Characterizing possible typical asymptotic behaviours of cellular automata

Transcription

1 Characterizing possible typical asymptotic behaviours of cellular automata Benjamin Hellouin de Menibus joint work with Mathieu Sablik Laboratoire d Analyse, Topologie et Probabilités Aix-Marseille University Automata Theory and Symbolic Dynamics Workshop, Vancouver Hellouin, Sablik (LATP) Characterization of limit measures ATSD Workshop 1 / 20

2 Definitions A, B finite alphabets; A the (finite) words; A Z the configurations; σ the shift action σ(a) i = a i 1 ; A cellular automaton is an action F : A Z A Z defined by a local rule f : A U A on some neighbourhood U. For A = {, } and U = { 1, 0, 1}: f }{{} Hellouin, Sablik (LATP) Characterization of limit measures ATSD Workshop 2 / 20

3 Definitions A, B finite alphabets; A the (finite) words; A Z the configurations; σ the shift action σ(a) i = a i 1 ; A cellular automaton is an action F : A Z A Z defined by a local rule f : A U A on some neighbourhood U. For A = {, } and U = { 1, 0, 1}: f }{{} Hellouin, Sablik (LATP) Characterization of limit measures ATSD Workshop 2 / 20

4 Definitions A, B finite alphabets; A the (finite) words; A Z the configurations; σ the shift action σ(a) i = a i 1 ; A cellular automaton is an action F : A Z A Z defined by a local rule f : A U A on some neighbourhood U. For A = {, } and U = { 1, 0, 1}: f }{{} Hellouin, Sablik (LATP) Characterization of limit measures ATSD Workshop 2 / 20

5 Definitions A, B finite alphabets; A the (finite) words; A Z the configurations; σ the shift action σ(a) i = a i 1 ; A cellular automaton is an action F : A Z A Z defined by a local rule f : A U A on some neighbourhood U. For A = {, } and U = { 1, 0, 1}: F Hellouin, Sablik (LATP) Characterization of limit measures ATSD Workshop 2 / 20

6 Simulations and typical asymptotic behaviour Traffic automaton Captive automaton 3-state cyclic automaton Additive automaton Hellouin, Sablik (LATP) Characterization of limit measures ATSD Workshop 3 / 20

7 Measure space M σ (A Z ) the σ-invariant probability measures on A Z. Examples µ([u]) the probability that a word u A appears, for µ M σ (A Z ). Bernoulli (i.i.d) measures Let (λ a ) a A such that λ a = 1. u A, µ([u]) = u 1 i=0 λ ui. Measures supported by a periodic orbit For a finite word w, δ w = 1 w 1 δ σi ( w w ). i=0 Markov measures with finite memory. Hellouin, Sablik (LATP) Characterization of limit measures ATSD Workshop 4 / 20

8 Action of an automaton on an initial measure F extends to an action F : M σ (A Z ) M σ (A Z ): F µ(u) = µ(f 1 U) for any borelian U. For an initial measure µ, F t µ describes the repartition at time t; Typical asymptotic behaviour is well described by the limit(s) of (F t µ) t N in the weak-* topology: F t µ t ν u A, F t µ([u]) ν([u]). Hellouin, Sablik (LATP) Characterization of limit measures ATSD Workshop 5 / 20

9 Examples of asymptotic behaviour Hellouin, Sablik (LATP) Characterization of limit measures ATSD Workshop 6 / 20

10 Examples of asymptotic behaviour Proposition Let µ be the uniform Bernoulli measure on {0, 1, 2} and F the 3-state cyclic automaton. F µ t 1 δ δ δ Hellouin, Sablik (LATP) Characterization of limit measures ATSD Workshop 6 / 20

11 Main question Question Which measures ν are reachable as the limit of the sequence (F t µ) t N for some cellular automaton F and initial measure µ? Hellouin, Sablik (LATP) Characterization of limit measures ATSD Workshop 7 / 20

12 Main question Question Which measures ν are reachable as the limit of the sequence (F t µ) t N for some cellular automaton F and initial measure µ? Answer All (take F = Id and µ = ν). Hellouin, Sablik (LATP) Characterization of limit measures ATSD Workshop 7 / 20

13 Main question Better question Which measures ν are reachable as the limit of the sequence (F t µ) t N for some cellular automaton F and simple initial measure µ (e.g. the uniform Bernoulli measure)? In a sense, this would correspond to the physically relevant measure for F. Hellouin, Sablik (LATP) Characterization of limit measures ATSD Workshop 7 / 20

14 Section 2 Necessary conditions: computability obstructions Hellouin, Sablik (LATP) Characterization of limit measures ATSD Workshop 8 / 20

15 Topological obstructions Topological obstruction The accumulation points of (F t µ) t N form a nonempty and compact set. Hellouin, Sablik (LATP) Characterization of limit measures ATSD Workshop 9 / 20

