M.-P. Beal and D. Perrin. Symbolic dynamics and nite automata. In G. Rozenberg and A. Salomaa, editors, Handbook of Formal Languages, volume 2, chapter 10. Springer Verlag, 1997.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... rst When are (A) and (A) equal to or close to the upper bound In particular, what happens for semiautomata, and for transitive semiautomata When is the sink free version of pow(A) again transitive These questions are originally motivated by the study of discrete dynamical systems, see [12, 1] for details and references on the topic. Brie y, let be an alphabet, and denote by the collection of all biin nite words over , usually referred to as con gurations in this context. We can associate every con guration X with its cover cov(X) the set of all nite factors of X. ....

M.-P. Beal and D. Perrin. Symbolic dynamics and nite automata. In G. Rozenberg and A. Salomaa, editors, Handbook of Formal Languages, volume 2, chapter 10. Springer Verlag, 1997.

....we present two typical sample sessions. Fairly detailed descriptions of earlier versions of the package can be found in [8, 9] 1. 1 Entropy of So c Shifts Suppose you wish to determine the entropy of the so c subshift associated with a particular one dimensional cellular automaton, see [5, 1] for more background information) Here is a short session in automata that shows the necessary calculations. The dialogue is captured the way it would appear in the plain text interface. For more elaborate examples using the notebook frontend, see http: www.cs.cmu.edu sutner. The rst command ....

....we calculate the syntactic semigroup of a regular language. Consider L = f x 2 fa; bg x 3 = a g, the set of all words having an a in the third position from the end. m = MinimizeFA[ IthSymbolFA[ a, 3 ] S,W,eq = SyntacticSG[ m, Equations True ] S SG[T[2, 3, 5, 7, 5, 7, 3, 2] T[1, 4, 6, 8, 6, 8, 4, 1], T[3, 5, 5, 3, 5, 3, 5, 3] T[4, 6, 6, 4, 6, 4, 6, 4] T[2, 7, 7, 2, 7, 2, 7, 2] T[1, 8, 8, 1, 8, 1, 8, 1] T[5, 5, 5, 5, 5, 5, 5, 5] T[6, 6, 6, 6, 6, 6, 6, 6] T[7, 7, 7, 7, 7, 7, 7, 7] T[8, 8, 8, 8, 8, 8, 8, 8] T[3, 3, 3, 3, 3, 3, 3, 3] T[4, 4, 4, 4, 4, 4, 4, 4] T[2, 2, 2, 2, 2, 2, ....

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M.-P. Beal and D. Perrin. Symbolic dynamics and nite automata. In G. Rozenberg and A. Salomaa, editors, Handbook of Formal Languages, volume 2, chapter 10. Springer Verlag, 1997.

.... rst When are (A) and (A) equal to or close to the upper bound In particular, what happens for semiautomata, and for transitive semiautomata When is the sink free version of pow(A) again transitive These questions are originally motivated by the study of discrete dynamical systems, see [12, 1] for details and references on the topic. Brie y, let be an alphabet, and denote by 1 the collection of all biin nite words over , usually referred to as con gurations in this context. We can associate every con guration X with its cover cov(X) the set of all nite factors of X. ....

M.-P. Beal and D. Perrin. Symbolic dynamics and nite automata. In G. Rozenberg and A. Salomaa, editors, Handbook of Formal Languages, volume 2, chapter 10. Springer Verlag, 1997.

....15 6. Zeta functions, subshifts of nite type and circular codes. In this section, we present a number of results on interrelated objects which are connected with cyclic permutation of words. We begin with notions classical in symbolic dynamics (see [25] or [24] for a general reference; see [13] or [22] for the link with nite automata) 6.1. Subshifts of nite type. A subshift is a set of biin nite words on a nite alphabet A which avoids a given set F of forbidden words. It is a topological space as a closed subset of the space A Z of functions from Z into the set A. The full shift ....

M.-P. B eal and D. Perrin, Symbolic dynamics and nite automata, in Handbook of Formal Languages, G. Rosenberg and A. Salomaa, eds., vol. 2, SpringerVerlag, 1997, ch. 10.

