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15
Symbolic Dynamics and Finite Automata
, 1999
"... this paper, based on notes by R. Beals and M. Spivak, methods of nite semigroups were introduced to obtain some of the results of G. Hedlund. ..."
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Cited by 26 (9 self)
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this paper, based on notes by R. Beals and M. Spivak, methods of nite semigroups were introduced to obtain some of the results of G. Hedlund.
Zeta functions of formal languages
 Trans. Amer. Math. Soc
, 1990
"... ABSTRACT. Motivated by symbolic dynamics and algebraic geometry over finite fields, we define cyclic languages and the zeta function of a language. The main result is that the zeta function of a cyclic language which is recognizable by a finite automation is rational. 1. ..."
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Cited by 15 (2 self)
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ABSTRACT. Motivated by symbolic dynamics and algebraic geometry over finite fields, we define cyclic languages and the zeta function of a language. The main result is that the zeta function of a cyclic language which is recognizable by a finite automation is rational. 1.
Linear Cellular Automata and Fischer Automata
 Parallel Computing
, 1997
"... Introduction Every linear cellular automaton ae can be associated with a regular language L(ae) of finite words: L(ae) is the collection of all finite subwords of configurations that arise after one application of the global map of the cellular automaton. Discussions of the language theoretic aspec ..."
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Cited by 14 (9 self)
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Introduction Every linear cellular automaton ae can be associated with a regular language L(ae) of finite words: L(ae) is the collection of all finite subwords of configurations that arise after one application of the global map of the cellular automaton. Discussions of the language theoretic aspects of linear cellular automata and sofic systems, in particular with respect to their relation to the topology of the space of configurations, can be found in [8], [10] and [7]. In this paper, we will study two measures of complexity associated with L(ae) that are based on minimal finite state machines of a certain type. The first is simply the size of the minimal automaton for L(ae), or, equivalently, the number of left quotients of this language. For the second measure, one can exploit the fact that the languages L(ae) are no
The Size of Power Automata
 Mathematical Foundations of Computer Science, volume 2136 of SLNCS
, 1994
"... We describe a class of simple transitive semiautomata that exhibit full exponential blowup during deterministic simulation. For arbitrary semiautomata we show that it is PSPACEcomplete to decide whether the size of the accessible part of their power automata exceeds a given bound. 1 Motivation C ..."
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Cited by 13 (6 self)
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We describe a class of simple transitive semiautomata that exhibit full exponential blowup during deterministic simulation. For arbitrary semiautomata we show that it is PSPACEcomplete to decide whether the size of the accessible part of their power automata exceeds a given bound. 1 Motivation Consider the following semiautomaton A = h[n]; ; i where [n] = f1; : : : ; ng, = fa; b; cg and the transition function is given by a a cyclic shift on [n]; b the transposition that interchanges 1 and 2, c sends 1 and 2 to 2, identity elsewhere. It is wellknown that A has a transition semigroup of maximal size n n , see [13]. In other words, every function f : [n] ! [n] is already of the form w for some word w. Note that a ; b can be replaced by any other pair of generators for the symmetric group on n points, and c can be replaced by any function whose range has cardinality n 1. It was shown by Salomaa that, for a threeletter alphabet , those are the only choices that produ...
A hierarchy of shift equivalent sofic shifts
"... We define new subclasses of the class of irreducible sofic shifts. These classes form an infinite hierarchy where the lowest class is the class of almost finite type shifts introduced by B. Marcus. We give effective characterizations of these classes with the syntactic semigroups of the shifts. We p ..."
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We define new subclasses of the class of irreducible sofic shifts. These classes form an infinite hierarchy where the lowest class is the class of almost finite type shifts introduced by B. Marcus. We give effective characterizations of these classes with the syntactic semigroups of the shifts. We prove that these classes define invariants shift equivalence (and thus for conjugacy). Finally, we extend the result to the case of reducible sofic shifts.
Profinite groups associated to sofic shifts are free
 Proc. London Math. Soc
"... Abstract. We show that the maximal subgroup of the free profinite semigroup associated by Almeida to an irreducible sofic shift is a free profinite group, generalizing an earlier result of the second author for the case of the full shift (whose corresponding maximal subgroup is the maximal subgroup ..."
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Cited by 5 (5 self)
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Abstract. We show that the maximal subgroup of the free profinite semigroup associated by Almeida to an irreducible sofic shift is a free profinite group, generalizing an earlier result of the second author for the case of the full shift (whose corresponding maximal subgroup is the maximal subgroup of the minimal ideal). A corresponding result is proved for certain relatively free profinite semigroups. We also establish some other analogies between the kernel of the free profinite semigroup and the Jclass associated to an irreducible sofic shift. 1.
PSEUDOVARIETIES DEFINING CLASSES OF SOFIC SUBSHIFTS CLOSED FOR TAKING SHIFT EQUIVALENT
, 2005
"... For a pseudovariety V of ordered semigroups, let S (V) be the class of sofic subshifts whose syntactic semigroup lies in V. It is proved that if V contains Sl − then S (V∗D) is closed for taking shift equivalent subshifts, and conversely, if S (V) is closed for taking conjugate subshifts then V con ..."
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Cited by 3 (3 self)
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For a pseudovariety V of ordered semigroups, let S (V) be the class of sofic subshifts whose syntactic semigroup lies in V. It is proved that if V contains Sl − then S (V∗D) is closed for taking shift equivalent subshifts, and conversely, if S (V) is closed for taking conjugate subshifts then V contains LSl − and S (V) = S (V∗D). Almost finite type subshifts are characterized as the irreducible elements of S (LInv), which gives a new proof that the class of almost finite type subshifts is closed for taking shift equivalent subshifts.
Linear Cellular Automata and Finite Automata
"... Linear cellular automata have a canonical representation in terms of labeled de Bruijn graphs. We will show that these graphs, construed as semiautomata, provide a natural setting for the study of cellular automata. For example, we give a simple algorithm to determine reversibility and surjectivity ..."
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Cited by 1 (0 self)
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Linear cellular automata have a canonical representation in terms of labeled de Bruijn graphs. We will show that these graphs, construed as semiautomata, provide a natural setting for the study of cellular automata. For example, we give a simple algorithm to determine reversibility and surjectivity of the global maps. We also comment on Wolfram’s question about the growth rates of the minimal finite state machines associated with iterates of a cellular automaton. 1
The syntactic graph of a sofic shift
 in STACS 2004
, 2004
"... invariant under shift equivalence ..."
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APPROACH FROM PROFINITE SEMIGROUP THEORY
"... Abstract: It is given a structural conjugacy invariant in the set of pseudowords whose finite factors are factors of a given subshift. Some profinite semigroup tools are developed for this purpose. With these tools a shift equivalence invariant of sofic subshifts is obtained, improving an invariant ..."
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Abstract: It is given a structural conjugacy invariant in the set of pseudowords whose finite factors are factors of a given subshift. Some profinite semigroup tools are developed for this purpose. With these tools a shift equivalence invariant of sofic subshifts is obtained, improving an invariant introduced by Béal, Fiorenzi and Perrin using different techniques. This new invariant is used to prove that some irreducible almost finite type subshifts with the same entropy and the same zeta function are not shift equivalent. shift, subshift, free profinite semigroup, sofic, conjugacy, shift equivalence [2000]20M05, 37B10, 20M35