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.

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

We describe a class of simple transitive semiautomata that exhibit full exponential blow-up during deterministic simulation. For arbitrary semiautomata we show that it is PSPACE-complete 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 well-known 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 three-letter alphabet , those are the only choices that produ...

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.

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 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