30 research outputs found
Cellular automata and strongly irreducible shifts of finite type
AbstractIf A is a finite alphabet and Γ is a finitely generated amenable group, Ceccherini-Silberstein, Machı̀ and Scarabotti have proved that a local transition function defined on the full shift AΓ is surjective if and only if it is pre-injective; this equivalence is the so-called Garden of Eden theorem. On the other hand, when Γ is the group of the integers, the theorem holds in the case of irreducible shifts of finite type as a consequence of a theorem of Lind and Marcus but it no longer holds in the two-dimensional case.Recently, Gromov has proved a GOE-like theorem in the much more general framework of the spaces of bounded propagation. In this paper we apply Gromov's theorem to our class of spaces proving that all the properties required in the hypotheses of this theorem are satisfied.We give a definition of strong irreducibility that, together with the finite-type condition, it allows us to prove the GOE theorem for the strongly irreducible shifts of finite type in AΓ (provided that Γ is amenable). Finally, we prove that the bounded propagation property for a shift is strictly stronger than the union of strong irreducibility and finite-type condition
Topological properties of cellular automata on trees
We prove that there do not exist positively expansive cellular automata
defined on the full k-ary tree shift (for k>=2). Moreover, we investigate some
topological properties of these automata and their relationships, namely
permutivity, surjectivity, preinjectivity, right-closingness and openness.Comment: In Proceedings AUTOMATA&JAC 2012, arXiv:1208.249
The syntactic graph of a sofic shift is invariant under shift equivalence
International audienceWe de ne a new invariant for shift equivalence of so fic shifts. This invariant, that we call the syntactic graph of a so fic shift, is the directed acyclic graph of characteristic groups of the non null regular D-classes of the syntactic semigroup of the shift
Bounds for the generalized repetition threshold
AbstractThe notion of the repetition threshold, which is the object of Dejean’s conjecture (1972), was generalized by Ilie et al. (2005) [8] to include the lengths of the avoided words. We give a lower and an upper bound on this generalized repetition threshold
Multipass automata and group word problems
We introduce the notion of multipass automata as a generalization of pushdown
automata and study the classes of languages accepted by such machines. The
class of languages accepted by deterministic multipass automata is exactly the
Boolean closure of the class of deterministic context-free languages while the
class of languages accepted by nondeterministic multipass automata is exactly
the class of poly-context-free languages, that is, languages which are the
intersection of finitely many context-free languages. We illustrate the use of
these automata by studying groups whose word problems are in the above classes
A hierarchy of irreducible sofic shifts
International audienceWe 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
Groups, Graphs, Languages, Automata, Games and Second-order Monadic Logic
In this paper we survey some surprising connections between group theory, the
theory of automata and formal languages, the theory of ends, infinite games of
perfect information, and monadic second-order logic
Periodic configurations of subshifts on groups
International audienceWe study the density of periodic configurations for shift spaces defined on (the Cayley graph of) a finitely generated group. We prove that in the case of a full shift on a residually finite group and in that of a group shift space on an abelian group, the periodic configurations are dense. In the one-dimensional case we prove the density for irreducible sofic shifts. In connection with this we study the surjunctivity of cellular automata and local selfmappings. Some related decision problems for shift spaces of finite type are also investigated