....Dynamics Symbolic Dynamics is a eld born at the beginning of the 20 s with the work in topology of Marston Morse (cf. 26] Later the theory was developed as a branch of ergodic theory. There are deep connections between the theory of automata and formal languages and Symbolic Dynamics (cf. [27], 6] and references therein) Several results from Symbolic Dynamics have a natural interpretation in terms of formal languages and conversely. We present in this section a short introduction to the basic concepts of Symbolic Dynamics and to their relations with the theory of Formal Languages ....

....to prove that the class of so c systems is closed under image and inverse image by a homomorphism. Then the notion of so c is invariant under conjugacy in the sense that any system that is conjugate to a so c system is again so c. A long standing open problem of the theory (cf. 31] 23] and [27] ) is the conjugacy problem for so c systems, i.e. to decide whether two given so c systems are conjugate or not. We should also mention that there is a strong connection between so c systems and the theory of constrained coding, a sub eld of coding theory (cf. 1] Moreover one has (cf. ....

D. Perrin. Symbolic dynamics and nite automata, invited lecture in Proc. MFCS'95, Lecture Notes in Comput. Sci., 969. 33

....and (m 1; a) local. A deterministic automaton is local if and only if the previous condition is satis ed with a null anticipation. It is then said to be m local. The following known property gives a polynomial time decision procedure to check whether a nite automaton is local (see for instance [7, 23, 9]) Proposition 2.1. Let A be an automaton which has a strongly connected graph or which is unambiguous. The two following properties are equivalent. i) the automaton A is local. ii) the automaton A has at most one cycle with a given label. A so c system which can be recognized by a local ....

.... = ac) b) d) on A = fa; b; c; dg. This permutation has a xed point in A, the point b, and it has a xed channel sequence b . It is known that any irreducible so c shift has a minimal representation, that is a minimal nite state machine which recognizes it (see for instance [7] 23] [9]) This minimal representation is a deterministic nite automaton 10 d 2 3 1 b b b c a c a 4 Fig. 2.6. Another channel invariant by permutation called the Fischer cover of the so c system. The so c shift is of nite type if and only if its minimal automaton is local. Let be a xed ....

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M.-P. B eal and D. Perrin, Symbolic dynamics and nite automata, in Handbook of Formal Languages, G. Rozenberg and A. Salomaa, eds., vol. 2, SpringerVerlag, 1997, ch. 10.

....some of them are given completely, even if sometimes condensed. Some others are only sketched or not even given here, as not being in the scope of this survey. The material presented is an extended version of a survey by the second author at the conference MFCS in September 1995 in Prague [51]. We would like to thank many people for their help during the preparation of this work and, in particular, Fr ed erique Bassino, V eronique Bruy ere, Aldo De Luca, and Paul Schupp. 2 Symbolic dynamical systems We present in this Section a short introduction to the concepts of symbolic ....

Dominique Perrin. Symbolic dynamics and nite automata. In Jiri Wiedermann and Petr Hajek, editors, Mathematical Foundations of Computer Science 1995, volume 969 of Lecture Notes in Computer Science, pages 94-104. Springer Verlag, 1995.

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B eal, M.-P., and Perrin, D. Symbolic dynamics and nite automata. In Handbook of Formal Languages, G. Rosenberg and A. Salomaa, Eds., vol. 2. Springer, 1997, ch. 10.

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M.-P. Beal and D. Perrin. Symbolic dynamics and nite automata. In G. Rozenberg and A. Salomaa, editors, Handbook of Formal Languages, volume 2, chapter 10. Springer Verlag, 1997.

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M.-P. Beal and D. Perrin. Symbolic dynamics and nite automata. In G. Rozenberg and A. Salomaa, editors, Handbook of Formal Languages, volume 2, chapter 10. Springer Verlag, 1997.

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M.-P. Beal and D. Perrin. Symbolic dynamics and nite automata. In G. Rozenberg and A. Salomaa, editors, Handbook of Formal Languages, volume 2, chapter 10. Springer Verlag, 1997.

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M.-P. Beal and D. Perrin. Symbolic dynamics and nite automata. In G. Rozenberg and A. Salomaa, editors, Handbook of Formal Languages, volume 2, chapter 10. Springer Verlag, 1997.

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