content of the output tape of M during the computation of f(n) after 1 step of computation, …, i steps of computation and T(M(n)) steps of computation. Hervé ZWIRN

  22. Computationalirreducibility (E-Turing machine):A Turing machine Mf will be called a E-Turing machine for f if: (i) Mf computes every f(n) (ii) during the computation of f(n), there exist increasing kn(i) for i=1 to n-1, such that f(i) is written on the output tape Rn,kn(i) at the right of the last symbol# Hervé ZWIRN

  23. Computationalirreducibility E-Turing machine for f n n-1 f(1) f(1) kn-1(1) kn(1) f(2) f(2) kn-1(2) kn(2) f(n-2) kn(n-2) f(n-1) f(n-2) kn-1(n-2) kn(n-1) kn-1(n-1) f(n-1) f(n) kn(n) Hervé ZWIRN

  24. Computationalirreducibility Tentative definition (CIR): A function f will be said CIR if and only if any Turing machine computing every f(n), is a E-Turing machine for f. Not a Robust Definition Need for more sophisticated concepts Hervé ZWIRN

  25. Computationalirreducibility (Asymptotically optimal Turing machine):We will say that a Turing machine Mf* for f is an asymptotically optimal Turing machine for f if for any other Turing machine M computing f: T(Mf*(n)) = O(T(M(n))) i.e. there are constantsc > 0, n0 > 0 such that n > n0, T(Mf* (n)) cT(M(n)). Asymptotically, no other Turing machine computing f computes faster than Mf* We assume that it is the case Hervé ZWIRN

  26. Computationalirreducibility (Efficient E-Turing machine):We will say that a E-Turing machine Mfeff for f is an efficient E-Turing machine for f if for any other E-Turing machine Mf for f: T(Mfeff(n)) = O(T(Mf(n))) i.e. there are constantsc > 0, n0 > 0 such that n > n0, T(Mfeff (n)) cT(Mf(n)). Asymptotically, no other E-Turing machine for f computes faster than Mfeff We assume that it is the case Hervé ZWIRN

  27. Computationalirreducibility A Turing Machine M will be said to be a P‑approximation of a E‑Turing machine for f if and only if there are a function F such that F(n)=O(T(Mf*(n)/n)) and a Turing machine P such that: (i) on input n, M computes a result rn such that P computes f(n) from rnin a number of steps F(n) and halts. (ii) during the computation, there exist increasing kn(i) for i=1 to n-1, such that P computes f(i) from i and Rn,kn(i) in a number of steps F(i) and halts. Hervé ZWIRN

  28. Computationalirreducibility P‑approximation of a E‑Turing machine for f n n P f(1) r1 kn(1) f(2) r2 kn(2) rn-1 f(n-1) kn(n-1) O(T(Mf*(n)/n)) f(n) rn kn(n) Hervé ZWIRN

  29. Computationalirreducibility Let M be a P‑approximation of a E‑Turing machine for f. Computation of f(n) based on the P‑approximationM: The computation of f(n) done initially through M with input n and continued when M has computed rn, by P which computes f(n) from n and rn in a time F(n) and halts. Hervé ZWIRN

  30. Computationalirreducibility Computation based on a P‑approximation of a E‑Turing machine for f n P f(1) r1 f(2) r2 f(n-1) rn-1 O(T(Mf*(n)/n)) f(n) rn Hervé ZWIRN

  31. Computationalirreducibility Strongly CIR (resp CIR) function: A function f(n) from N to N will be said to be strongly CIR(respCIR) if and only if for any Turing machine M computing every f(n) there is a P-approximation of a E‑Turing machine for f, M’, such that for every n (resp. for infinitely many n), the computation of f(n) by M is based on M’. If a function is strongly CIR, for each n there is no other way to compute f(n) than to compute before all the values f(i) for i<n (or values that are near in the sense given in the definition of the approximation of a E-Turing machine). There is no shortcut allowing to get directly the value of f(n) without having computed before f(n-1) or a value that is near f(n-1) and so forth for the previous values. Hervé ZWIRN

  32. Computationalirreducibility Theorem if f is CIR no Turing machine computing every f(n) can compute f(n) faster than an efficient E­-Turing machine for f. More precisely, if Mfis aTuring machine computing every f(n) and if f is CIR then T(Mfeff(n)) = O(T(Mf(n))). Hervé ZWIRN

  33. Computationalirreducibility Possible candidates: Langton’sant Rule 110 The number of configurations of index < n still alive after n steps in the game of life The function f défined as : f(1) = the first digit of  f(n) = the digit of  afterhavingskipped f(n-1) digits from the digit f(n-1) Open problem: Prove that any of the above possible candidates is CIR. Hervé ZWIRN

  34. Emergence Whatisnecessary for Emergence? 2 levels: individual / collectif micro / macro Knowledge of the rules for the lowlevel Apparent irreducibility of the phenomenonappearingat the upperlevel to the lowlevelrules Hervé ZWIRN

  35. Emergence Objective (weak) emergence Objective emergenceshouldbeindependant of ourhumancapacities. Non-epistemiccriterion. Emergence appears when the dynamics of the low level is CIR Hervé ZWIRN

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