Bounds on non-surjective cellular automata

Transkriptio

1 Bounds on non-surjective cellular automata Jarkko Kari Pascal Vanier Thomas Zeume University of Turku LIF Marseille Universität Hannover 27 august 2009 J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august 2009 / 2

2 Content Denitions Cellular Automata Unbalanced words Orphans Diamonds 2 Linear algebra approach Unbalanced words Orphans Diamonds General range 3 Algorithms 4 Conclusion J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august / 2

3 One-dimensional Cellular Automata S = { }, N = {0, } f J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august / 2

9 We consider here non-surjective, one dimensional cellular automata: F : S Z S Z Where S = {,..., n} the set of states, hence S = n. J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august / 2

10 We consider here non-surjective, one dimensional cellular automata: F : S Z S Z Where S = {,..., n} the set of states, hence S = n. The range of the local rule f : S r S is r: t f r J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august / 2

11 We consider here non-surjective, one dimensional cellular automata: F : S Z S Z Where S = {,..., n} the set of states, hence S = n. The range of the local rule f : S r S is r: t f r We will focus on range 2 cellular automata. J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august / 2

12 Unbalanced words On the average, words of length k have n preimages of length k +. If the CA is not surjective some words are unbalanced: their number of preimages is dierent from n. J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august / 2

13 Unbalanced words On the average, words of length k have n preimages of length k +. If the CA is not surjective some words are unbalanced: their number of preimages is dierent from n. Combinatorial upper bound on the size of shortest unbalanced words: n2 log 2 n 2 J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august / 2

14 Orphans Non-surjective CA have congurations with no preimage. By compactness, they also have orphans : nite congurations with no preimage. J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august / 2

15 Orphans Non-surjective CA have congurations with no preimage. By compactness, they also have orphans : nite congurations with no preimage. Combinatorial upper bound on the size of shortest orphans: 2 n. J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august / 2

16 Diamonds By the Garden-of-Eden theorem, if f is non-surjective there exists a diamond: two dierent words with a common prex, a common sux and the same image by f. f J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august / 2

17 Diamonds By the Garden-of-Eden theorem, if f is non-surjective there exists a diamond: two dierent words with a common prex, a common sux and the same image by f. f length The length of a diamond is the length of the diering part. J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august / 2

18 Diamonds By the Garden-of-Eden theorem, if f is non-surjective there exists a diamond: two dierent words with a common prex, a common sux and the same image by f. f length The length of a diamond is the length of the diering part. Combinatorial upper bound on the length of shortest diamonds : n2 2. J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august / 2

19 Summary Combinatorial Orphans Diamonds Unbalanced words 2 n n2 log 2 n 2 n2 2 J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august / 2

20 Summary Combinatorial Linear algebraic Orphans 2 n n 2 Diamonds Unbalanced words n2 log 2 n 2 2n n n2 2 n J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august / 2

21 Linear algebra approach We interpret states as unit coordinate vectors of R n : (, 0, 0,..., 0) 2 (0,, 0,..., 0). n (0, 0, 0,..., ) Sets of states are 0/-vectors obtained by adding up their elements : (0, 0, 0, 0, 0,..., 0) S (,,,,,..., ) {, 2, 4} (,, 0,, 0,..., 0) J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august / 2

22 The weight of a vector x = (x,..., x n ) R n is and is a linear form. φ(x) = x + + x n Furthermore, if x is a 0/-vector, representing a set of states, then φ(x) coincides with the cardinality. J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august / 2

23 For each a S we dene the linear function f a : R n R n that maps each s as follows: f a : s {x S f (s, x) = a} f J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august 2009 / 2

24 For each a S we dene the linear function f a : R n R n that maps each s as follows: f a : s {x S f (s, x) = a} f f f (0,, 0) = (0,, ) J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august 2009 / 2

25 For each a S we dene the linear function f a : R n R n that maps each s as follows: f a : s {x S f (s, x) = a} f f f (,, ) = (,, 2) J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august 2009 / 2

26 For each a S we dene the linear function f a : R n R n that maps each s as follows: f a : s {x S f (s, x) = a} f f f (,, ) = (,, 2) The matrix of f a is obtained from the local rule table by replacing each entry a by and each entry dierent of a by 0. J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august 2009 / 2

27 For each a S we dene the linear function f a : R n R n that maps each s as follows: f a : s {x S f (s, x) = a} f f f f (,, ) = (,, 3) For any word w = a... a k, we denote f w = f ak f a. J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august 2009 / 2

28 For each a S we dene the linear function f a : R n R n that maps each s as follows: f a : s {x S f (s, x) = a} 2 f f f f 2 (,, ) = (, 3, ) f 2 f w counts preimages: For Q S, φ(f w (Q)) gives the number of preimages of w starting with some state of Q. J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august 2009 / 2

29 Observe the following: A word w is unbalanced if and only if: φ(f w (,..., )) n A word w is an orphan if and only if: φ(f w (,..., )) = 0 Note that V k = {x φ(x) = k} = (k, 0,..., 0) + ker φ is an ane subspace of R n of dimension n. J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august / 2

30 Observe the following: A word w is unbalanced if and only if: φ(f w (,..., )) n A word w is an orphan if and only if: φ(f w (,..., )) = 0 Note that V k = {x φ(x) = k} = (k, 0,..., 0) + ker φ is an ane subspace of R n of dimension n. Unbalanced word: nding the shortest composition of f a taking (,..., ) outside of the ane subspace V n of R n. J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august / 2

31 Theorem Let A be an ane space, if there is a word w such that f w (x) A, there exist such a word of length at most dim A +. J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august / 2

