Nicolas Ollinger

Permutive one-way cellular automata and the finiteness problem for automaton groups

The decidability of the finiteness problem for automaton groups is a well-studied open question on Mealy automata. We connect this question of algebraic nature to the periodicity problem of one-way cellular automata, a dynamical question known to be undecidable in the general case. We provide a first undecidability result on the dynamics of one-way permutive cellular automata, arguing in favor of the undecidability of the finiteness problem for reset Mealy automata.

A small minimal aperiodic reversible Turing machine

We present a complete and reversible Turing machine which is aperiodic.We prove that its associated dynamical system is time symmetric and minimal.The column shift of the presented machine is proved to be substitutive.We prove the undecidability of the existence of a periodic orbit in Turing machines. A simple reversible Turing machine with four states, three symbols and no halting configuration is constructed that has no periodic orbit, simplifying a construction by Blondel, Cassaigne and Nichitiu and positively answering a conjecture by Kari and Ollinger. The constructed machine has other interesting properties: it is symmetric both for space and time and has a topologically minimal associated dynamical system whose column shift is associated to a substitution. Using a particular embedding technique of an arbitrary reversible Turing machine into the one presented, it is proven that the problem of determining if a given reversible Turing machine without halting state has a periodic orbit is undecidable.

The Transitivity Problem of Turing Machines

A Turing machine is topologically transitive if every partial configuration — that is a state, a head position, plus a finite portion of the tape — can reach any other partial configuration, provided that they are completed into proper configurations. We study topological transitivity in the dynamical system models of Turing machines with moving head, moving tape and for the trace-shift and we prove its undecidability. We further study minimality, the property of every configuration reaching every partial configuration.

Substitutions and strongly deterministic tilesets

Substitutions generate hierarchical colorings of the plane. Despite the non-locality of substitution rules, one can extend them by adding compatible local matching rules to obtain locally checkable colorings as the set of tilings of finite tileset. We show that for 2×2 substitutions the resulting tileset can furthermore be chosen strongly deterministic, a tile being uniquely determined by any two adjacent edges. A tiling by a strongly deterministic tileset can be locally reconstructed starting from any infinite path that cross every line and column of the tiling.

Undecidability of the Surjectivity of the Subshift Associated to a Turing Machine

We consider Turing machines (TM) from a dynamical system point of view, and in this context, we associate a subshift by taking the sequence of symbols and states that the head has at each instant. Taking a subshift that select only a part of the state of a system is a classical technic in dynamical systems that plays a central role in their analysis. Surjectivity of Turing machines is equivalent to their reversibility and it can be simply identified from the machine rule. Nevertheless, the associated subshift can be surjective even if the machine is not, and the property results to be undecidable in the symbolic system.

On Aperiodic Reversible Turing Machines (Invited Talk).

A complete reversible Turing machine bijectively transforms configurations consisting of a state and a bi-infinite tape of symbols into another configuration by updating locally the tape around the head and translating the head on the tape. We discuss a simple machine with 4 states and 3 symbols that has no periodic orbit and how that machine can be embedded into other ones to prove undecidability results on decision problems related to dynamical properties of Turing machines.

Fast-Parallel Algorithms for Freezing Totalistic Asynchronous Cellular Automata.

In this paper we study the family of two-state Totalistic Freezing Cellular Automata (FTCA) defined over the triangular grids with von Neumann neighborhoods. We say that a Cellular Automaton is Freezing and Totalistic if the active cells remain unchanged, and the new value of an inactive cell depends only of the sum of its active neighbors. We study the family of FTCA in the context of asynchronous updating schemes (calling them FTACA), meaning that at each time-step only one cell is updated. The sequence of updated sites is called a sequential updating schemes. Given configuration, we say that a site is stable if it remains in the same state over any sequential updating scheme.

Knight Tiles: Particles and Collisions in the Realm of 4-Way Deterministic Tilings

Particles and collisions are convenient construction tools to compute inside tilings and enforce complex sets of tilings with simple tilesets. Locally enforceable particles being incompatible with expansivity in the orthogonal direction, a compromise has to be found to combine both notions in a same tileset. This paper introduces knight tiles: a framework to construct 4-way deterministic tilings, that is tilings completely determined by any infinite diagonal of tiles, for which local particles and collisions with many slopes can still be constructed while being expansive in infinitely many directions. The framework is then illustrated by an elegant yet simple construction to mark a diagonal with a 4-way deterministic knight tileset.