16 Measures and computability f : N N is computable if there exists a Turing machine that, on any input n N, stops and outputs f(n) (up to encoding). Hellouin, Sablik (LATP) Characterization of limit measures ATSD Workshop 10 / 20

17 Measures and computability f : N N is computable if there exists a Turing machine that, on any input n N, stops and outputs f(n) (up to encoding). A probability measure µ M σ (A Z ) is: computable if u µ([u]) is computable, i.e. if there exists f : A N Q computable such that µ([u]) f(u, n) < 2 n. ( can be simulated by a probabilistic Turing machine) Examples of computable measures Any periodic orbit measure; Any Bernoulli or Markov measure with computable parameters. Hellouin, Sablik (LATP) Characterization of limit measures ATSD Workshop 10 / 20

18 Measures and computability f : N N is computable if there exists a Turing machine that, on any input n N, stops and outputs f(n) (up to encoding). A probability measure µ M σ (A Z ) is: semi-computable if there exists a computable function f : A N Q such that µ([u]) f(u, n) n 0. ( limit of a computable sequence of measures) Examples of computable measures Any periodic orbit measure; Any Bernoulli or Markov measure with computable parameters. Hellouin, Sablik (LATP) Characterization of limit measures ATSD Workshop 10 / 20

19 Computability obstruction Action of an automaton on a computable measure If µ is computable, then F t µ is computable; If µ is computable, and F t µ t ν, then ν is semi-computable. Hellouin, Sablik (LATP) Characterization of limit measures ATSD Workshop 11 / 20

20 Section 3 Sufficient conditions: construction of limit measures Hellouin, Sablik (LATP) Characterization of limit measures ATSD Workshop 12 / 20

21 State of the art Theorem [Boyet, Poupet, Theyssier 06] There is an automaton F such that the language of words u satisfying F µ([u]) t 0 in not computable for any nondegenerate Bernoulli measure µ. Theorem [Boyer, Delacourt, Sablik 10] Let µ be the uniform Bernoulli measure. For a large class of subshifts U A Z (under computability conditions), there is an automaton F such that U = supp(ν). ν V(F t µ) Hellouin, Sablik (LATP) Characterization of limit measures ATSD Workshop 13 / 20

22 Main result Action of an automaton on a computable measure If µ is computable, and F t µ t ν, then ν is semi-computable. Motto: The only obstruction is the computability obstruction E. Jeandel, June 3rd 2013 Hellouin, Sablik (LATP) Characterization of limit measures ATSD Workshop 14 / 20

23 Main result Action of an automaton on a computable measure If µ is computable, and F t µ t ν, then ν is semi-computable. Motto: Theorem The only obstruction is the computability obstruction Let ν be a semi-computable measure. There exists: an alpabet B A a cellular automaton F : B B such that, for any ergodic and full-support measure µ M σ (B Z ), F t µ t ν E. Jeandel, June 3rd 2013 Hellouin, Sablik (LATP) Characterization of limit measures ATSD Workshop 14 / 20

24 Approximation by periodic orbits Proposition Measures supported by periodic orbits are dense in M σ (A Z ). Example: Uniform Bernoulli measure w 0 = 01 w 1 = 0011 w 2 = w 3 = Hellouin, Sablik (LATP) Characterization of limit measures ATSD Workshop 15 / 20

25 Approximation by periodic orbits Proposition Measures supported by periodic orbits are dense in M σ (A Z ). Example: Uniform Bernoulli measure w 0 = 01 w 1 = 0011 w 2 = w 3 = Proposition If ν M σ (A Z ) is semi-computable, there is a computable sequence of words (w n ) n N such that δ wn ν. Our construction will compute each w n and approach the measure δ wn by writing concatenated copies of w n on all the configuration. Hellouin, Sablik (LATP) Characterization of limit measures ATSD Workshop 15 / 20

26 x x x x x x x x x x x x x x x x x x x x Hellouin, Sablik (LATP) Characterization of limit measures ATSD Workshop 16 / 20

27 Section 4 Extensions and related results Hellouin, Sablik (LATP) Characterization of limit measures ATSD Workshop 17 / 20

28 Extensions and related results Questions 1. Implementation of the construction? No (but for good reasons) Implementation Non-trivial Turing machine satisfying space constraints; Large number of states; (for B = 2, at least 2244 times more than the corresponding Turing machine) ( ) Speed of convergence O in the best case. 1 log t Hellouin, Sablik (LATP) Characterization of limit measures ATSD Workshop 18 / 20

29 Extensions and related results Questions 1. Implementation of the construction? No (but for good reasons) 2. No auxiliary states? Yes, if the target measure is not full-support Theorem Let ν be a non full-support, semi-computable measure. Then there exists an automaton F : A A such that, for any measure µ M σ (A Z ) σ-mixing and full-support, F t µ t ν. Hellouin, Sablik (LATP) Characterization of limit measures ATSD Workshop 18 / 20

