On Factor Universality in Symbolic Spaces

The study of factoring relations between subshifts or cellular automata is central in symbolic dynamics. Besides, a notion of intrinsic universality for cellular automata based on an operation of rescaling is receiving more and more attention in the literature. In this paper, we propose to study the factoring relation up to rescalings, and ask for the existence of universal objects for that simulation relation. In classical simulations of a system S by a system T, the simulation takes place on a specific subset of configurations of T depending on S (this is the case for intrinsic universality). Our setting, however, asks for every configurations of T to have a meaningful interpretation in S. Despite this strong requirement, we show that there exists a cellular automaton able to simulate any other in a large class containing arbitrarily complex ones. We also consider the case of subshifts and, using arguments from recursion theory, we give negative results about the existence of universal objects in some classes

Cellular automaton supercolliders

Gliders in one-dimensional cellular automata are compact groups of non-quiescent and non-ether patterns (ether represents a periodic background) translating along automaton lattice. They are cellular-automaton analogous of localizations or quasi-local collective excitations travelling in a spatially extended non-linear medium. They can be considered as binary strings or symbols travelling along a one-dimensional ring, interacting with each other and changing their states, or symbolic values, as a result of interactions. We analyse what types of interaction occur between gliders travelling on a cellular automaton `cyclotron' and build a catalog of the most common reactions. We demonstrate that collisions between gliders emulate the basic types of interaction that occur between localizations in non-linear media: fusion, elastic collision, and soliton-like collision. Computational outcomes of a swarm of gliders circling on a one-dimensional torus are analysed via implementation of cyclic tag systems

Topological chaos: what may this mean ?

We confront existing definitions of chaos with the state of the art in topological dynamics. The article does not propose any new definition of chaos but, starting from several topological properties that can be reasonably called chaotic, tries to sketch a theoretical view of chaos. Among the main ideas in this article are the distinction between overall chaos and partial chaos, and the fact that some dynamical properties may be considered more chaotic than others

The transitivity problem of Turing machines

International audienceA 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 it is completed into a proper configuration. We characterize topological transitivity and study its computational complexity in the dynamical system models of Turing machines with moving head, moving tape and for the trace-shift. We further study minimality, the property of every configuration reaching every partial configuration

Zigzags in Turing machines ⋆

Abstract. We study one-head machines through symbolic and topological dynamics. In particular, a subshift is associated to the system, and we are interested in its complexity in terms of realtime recognition. We emphasize the class of one-head machines whose subshift can be recognized by a deterministic pushdown automaton. We prove that this class corresponds to particular restrictions on the head movement, and to equicontinuity in associated dynamical systems

The Group of Reversible Turing Machines

Part 2: Regular PapersInternational audienceWe consider Turing machines as actions over configurations in Σ 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

Decidable Properties for Regular Cellular AutomataFourth IFIP International Conference on Theoretical Computer Science- TCS 2006

We investigate decidable properties for regular cellular automata. In particular, we show that regularity itself is an undecidable property and that nilpotency, equicontinuity and positively expansiveness became decidable if we restrict to regular cellular automata