Part 2: Regular PapersInternational audienceWe consider Turing machines as actions over configurations in ${\mathrm{\Sigma <!--\; \Sigma \; -->}}^{{\mathbb{Z}}^d}$which only change them locally around a marked position that can move and carry a particular state. In this setting we study the monoid of Turing machines and the group of reversible Turing machines. We also study two natural subgroups, namely the group of finite-state automata, which generalizes the topological full groups studied in the theory of orbit-equivalence, and the group of oblivious Turing machines whose movement is independent of tape contents, which generalizes lamplighter groups and has connections to the study of universal reversible logical gates. Our main results are that the group of Turing machines in one dimension is neither amenable nor residually finite, but is locally embeddable in finite groups, and that the torsion problem is decidable for finite-state automata in dimension one, but not in dimension two

International audienceWe study property testing in the context of distributed computing , under the classical CONGEST model. It is known that testing whether a graph is triangle-free can be done in a constant number of rounds, where the constant depends on how far the input graph is from being triangle-free. We show that, for every connected 4-node graph H, testing whether a graph is H-free can be done in a constant number of rounds too. The constant also depends on how far the input graph is from being H-free, and the dependence is identical to the one in the case of testing triangle-freeness. Hence, in particular, testing whether a graph is K4-free, and testing whether a graph is C4-free can be done in a constant number of rounds (where K k denotes the k-node clique, and C k denotes the k-node cycle). On the other hand, we show that testing K k-freeness and C k-freeness for k ≥ 5 appear to be much harder. Specifically , we investigate two natural types of generic algorithms for testing H-freeness, called DFS tester and BFS tester. The latter captures the previously known algorithm to test the presence of triangles, while the former captures our generic algorithm to test the presence of a 4-node graph pattern H. We prove that both DFS and BFS testers fail to test K k-freeness and C k-freeness in a constant number of rounds for k ≥ 5

We discuss the set of subgroups of the automorphism group of a full shift, and submonoids of its endomorphism monoid. We prove closure under direct products in the monoid case, and free products in the group case. We also show that the automorphism group of a full shift embeds in that of an uncountable sofic shift. Some undecidability results are obtained as corollaries.

Abstract. In this article, we prove that for all pairs of primitive Pisot or uniform substitutions with the same dominating eigenvalue, there exists a finite set of block maps such that every block map between the corresponding subshifts is an element of this set, up to a shift. IntroductionTo a substitution τ with a fixed point x, we can attach a subshift X τ in a natural way: by taking the orbit closure of x. If the substitution is primitive, the resulting subshift is independent of the choice of x, and is uniformly recurrent (also called minimal), which is an interesting dynamical property. Not all uniformly recurrent subshifts arise this way, and intuitively the ones generated by substitutions have a simple self-similar structure. In this article, we study the sets of morphisms (shift-commuting continuous functions) between pairs of such subshifts, with the goal of showing that also these sets are in some sense quite simple.Morphisms between associated subshifts of substitutions have also been studied in for example [7], where it was proved that all endomorphisms of such subshifts are automorphisms. In [4], it was shown that every endomorphism of a binary uniform primitive substitution is a shift map, possibly composed with a bit flip. In [11], this was generalized by proving that for certain pairs of primitive uniform substitutions, up to powers of the shift, there are finitely many block maps between their subshifts. In fact, [11] considers not only continuous, but measurable functions, so their results are stronger; we only consider the topological case in this paper, but see Section 8 for some related questions. There is also some work on the non-uniform case. For example, in [14] it was shown that the endomorphisms of Sturmian substitutions are shift maps. Also, results of a similar flavor (though not directly related) were obtained in [5,3], where an upper bound was obtained for the number of letters in a substitution ρ whose subshift is topologically conjugate to that of a fixed substitution τ . ⋆ Research supported by the Academy of Finland Grant 131558We extend the (topological) results of [11] to a more general class of substitution pairs, namely, pairs (τ, ρ) of primitive substitutions of 'balanced growth' with the same dominating eigenvalue (asymptotic growth rate of words). The balanced growth property is implied by both uniformness and the Pisot property, and defined in Section 6. Our main theorem is the following.

Abstract. It is well-known that the Toffoli gate and the negation gate together yield a universal gate set, in the sense that every permutation of {0, 1} n can be implemented as a composition of these gates. Since every bit operation that does not use all of the bits performs an even permutation, we need to use at least one auxiliary bit to perform every permutation, and it is known that one bit is indeed enough. Without auxiliary bits, all even permutations can be implemented. We generalize these results to non-binary logic: If A is a finite set of odd cardinality then a finite gate set can generate all permutations of A n for all n, without any auxiliary symbols. If the cardinality of A is even then, by the same argument as above, only even permutations of A n can be implemented for large n, and we show that indeed all even permutations can be obtained from a finite universal gate set. We also consider the conservative case, that is, those permutations of A n that preserve the weight of the input word. The weight is the vector that records how many times each symbol occurs in the word. It turns out that no finite conservative gate set can, for all n, implement all conservative even permutations of A n without auxiliary bits. But we provide a finite gate set that can implement all those conservative permutations that are even within each weight class of A n .

We compute an explicit presentation of the (topological) automorphism group of a particular Toeplitz subshift with subquadratic complexity. The automorphism group is a non-finitely generated subgroup of rational numbers, or alternatively the 5-adic integers, under addition, the shift map corresponding to the rational number 1. The group is ( (5/2) i | i ∈ N , +) ≤ (Q, +).

We study the central objects of symbolic dynamics, that is, subshifts and block maps, from the perspective of basic category theory, and present several natural categories with subshifts as objects and block maps as morphisms. Our main goals are to find universal objects in these symbolic categories, to classify their block maps based on their category theoretic properties, to prove category theoretic characterizations for notions arising from symbolic dynamics, and to establish as many natural properties (finite completeness, regularity etc.) as possible. Existing definitions in category theory suggest interesting new problems in symbolic dynamics. Our main technical contributions are the solution to the dual problem of the Extension Lemma and results on certain types of conserved quantities, suggested by the concept of a coequalizer.

Artículo de publicación ISIWe study the dynamics of majority automata networks when the vertices are updated according to a block sequential updating scheme. In particular, we show that the complexity of the problem of predicting an eventual state change in some vertex, given an initial configuration, is PSPACE-complete.CONICYT-Becas Chile 72130083 FONDECYT 1140090 ECOS C12E05 Basal project PFB-03 Centro de Modelamiento Matematico CNRS UMI 2807 Universidad de Chile Academy of Finland 13155