30 Extensions and related results Questions 1. Implementation of the construction? No (but for good reasons) 2. No auxiliary states? Yes, if the target measure is not full-support Idea: use forbidden words to encode auxiliary states. Remark If F t µ ν where ν is a full support measure, then F is a surjective automaton and the uniform Bernoulli measure is invariant. Hellouin, Sablik (LATP) Characterization of limit measures ATSD Workshop 18 / 20

31 Extensions and related results Questions 1. Implementation of the construction? No (but for good reasons) 2. No auxiliary states? Yes, if the target measure is not full-support 3. Sets of accumulation points? Yes, with a computability condition on compact sets Theorem Let V be a nonempty, compact, connected, Σ 2 -computable set of measures. Then there exists an automaton F : A A such that, for any measure µ M σ (A Z ) σ-mixing and full-support, The set of accumulation points of (F t µ) t N is V. Hellouin, Sablik (LATP) Characterization of limit measures ATSD Workshop 18 / 20

32 Extensions and related results Questions 1. Implementation of the construction? No (but for good reasons) 2. No auxiliary states? Yes, if the target measure is not full-support 3. Sets of accumulation points? Yes, with a computability condition on compact sets 4. Cesaro mean convergence? Yes Hellouin, Sablik (LATP) Characterization of limit measures ATSD Workshop 18 / 20

33 Extensions and related results Questions 1. Implementation of the construction? No (but for good reasons) 2. No auxiliary states? Yes, if the target measure is not full-support 3. Sets of accumulation points? Yes, with a computability condition on compact sets 4. Cesaro mean convergence? Yes 5. Characterization of the support? In progress Hellouin, Sablik (LATP) Characterization of limit measures ATSD Workshop 18 / 20

34 Extensions and related results Questions 1. Implementation of the construction? No (but for good reasons) 2. No auxiliary states? Yes, if the target measure is not full-support 3. Sets of accumulation points? Yes, with a computability condition on compact sets 4. Cesaro mean convergence? Yes 5. Characterization of the support? In progress 6. Properties of the limit measure? Mostly undecidable Hellouin, Sablik (LATP) Characterization of limit measures ATSD Workshop 18 / 20

35 Extensions and related results Questions 1. Implementation of the construction? No (but for good reasons) 2. No auxiliary states? Yes, if the target measure is not full-support 3. Sets of accumulation points? Yes, with a computability condition on compact sets 4. Cesaro mean convergence? Yes 5. Characterization of the support? In progress 6. Properties of the limit measure? Mostly undecidable 7. Using the initial measure as an argument or an oracle? Some simple cases Hellouin, Sablik (LATP) Characterization of limit measures ATSD Workshop 18 / 20

36 Computation in the space of measures Let us consider the operator µ accumulation points of (F t µ) t R The previous construction gave us operators that were essentially constant (on a large domain). Question Which operators M σ (A Z ) M σ (A Z ) (ou M σ (A Z ) P(M σ (A Z ))) can be realized in this way? Hellouin, Sablik (LATP) Characterization of limit measures ATSD Workshop 19 / 20

37 Computation in the space of measures Let us consider the operator µ accumulation points of (F t µ) t R The previous construction gave us operators that were essentially constant (on a large domain). Question Which operators M σ (A Z ) M σ (A Z ) (ou M σ (A Z ) P(M σ (A Z ))) can be realized in this way? Theorem Let ν : R M σ (A Z ) be a semi-computable operator. There is: an alphabet B A, an automaton F : B Z B Z such that, for any full-support and exponentially σ-mixing measure µ, F µ t ν ( µ ( )) I. t Hellouin, Sablik (LATP) Characterization of limit measures ATSD Workshop 19 / 20

38 Some examples Let M M σ (B Z ) be the set of full-support, exponentially σ-mixing measures. Example 1: Density classification There exists an automaton F : B Z B Z realizing the operator: M M σ ({0, 1} Z ) { δ0 if µ( I ) < 1 2 µ δ 1 otherwise. Hellouin, Sablik (LATP) Characterization of limit measures ATSD Workshop 20 / 20

39 Some examples Let M M σ (B Z ) be the set of full-support, exponentially σ-mixing measures. Example 1: Density classification There exists an automaton F : B Z B Z realizing the operator: M M σ ({0, 1} Z ) { δ0 if µ( I ) < 1 2 µ δ 1 otherwise. Example 2: A simple oracle There exists an automaton F : B Z B Z realizing the operator: M M σ ({0, 1} Z ) µ Ber(µ( I )) Hellouin, Sablik (LATP) Characterization of limit measures ATSD Workshop 20 / 20

40 Implementation of a simple case Fibonacci word Consider the morphism : ϕ : A A Then the sequence ϕ n (0) converges to an infinite word called Fibonacci word: and it is uniquely ergodic. ϕ (0) = Hellouin, Sablik (LATP) Characterization of limit measures ATSD Workshop 21 / 20