https://ir.lib.hiroshima-u.ac.jp/files/public/2/25576/20141016152033521657/TCS_395_101.pdf

Reversible computing and cellular automata—A survey

Cellular automata (CAs) are dynamical systems which exhibit complex global behavior from simple local interaction and computation. Since the inception of cellular automaton (CA) by von Neumann in 1950s, it has attracted the attention of several researchers over various backgrounds and fields for modeling different physical, natural as well as real-life phenomena. Classically, CAs are uniform. However, non-uniformity has also been introduced in update pattern, lattice structure, neighborhood dependency and local rule. In this survey, we tour to the various types of CAs introduced till date, the different characterization tools, the global behavior of CAs, like universality, reversibility, dynamics etc. Special attention is given to non-uniformity in CAs and especially to non-uniform elementary CAs, which have been very useful in solving several real-life problems.

A survey of cellular automata: types, dynamics, non-uniformity and applications

Reversible computing is a paradigm that has a close relation to physical reversibility. Since microscopic physical laws are reversible, and future computing devices will surely be implemented directly by physical phenomena in the nano-scale level, it is an important problem to know how reversibility can be effectively utilized in computing. Reversible computing systems are defined as systems for which each of their computational configurations has at most one predecessor. Hence, they are “backward deterministic” systems. Though the definition is thus rather simple, these systems reflect physical reversibility very well, and they are suited for investigating how computing systems can be realized in reversible physical environments. In this chapter, we argue reversibility in physics and computing, the significance of reversible computing, and the scope of this volume. Various models of reversible computing ranging from a microscopic level to a macroscopic one are dealt with from the viewpoint of the theory of automata and computing. Terminologies and notations on logic, mathematics, and formal languages used in this volume are also summarized.

Theory of Reversible Computing

This paper examines the claim that cellular automata (CA) belonging to Class III (in Wolfram’s classification) are capable of (Turing universal) computation. We explore some chaotic CA (believed to belong to Class III) reported over the course of the CA history, that may be candidates for universal computation, hence spurring the discussion on Turing universality on both Wolfram’s classes III and IV.

https://repository.uaeh.edu.mx/bitstream/bitstream/handle/123456789/8015/jca_seck.pdf;sequence=1

Computation and Universality: Class IV versus Class III Cellular Automata

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 traveling in a spatially extended nonlinear medium They can be considered as binary strings or symbols traveling along a one-dimensional ring, interacting with each other and changing their states, or symbolic values, as a result of interactions We analyze what types of interaction occur between gliders traveling 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 nonlinear media: fusion, elastic collision, and soliton-like collision Computational outcomes of a swarm of gliders circling on a one-dimensional torus are analyzed via implementation of cyclic tag systems

Cellular automaton supercolliders

/pdf/procedures-for-calculating-reversible-one-dimensional-483xyl2152.pdf

Procedures for calculating reversible one-dimensional cellular automata

Rule 110 is a complex elementary cellular automaton able of supporting universal computation and complicated collision-based reactions between gliders. We propose a representation for coding initial conditions by means of a finite subset of regular expressions. The sequences are extracted both from de Bruijn diagrams and tiles specifying a set of phases fi_1 for each glider in Rule 110. The subset of regular expressions is explained in detail.

Determining a regular language by glider-based structures called phases fi_1 in Rule 110

This paper implements the cyclic tag system (CTS) in Rule 110 developed by Cook in [1, 2] using regular expressions denominated phases fi_1 [3]. The main problem in CTS is coding the initial condition based in a system of gliders. In this way, we develop a method to control the periodic phases of the strings representing all gliders until now known in Rule 110, including glider guns. These strings form a subset of regular expressions implemented in a computational system to facilitate the construction of CTS. Thus, these phases are useful to establish distances and positions for every glider and then to delineate more sophisticated components or packages of gliders. In this manuscript, it is possible to find differences with the results exposed in Wolfram's book [2], inclusively some mistakes which avoid to obtain an appropriated realization of CTS in Rule 110; fortunately, these irregularities were discussed and clarified by Cook.

Reproducing the cyclic tag system developed by Matthew Cook with Rule 110 using the phases f1_1

Cellular automata are discrete dynamical systems based on simple local interactions among its components, but sometimes they are able to yield quite a complex global behavior. A special kind of cellular automaton is the one where the global behavior is invertible, this type of cellular automaton is called reversible. In this paper we expose the graph representation provided by de Bruijn diagrams of reversible one-dimensional cellular automata and we define the distinct types of paths between self-loops in such diagrams. With this, we establish the way in which a reversible one-dimensional cellular automaton generates sequences composed by subsequences produced by the undefined repetition of a single state. Using this graph presentation, we define Welch diagrams which will be useful for proving that all the extensions of the ancestors in reversible one-dimensional cellular automata are equivalent to the full shift. In this way an important result of this paper is that we understand and classify the behavior of a reversible automaton analyzing the extensions of the ancestors of a given sequence by means of symbolic dynamics tools. A final example illustrates the results exposed in the paper.

Sergio V. Chapa Vergara

Papers

Procedures for calculating reversible one-dimensional cellular automata

Determining a regular language by glider-based structures called phases fi_1 in Rule 110

Reproducing the cyclic tag system developed by Matthew Cook with Rule 110 using the phases f1_1

Determining a regular language by glider-based structures called phases fi_1 in Rule 110

Extensions in reversible one-dimensional cellular automata are equivalent with the full shift