32 Theorem Let A be an ane space, if there is a word w such that f w (x) A, there exist such a word of length at most dim A +. Corollary There exists an unbalanced word of length at most n. Proof. By taking A = V n. J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august / 2

33 Theorem Let A be an ane space, if there is a word w such that f w (x) A, there exist such a word of length at most dim A +. Proof. Consider A 0 A... ane subspaces dened by: A 0 = {x}, dim A 0 = 0 (A i a S f a (A i ) ) A i+ = A J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august / 2

34 Theorem Let A be an ane space, if there is a word w such that f w (x) A, there exist such a word of length at most dim A +. Proof. Consider A 0 A... ane subspaces dened by: A 0 = {x}, dim A 0 = 0 (A i a S f a (A i ) ) A i+ = A Let m be the smallest i such that A i A: J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august / 2

35 Theorem Let A be an ane space, if there is a word w such that f w (x) A, there exist such a word of length at most dim A +. Proof. Consider A 0 A... ane subspaces dened by: A 0 = {x}, dim A 0 = 0 (A i a S f a (A i ) ) A i+ = A Let m be the smallest i such that A i A: A i = A i+ means that f a (A i ) A i for all a, so A i = A j for all j > i. J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august / 2

36 Theorem Let A be an ane space, if there is a word w such that f w (x) A, there exist such a word of length at most dim A +. Proof. Consider A 0 A... ane subspaces dened by: A 0 = {x}, dim A 0 = 0 (A i a S f a (A i ) ) A i+ = A Let m be the smallest i such that A i A: A i = A i+ means that f a (A i ) A i for all a, so A i = A j for all j > i. Hence all inclusions are proper: A 0 A A m, so 0 = dim A 0 < < dim A m Since dim A m dim A, we have a word of the desired length. J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august / 2

37 Lemma (Balance) For any vector x of R n, the following relation holds: φ (f a (x)) = n φ (x) a S J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august / 2

38 Lemma (Balance) For any vector x of R n, the following relation holds: φ (f a (x)) = n φ (x) a S Corollary There is a word w such that φ(f w (x)) > φ(x) if and only if there is a word v, v = w such that φ(f v (x)) < φ(x). J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august / 2

39 Lemma (Balance) For any vector x of R n, the following relation holds: φ (f a (x)) = n φ (x) a S Proof. By linearity of φ, it is enough to consider x representing a single state s. For every b S the word sb is a preimage of exactly one a S, so a S φ(f a(s)) = n. Corollary There is a word w such that φ(f w (x)) > φ(x) if and only if there is a word v, v = w such that φ(f v (x)) < φ(x). J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august / 2

40 We can use the method used for the unbalanced words! Theorem There exists an orphan of length at most n 2. J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august / 2

41 We can use the method used for the unbalanced words! Theorem There exists an orphan of length at most n 2. Proof. Starting with x 0 = (,..., ), we nd words w,..., w j of length at most n such that: where w w x 0 2 w j x x j n = φ(x 0 ) > φ(x ) > > φ(x j ) = 0 The word w w 2... w j is an orphan of length at most jn n 2. J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august / 2

42 To nd a diamond, we start with some pair {a, b}, a b, such that f (s, a) = f (s, b), for some s S. The weight of the corresponding vector is 2. J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august / 2

43 We repeatedly nd words of length at most n that increase the weight. J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august / 2

44 n n > n We stop once the vector has weight > n. The length of the corresponding word is at most n n. J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august / 2

45 n n > n If the nal vector is not a 0/-vector, then we have found a diamond of length at most n n. J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august / 2

46 n n n n > n > n We do the same from right to left. J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august / 2

47 n n n n > n > n There are more than n pairs of states in the two nal sets, so two pairs have the same image under f. J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august / 2

48 n n n n > n > n There are more than n pairs of states in the two nal sets, so two pairs have the same image under f. Theorem There exist a diamond of length at most 2n n. J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august / 2

49 We have proved the following bounds: range 2 general r Diamonds 2n n 2n 3(r ) 2 Orphans n 2 n 2(r ) Unbalanced word n n r J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august / 2

50 Algorithms This method gives us polynomial time algorithms to nd orphans, unbalanced words and diamonds. Note: The words found are not necessarily the shortest solution, but they are always below our bound. J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august / 2

51 A x All we need is a polynomial algorithm to nd the shortest sequence of linear transformations f a taking a vector x outside an ane space A. J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august / 2

52 A x We apply all transformations in breadth-rst order. J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august / 2

53 A x For each vector constructed, we check wether it is a linear combination of the previously discovered ones. If so, the node does not need further processing. J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august / 2

54 A x A shortest path out of A is guaranteed to be found. Up to n 2 vectors are processed: each node has n successors, and the number of linearly independent vectors found is at most n. J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august / 2

55 Conclusion Lower bounds Upper bounds Orphans 2n n 2 Diamonds 6 2n 3 2 Unbalanced words? n J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august / 2

56 Conclusion Lower bounds Upper bounds Orphans 2n 2n? Diamonds n? n? Unbalanced words?? J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august / 2

57 Conclusion Lower bounds Upper bounds Orphans 2n n 2 Diamonds 6 2n 3 2 Unbalanced words? n Polynomial time algorithms for nding a shortest orphan or diamond? J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august / 2

58 Conclusion Lower bounds Upper bounds Orphans 2n n 2 Diamonds 6 2n 3 2 Unbalanced words? n Polynomial time algorithms for nding a shortest orphan or diamond? Is there a polynomial bound for k such that at least half of the words of length k are orphans? J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august / 2

59 Thank you J. Kari, P. Vanier, T. Zeume (UTU) Bounds on non-surjective cellular automata 27 august / 2