Katsunobu Imai

Self-reproduction in a reversible cellular space

Abstract We investigate a problem whether self-reproduction is possible in a two-dimensional “reversible” cellular space, and give an affirmative answer. A reversible (or injective) cellular automaton (RCA) is a CA such that every configuration has at most one predecessor. In order to design an RCA we use a framework of partitioned cellular automaton (PCA). A PCA with von Neumann neighborhood is a special type of CA whose cell is divided into five parts. We designed here a reversible PCA SR 8 having 8 states in each part (thus one cell has 8 5 states). In this cellular space, encoding the shape of an object into a “gene” represented by a command sequence, copying the gene, and interpreting the gene to create an object, are all performed reversibly. We show that, by using these operations, various objects can reproduce themselves in a very simple manner.

A computation-universal two-dimensional 8-state triangular reversible cellular automaton

A reversible cellular automaton (RCA) is a cellular automaton (CA) whose global function is injective and every configuration has at most one predecessor. Margolus showed that there is a computation-universal two-dimensional 2-state RCA. But his RCA has a non-uniform neighbor, so Morita and Ueno proposed 16-state computation-universal RCA using partitioned cellular automata (PCA). Because PCA can be regarded as a subclass of standard CA, their models have a standard neighbor. In this paper, we show that the number of states of Morita and Uenos models can be reduced. To decrease the number of states from their models with preserving isotropic and bit-preserving properties, we used a triangular 3-neighbor, and thus an 8-state RCA can be possible. This is the smallest state two-dimensional RCA under the condition of isotropic property in the framework of PCA. We show that our model can simulate basic circuit elements such as unit wires, delay elements, crossing wires, switch gates and inverse switch gates, and it is possible to construct a Fredkin gate by combining these elements. Since Fredkin gate is known to be a universal logic gate, our model has computation-universality.

Self-reproduction in three-dimensional reversible cellular space

Due to inevitable power dissipation, it is said that nano-scaled computing devices should perform their computing processes in a reversible manner. This will be a large problem in constructing three-dimensional nano-scaled functional objects. Reversible cellular automata (RCA) are used for modeling physical phenomena such as power dissipation, by studying the dissipation of garbage signals. We construct a three-dimensional self-inspective self-reproducing reversible cellular automaton by extending the two-dimensional version SR8. It can self-reproduce various patterns in three-dimensional reversible cellular space without dissipating garbage signals.

Universal computing in reversible and number-conserving two-dimensional cellular spaces

A number-conserving reversible cellular automaton (NC-RCA) is a computing model that reflects both reversibility and mass or energy conservation law in physics. We show that, despite strict constraints of reversibility and number-conservation, there exist simple two-dimensional NC-RCAs that are capable of universal computation. These automata are classified into two types. Automaton of the first type implements a Fredkin gate in its space-time dynamic. That is a Fredkin gate, as a universal logical element, is embedded in its cellular space. Automaton of the second type, incorporate a novel logical element, called a “rotary element” in its lattice dynamic. NC-RCAs of the second type simulate any reversible two-counter machine in their space-time configuration. In both types of cellular automata the computation is realized via appropriate control of “moving particles” in the cellular space.

Firing squad synchronization problem in reversible cellular automata

We studied the Firing Squad Synchronization Problem (FSSP) on reversible (i.e., backward deterministic) cellular automata (RCA). First we proved that, in the case of RCA, there is no solution under the usual condition for FSSP where the firing state is only one. So we defined a little weaker condition suitable for RCA in which finite number of firing states are allowed. We showed that Minskys solution in time 3n can be embedded in an RCA with 99 states that satisfies the new condition. We used the framework of partitioned CA (PCA), which is regarded as a subclass of CA, for making ease of constructing RCA.

Universality of Reversible Hexagonal Cellular Automata

We define a kind of cellular automaton called a hexagonal partitioned cellular automaton (HPCA), and study logical universality of a reversible HPCA. We give a specific 64-state reversible HPCA H 1 , and show that a Fredkin gate can be embedded in this cellular space. Since a Fredkin gate is known to be a universal logic element, logical universality of H 1 is concluded. Although the number of states of H 1 is greater than those of the previous models of reversible CAs having universality, the size of the configuration realizing a Fredkin gate is greatly reduced, and its local transition function is still simple. Comparison with the previous models, and open problems related to these model are also discussed.

Number-Conserving Reversible Cellular Automata and Their Computation-Universality

We introduce a new model of cellular automaton called a one-dimensional number-conserving partitioned cellular automaton (NC-PCA). An NC-PCA is a system such that a state of a cell is represented by a triple of non-negative integers, and the total (i. e., sum) of integers over the configuration is conserved throughout its evolving (computing) process. It can be thought as a kind of modelization of the physical conservation law of mass (particles) or energy. We also define a reversible version of NC-PCA, and prove that a reversible NC-PCA is computation-universal. It is proved by showing that a reversible two-counter machine, which has been known to be universal, can be simulated by a reversible NC-PCA.

Constructible functions in cellular automata and their applications to hierarchy results

We investigate time-constructible functions in one-dimensional cellular automata (CA). It is shown that (i) if a function t(n) is computable by an O(t(n)n)-time Turing machine, then t(n) is time constructible by CA and (ii) if two functions are time constructible by CA, then the sum, product, and exponential functions of them are time constructible by CA. As an application, it is shown that if t1(n) and t2(n) are time constructible functions such that limnt1(n)/t2(n)=0 and t1(n)n, then there is a language which can be recognized by a CA in t2(n) time but not by any CA in t1(n) time.

On DNA-Based Gellular Automata

We propose the notion of gellular automata and their possible implementations using DNA-based gels. Gellular automata are a kind of cellular automaton in which cells in space are separated by gel materials. Each cell contains a solution with designed chemical reactions whose products dissolve or construct gel walls separating the cells. We first introduce the notion of gellular automata and their computational models. We then give examples of gellular automata and show that computational universality is achieved through the implementation of rotary elements by gellular automata. We finally examine general strategies for implementing gellular automata using DNA-based gels and report results of preliminary experiments.

Fluctuation-driven computing on number-conserving cellular automata

A number-conserving cellular automaton (NCCA) is a cellular automaton in which the states of cells are denoted by integers, and the sum of all of the numbers in a configuration is conserved throughout its evolution. NCCAs have been widely used to model physical systems that are ruled by conservation laws of mass or energy. Imai et al. [13] showed that the local transition function of NCCA can be effectively translated into the sum of a binary flow function over pairs of neighboring cells. In this paper, we explore the computability of NCCAs in which the pairwise number flows are performed at fully asynchronous timings. Despite the randomness that is associated with asynchronous transitions, useful computation still can be accomplished efficiently in the cellular automata through the active exploitation of fluctuations [18]. Specifically, certain numbers may flow randomly fluctuating between forward and backward directions in the cellular space, as if they were subject to Brownian motion. Because random fluctuations promise a powerful resource for searching through a computational state space, the Brownian-like flow of the numbers allows for efficient embedding of logic circuits into our novel asynchronous NCCA.