Automata that Construct as well as Compute

This article describes in detail an implementation of John von Neumann's self-reproducing machine. Self-reproduction is achieved as a special case of construction by a universal constructor. The th...

A class of fine-grained (FG) parallel models and algorithms is defined as a generalization of Cellular Automata (CA). It comprises all CA-modifications, in which two main CA-properties (locality and parallelism of intercell interaction) are preserved, no constraint being imposed on state alphabets and transition functions. A set of methods for composing a complex FG-algorithm out of a number of simple ones is proposed. To make compatible FG-algorithms with different alphabets, a number of algebraic operations on cellular arrays are introduced. The set of proposed composition methods has a two-level structure: the lower level comprises composition of cell transition functions, while the higher level deals with global operators on cellular arrays. For each type of proposed methods an example is given and the domain of application is determined.

Toward a viable, self-reproducing universal computer

We investigate the relative computing power of Turing machines with differences in the number of work tapes, heads pro work tape, instruction repertoire etc. We concentrate on the k-tape, k-head and k-head jump models as well as the 2-way multihead finite automata with and without jumps. Differences in computing power between machines of unlike specifications emerge under the real-time restriction. In particular it is shown that k+1 heads are more powerful than k heads for real-time Turing machines.

Computing pioneer and polymath John von Neumann introduced the concept of a Universal Constructor as part of his effort to develop a mathematical theory describing living organisms. A Universal Constructor is a kinematic machine able to manipulate and assemble primitive building blocks. Von Neumann showed how this hypothetical constructor, being itself composed of the same primitive blocks, could self-reproduce and evolve. Remarkably, although this model system pre-dates the discovery of the genetic code, it applies to cell molecular biology as well as man-made machines. This chapter describes some key laboratory demonstrations related to universal construction and machine self-reproduction, and discusses parallels between reproduction processes in machines and biological cells.

The main thesis of this chapter is that a dynamical viewpoint allows us to better understand some important foundational issues of computation theory. Effective procedures are traditionally studied from two different but complementary points of view. The first approach is concerned with individuating those numeric functions that are effectively calculable. This approach reached its systematization with the theory of the recursive functions (Gödel, Church Kleene).This theory is not directly concerned with computing devices or computations. Rather, the effective calculability of a recursive function is guaranteed by the algorithmic nature of its definition. In contrast, the second approach focuses on a family of abstract mechanisms, which are then typically used to compute or recognize numeric functions, sets of numbers, or numbers. These devices can be divided into two broad categories: automata or machines (Turing and Post), and systems of rules for symbol manipulation (Post). The mechanisms that have been studied include: a. Automata or Machines 1. gate-nets and McCulloch-Pitts nets 2. finite automata (Mealy and Moore machines) 3. push-down automata 4. stack automata 5. Turing machines 6. register machines 7. wang machines 8. cellular automata b. Systems of Rules 9. monogenic production systems in general 10. monogenic Post canonical systems 11. monogenic Post normal systems 12. tag systems. I call any device studied by computation theory a computational system. Computation theory is traditionally interested in studying the relations between each type of computational system and the others, and in establishing what class of numeric functions each type can compute. Accordingly one proves two kinds of theorem: (1) that systems of a given type emulate systems of another type (examples: Turing machines emulate register machines and cellular automata; cellular automata emulate Turing machines, etc.), and (2) that a certain type of system is complete relative to the class of the (partial) recursive functions or, in other words, that this type of system can compute all and only the (partial) recursive functions (examples of complete systems: Turing machines, register machines, cellular automata, tag systems, etc.). All different types of computational systems have much in common. Nevertheless, it is not at all clear exactly which properties these mechanisms share.

This chapter describes a Turing machine built from patterns in the Conway’s Game of Life cellular automaton. It outlines the architecture of the construction, the structure of its parts and explains how the machine works. It also illustrates the principle choices made during the design0 [17]. Background information about Turing machines, minimal universal Turing machines (those that simulate any other Turing machine) and non-erasing Turing machines can be found in [20,7,15,14,18]. The importance of Turing machines is the existence of universal Turing machines. Thus a machine that can simulate any Turing machine can simulate a universal Turing machine. It has been proved that Turing machines can be simulated by many types of machine: cellular automata (as one can see in this and other chapters of this book), random access machines [4], register machines [1] and others. In particular Minsky [16] describes a register machine which can simulate a Turing machine. The registers have the unusual property of being able to store positive numbers of any size. Remarkably, a long time ago Conway described [1] a method of constructing a register of this form in the Game of Life. This is discussed later in this chapter.

It is speculated that there is a relationship between 1/f noise and computational universality in cellular automata. We use genetic algorithms to search for one-dimensional and two-state, five-neighbor cellular automata which have 1/f-type spectrum. A power spectrum is calculated from the evolution starting from a random initial configuration. The fitness is estimated from the power spectrum in consideration of the similarity to 1/f-type spectrum. The result shows that the rule with the highest average fitness has a propagating structure like other computationally universal cellular automata, although computational universality of the rule has not been proved yet.

