27th IFIP WG 1.5 International Workshop on Cellular Automata and Discrete Complex Systems (AUTOMATA 2021)

The fixed point construction is a method for designing tile sets and cellular automata with highly nontrivial dynamical and computational properties. It produces an infinite hierarchy of systems where each layer simulates the next one. The simulations are implemented entirely by computations of Turing machines embedded in the tilings or spacetime diagrams. We present an overview of the construction and list its applications in the literature.</p

Hierarchy and Expansiveness in Two-Dimensional Subshifts of Finite Type

Using a deterministic version of the self-similar (or hierarchical, or fixed-point ) method for constructing 2-dimensional subshifts of finite type (SFTs), we construct aperiodic 2D SFTs with a unique direction of non-expansiveness and prove that the emptiness problem of SFTs is undecidable even in this restricted case. As an additional application of our method, we characterize the sets of directions that can be the set of non-expansive directions of 2D SFTs.Comment: 72 pages, main body of the author's PhD Thesis, most of the results obtained in collaboration with Pierre Guillo

Hierarchy and Expansiveness in Two-Dimensional Subshifts of Finite Type

Subshifts are sets of configurations over an infinite grid defined by a set of forbidden patterns. In this thesis, we study two-dimensional subshifts offinite type (2D SFTs), where the underlying grid is Z2 and the set of for-bidden patterns is finite. We are mainly interested in the interplay between the computational power of 2D SFTs and their geometry, examined through the concept of expansive subdynamics. 2D SFTs with expansive directions form an interesting and natural class of subshifts that lie between dimensions 1 and 2. An SFT that has only one non-expansive direction is called extremely expansive. We prove that in many aspects, extremely expansive 2D SFTs display the totality of behaviours of general 2D SFTs. For example, we construct an aperiodic extremely expansive 2D SFT and we prove that the emptiness problem is undecidable even when restricted to the class of extremely expansive 2D SFTs. We also prove that every Medvedev class contains an extremely expansive 2D SFT and we provide a characterization of the sets of directions that can be the set of non-expansive directions of a 2D SFT. Finally, we prove that for every computable sequence of 2D SFTs with an expansive direction, there exists a universal object that simulates all of the elements of the sequence. We use the so called hierarchical, self-simulating or fixed-point method for constructing 2D SFTs which has been previously used by Ga´cs, Durand, Romashchenko and Shen.Siirretty Doriast

Cellular automata with complicated dynamics

A subshift is a collection of bi-infinite sequences (configurations) of symbols where some finite patterns of symbols are forbidden to occur. A cellular automaton is a transformation that changes each configuration of a subshift into another one by using a finite look-up table that tells how any symbol occurring at any possible context is to be changed. A cellular automaton can be applied repeatedly on the configurations of the subshift, thus making it a dynamical system. This thesis focuses on cellular automata with complex dynamical behavior, with some different definitions of the word “complex”. First we consider a naturally occurring class of cellular automata that we call multiplication automata and we present a case study with the point of view of symbolic, topological and measurable dynamics. We also present an application of these automata to a generalized version of Mahler’s problem. For different notions of complex behavior one may also ask whether a given subshift or class of subshifts has a cellular automaton that presents this behavior. We show that in the class of full shifts the Lyapunov exponents of a given reversible cellular automaton are uncomputable. This means that in the dynamics of reversible cellular automata the long term maximal propagation speed of a perturbation made in an initial configuration cannot be determined in general from short term observations. In the last part we construct, on all mixing sofic shifts, diffusive glider cellular automata that can decompose any finite configuration into two distinct components that shift into opposing direction under repeated action of the automaton. This implies that every mixing sofic shift has a reversible cellular automaton all of whose directions are sensitive in the sense of the definition of Sablik. We contrast this by presenting a family of synchronizing subshifts on which all reversible cellular automata always have a nonsensitive direction

Turku Centre for Computer Science – Annual Report 2013

Due to a major reform of organization and responsibilities of TUCS, its role, activities, and even structures have been under reconsideration in 2013. The traditional pillar of collaboration at TUCS, doctoral training, was reorganized due to changes at both universities according to the renewed national system for doctoral education. Computer Science and Engineering and Information Systems Science are now accompanied by Mathematics and Statistics in newly established doctoral programs at both University of Turku and &Aring;bo Akademi University. Moreover, both universities granted sufficient resources to their respective programmes for doctoral training in these fields, so that joint activities at TUCS can continue. The outcome of this reorganization has the potential of proving out to be a success in terms of scientific profile as well as the quality and quantity of scientific and educational results.&nbsp; International activities that have been characteristic to TUCS since its inception continue strong. TUCS&rsquo; participation in European collaboration through EIT ICT Labs Master&rsquo;s and Doctoral School is now more active than ever. The new double degree programs at MSc and PhD level between University of Turku and Fudan University in Shaghai, P.R.China were succesfully set up and are&nbsp; now running for their first year. The joint students will add to the already international athmosphere of the ICT House.&nbsp; The four new thematic reseach programmes set up acccording to the decision by the TUCS Board have now established themselves, and a number of events and other activities saw the light in 2013. The TUCS Distinguished Lecture Series managed to gather a large audience with its several prominent speakers. The development of these and other research centre activities continue, and&nbsp; new practices and structures will be initiated to support the tradition of close academic collaboration.&nbsp; The TUCS&rsquo; slogan Where Academic Tradition Meets the Exciting Future has proven true throughout these changes. Despite of the dark clouds on the national and European economic sky, science and higher education in the field have managed to retain all the key ingredients for success. Indeed, the future of ICT and Mathematics in Turku seems exciting.</p