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We study the following decision problem: is the language recognized by a quantum finite automaton empty or nonempty? We prove that this problem is decidable or undecidable depending on whether recognition is defined by strict or nonstrict thresholds. This result is in contrast with the corresponding situation for probabilistic finite automata, for which it… (More)

In this paper, we study the structure of the set of tilings produced by any given tile-set. For better understanding this structure, we address the set of finite patterns that each tiling contains. This set of patterns can be analyzed in two different contexts: the first one is combi-natorial and the other topological. These two approaches have independent… (More)

We show that several problems which are known to be undecidable for probabilistic automata become decidable for quantum finite automata. Our main tool is an algebraic result of independent interest: we give an algorithm which, given a finite number of invertible matrices, computes the Zariski closure of the group generated by these matrices.

We know that tilesets that can tile the plane always admit a quasi-periodic tiling [4, 8], yet they hold many uncomputable properties [3, 11, 21, 25]. The quasi-periodicity function is one way to measure the regularity of a quasi-periodic tiling. We prove that the tilings by a tileset that admits only quasi-periodic tilings have a recursively (and… (More)

In this paper we emphasize the links between model theory and tilings. More precisely, after giving the definitions of what tilings are, we give a natural way to have an interpretation of the tiling rules in first order logics. This opens the way to map some model theoretical properties onto some properties of sets of tilings, or tilings themselves.

In this paper we are interested in computability aspects of subshifts and in particular Turing degrees of 2-dimensional SFTs (i.e. tilings). To be more precise, we prove that given any Π 0 1 class P of {0, 1} N there is a SFT X such that P × Z 2 is recursively homeomorphic to X \ U where U is a computable set of points. As a consequence, if P contains a… (More)

Dimitri Grigoriev has shown that for any family of N vectors in the d-dimensional linear space E = (F 2) d , there exists a vector in E which is orthogonal to at least N/3 and at most 2N/3 vectors of the family. We show that the range [N/3, 2N/3] can be replaced by the much smaller range [N/2 − √ N/2, N/2 + √ N /2] and we give an efficient, deterministic… (More)

Subshifts of finite type are sets of colorings of the plane defined by local constraints. They can be seen as a discretization of continuous dynamical systems. We investigate here the hardness of deciding factorization, conjugacy and embedding of subshifts of finite type (SFTs) in dimension d > 1. In particular, we prove that the factorization problem is Σ… (More)