Emmanuel Jeandel

Decidable and Undecidable Problems about Quantum Automata

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 is known that strict and nonstrict thresholds both lead to undecidable problems.

Quantum automata and algebraic groups

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.

Topological Automata

We give a new, topological definition of automata that extends previous definitions of probabilistic and quantum automata. We then are able to prove in a unified framework that deterministic or non-deterministic probabilistic and quantum automata recognise only regular languages with an isolated threshold.

STRUCTURAL ASPECTS OF TILINGS

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 combinatorial and the other topological. These two approaches have independent merits and, once combined, provide somehow surprising results. The particular case where the set of produced tilings is countable is deeply investigated while we prove that the uncountable case may have a completely different structure. We introduce a pattern preorder and also make use of Cantor-Bendixson rank. Our first main result is that a tile-set that produces only periodic tilings produces only a finite number of them. Our second main result exhibits a tiling with exactly one vector of periodicity in the countable case.

Universality in Quantum Computation

We introduce several new definitions of universality for sets of quantum gates, and prove separation results for these definitions. In particular, we prove that realisability with ancillas is different from the classical notion of completeness. We give a polynomial time algorithm of independent interest which decides if a subgroup of a classical group (SO n , SU n , SL n ...) is Zariski dense, thus solving the decision problem for the completeness. We also present partial methods for the realisability with ancillas.

Hardness of Conjugacy, Embedding and Factorization of multidimensional Subshifts of Finite Type

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 Sigma^0_3-complete and that the conjugacy and embedding problems are Sigma^0_1-complete in the arithmetical hierarchy.

Playing with Conway's problem

The centralizer of a language is the maximal language commuting with it. The question, raised by Conway in [J.H. Conway, Regular Algebra and Finite Machines, Chapman Hall, 1971], whether the centralizer of a rational language is always rational, recently received a lot of attention. In Kunc [M. Kunc, The power of commuting with finite sets of words, in: Proc. of STACS 2005, in: LNCS, vol. 3404, Springer, 2005, pp. 569-580], a strong negative answer to this problem was given by showing that even complete co-recursively enumerable centralizers exist for finite languages. Using a combinatorial game approach, we give here an incremental construction of rational languages embedding any recursive computation in their centralizers.

Hardness of conjugacy, embedding and factorization of multidimensional subshifts

Knowing whether two SFTs of dimension higher than two are conjugate or if one embeds in the other is Σ 1 0 -complete.Deciding whether one of two multidimensional SFTs, effective or sofic shifts factor onto the other is Σ 3 0 -complete.For effective subshifts, all the results are also true in dimension one. 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 in dimensions d 1 for subshifts of finite type and sofic shifts and in dimensions d ? 1 for effective shifts. In particular, we prove that the conjugacy, factorization and embedding problems are Σ 3 0 -complete for sofic and effective subshifts and that they are Σ 1 0 -complete for SFTs, except for factorization which is also Σ 3 0 -complete.

Topological automata

We give here a new, topological, definition of automata that extends previous definitions of probabilistic and quantum automata. We then prove in an unified framework that deterministic or non-deterministic probabilistic and quantum automata with an isolatedthreshold recognize only regular languages.