Cellular spaces computationally equivalent to any given Turing machine are exhibited which are simple in the sense that each cell has only a small number of states and a small neighborhood. Neighborhood reduction theorems are derived in this interest, and several simple computationuniversal cellular spaces are presented. Conditions for computation-universality of a cellular space are investigated, and, in particular, the conjecture that unbounded but boundable propagation in a space is a sufficient condition is refuted. Finally, the computation-universal spaces derived in the study are used to introduce, via recursive function theory, examples of simple self-reproducing universal Turing machine configurations in one and two dimensions.

Non-linear oscillators with Kuramoto-like local coupling: Complexity analysis and spatiotemporal pattern generation

Artificial genetic circuits have been recently proposed to realize unprecedented biological systems which do not exist in nature by combining several genes. For example, recombinant Escherichia coli provides novel artificial functions such as blinking periodically or invading cancer cells. In protein engineering, new functional proteins have been created by site-directed mutagenesis methods. The main purpose of our research is to implement the Cellular Automata (CA) in vivo employing these methods. Computational capacity of CA is equivalent to the universal Turing Machine. CA are multi-cellular systems in which uniform cells are allocated on lattice grid. Each cell has a finite number of states. The state transition is determined by the current state and the states of neighbor cells. Three mechanisms are required to implement CA in bacteria: (1) Sending and receiving signals between cells, (2) Sustaining the state, (3) Sensing input signals and changing the state following the state transition rules. To encode signals for the mechanism (1), we use small molecules inducing transcription. For the mechanism (2), the toggle switch circuit was employed to represent a finite number of states using gene expression. If one gene does express and the other does not, the state is 1, and vice versa, 0. For the mechanism (3), the state transition functions were implemented as logic gates using transcriptional regulatory proteins which bind to specific signal molecules (chemicals). In this paper, we applied this method to in vivo implementation of one-dimensional CA and corroborated whether it behaves correctly. In principle, this method can be applied to any dimensional CA by employing more signal molecules Permission to make digital or hard copies of all or part of this work for to construct more complex logic gate. Furthermore, we succeeded to execute one step calculation of one-dimensional CA in vivo in a laboratory experiment.

A concept of cellular automata system (CA-system) is introduced as a model of comp[lex phenomena in which several interacting species are involved. CA system suggests a common work of several CA where each processes its own cellular array using in its transition rules cell states of others CA of the system. Taking into account that multi core computers with shared memory are nowadays widely used, a temptation to accelerate the computation by allocating each CA of the system onto one of computer cores is quite natural. Hence, it would be helpful to know what speedup can be obtained by such a parallelization. The paper is aimed to get an answer to this question by determining the conditions, when multicore parallel implementation of CA systems is expedient and correct, and develop the parallelization algorithms for typical CA systems. The results are illustrated by simulation experiments.

A new class of universal computers, cellular array machines, is introduced. The development of reprogrammable logic leading up to the development of field programmable gate arrays is reviewed. An architecture for constructing modular and scalable general-purpose computers from reprogrammable logic is presented. It is shown that such a machine can be programmed by translating Occam into asynchronous digital logic circuits. Examples are given of circuit components which implement Occam operators.

Turing machines based on quantum logic can solve undecidableproblems. In this paper we will give recursion-theoreticalcharacterization of the computational power of this kind of quantumTuring machines. In detail, for the unsharp case, it is proved that&#931<sup>0</sup><sub>1</sub>&#8746&#928<sup>0</sup><sub>1</sub>&#8838L<sup>T</sup><sub>d</sub>(&#949,&#931)(L<sup>T</sup><sub>w</sub>(&#949,&#931))&#8838&#928<sup>0</sup><sub>2</sub>when the truth value lattice is locally finite and the operation &#8743is computable, whereL<sup>T</sup><sub>d</sub>(&#949,&#931)(L<sup>T</sup><sub>w</sub>(&#949,&#931))denotes theclass of quantum language accepted by these Turing machine indepth-first model (respectively, width-first model);for the sharp case, we can obtain similar results for usual orthomodular lattices.

Since Richard Feynman introduced the notion of computation in 1982, various models of quantum computers have been proposed (R. Feynman, 1992). These models include Turing machines and circuits. We define another computational model, one dimensional cellular automata, and demonstrate that any Turing machine can be efficiently simulated by a one dimensional cellular automaton with constant slowdown. This can be accomplished by consideration of a restricted class of one dimensional cellular automata called one dimensional partitioned cellular automata. We also show that any one dimensional partitioned cellular automaton can be simulated by a Turing machine with linear slowdown, but the problem of efficiently simulating an arbitrary one dimensional cellular automaton with a Turing machine is left open. From this discussion, some interesting facts concerning these models are easily deduced.