Constructibility of Signal-Crossing Solutions in von Neumann 29-State Cellular Automata

In von Neumann 29-state cellular automata, the crossing of signals is an important problem, with three solutions reported in the literature. These solutions greatly impact automaton design, especially self-replicators. This paper examines these solutions, with emphasis upon their constructibility. We show that two of these solutions are difficult to construct, and offer an improved design technique. We also argue that solutions to the signal-crossing problem have implications for machine models of biological development, especially with regard to the cell cycle. 1 Von Neumann 29-State Cellular Automata Signal-Crossing John von Neumann developed cellular automata theory, yielding an environment in which to demonstrate his thesis that machines may be designed having the property of self-replication [1]. Von Neumann cellular automata are characterized by a two-dimensional, rectilinear lattice network of finite state automata (the cells), each identical in form, function, and association, as specified by a set of states, a set of rules for the transition of cells between states (the state transition function), and a grouping function that places each cell at the center of a neighborhood of adjacent cells (specifying the set of cells operated upon by the state transition function in the computation of state transitions). All cells transition their state synchronously. States are grouped into five categories; a ground state, the transition states, the confluent states (C), the ordinary transmission states (D), and the special transmission states (M). The last three categories have an activity property, while the last two categories have the property of direction. Activity corresponds to carried data, it being transmitted between states at the rate of one bit per application of the state transition function. Confluent states have the additional property of a one-cycle delay, and so hold two bits of data. The direction property indicates the flow of data between states. Ordinary and special transmission states have an antagonistic relationship, with mutually directed active cells of each causing the annihilation of the other, to yield the ground state. Active special transmission states also yield confluent state annihilation. Confluent states accept data from ordinary transmission states, perform a logical AND on 1 AKA Amar Mukhopadhyay the inputs, and transmit data to both ordinary and special transmission states. Ordinary and special transmission states logically OR inputs. An ordinary transmission state accepts input only from like states, and from adjacent confluent states. Special transmission states accept input likewise. Confluent states pass data to any adjacent transmission state not pointed at the confluent state. Data are not transmitted to transmission states against the direction of those transmission states. For instance, two ordinary transmission states pointing at each other do not exchange data. Instead, the data is simply lost. Data held by a confluent state is lost if there is no adjacent transmission state not pointing at the confluent state. Patterns of cells are called configurations, with those that implement specific functionality being called organs. Configurations can be compared in terms of their constructibility. Constructibility is both an absolute measure, and a relative measure. Some configurations are not constructible, while other configurations are constructible. In absolute terms, constructibility is the property that a configuration can be obtained through the act of another configuration. In relative terms, constructibility is an inverse measure of effort. In von Neumann 29-state cellular automata, the organ that facilitates configuration construction is known as the construction arm. 2 The Signal-Crossing Problem and Available Solutions A problem arises within any two-dimensional system respecting the mechanisms of translocation the crossing problem. The familiar example is roadway transportation, the solutions being stop-and-go intersections, bridges, and traffic circles. In cellular automata, we have the signal-crossing problem. This owes to the fixed-position nature of the component finite state automata, where the translocation is of data (in the form of signals). In such cases, translocation is called communication. Signals are an ordered sequence of data (bits), whether of fixed or arbitrary length, that are communicated between organs. The literature reports three solutions to the signal-crossing problem within von Neumann 29-state cellular automata. These signal-crossing organs are the Coded Channel (CC), the Mukhopadhyay Crossing Organ (MCO) [2], and the Real-Time Crossing Organ (RTCO). We are here concerned with the properties of these signal-crossing organs, particularly the latter two. The MCO and RTCO are general signal-crossing solutions, able to serve the crossing needs of any two signals, regardless of length. The CC is a more constrained signal-crossing solution, capable of serving only signals of varying fixed length, though extendable to service an arbitrarily large number of signals. While the MCO and RTCO are indiscriminate in the signals they service, the CC discriminates between signals, via selective acceptance. The function of the CC is programmable, while neither the MCO nor the RTCO is programmable. We now consider signal-crossing organ architecture [3]. The CC has two-layers, with an internal channel (or signal path) positioned between inputs and outputs. The internal channel is of finite length, and is non-cyclic. The first CC layer accepts signal input and translates it into a code carried by the internal channel. The second layer of the CC translates this code into signal output. The CC may accept any input signal a multiple number of times, and may generate the corresponding output signal any number of times. Linearity of the internal channel requires input acceptance prior to output generation. Each input may accept more than one signal, while each output generates only one signal. If corruption of channel code occurs, unwanted output signal generation may result. Thus, signal inputs ought occur with sufficient relative delay. The CC is best applied where it is known that signals are incident only upon the complete servicing of any previously accepted signal. In the simplest implementation, shown in figure 1, the CC expresses a bijection of two inputs to two outputs. It is especially easy to see in this case that signal input can be both crossed and duplicated on output. If the input signals A and B are accepted in that order, with input acceptors coming before output generators, and the order of outputs is B then A, we have that the signals are crossed and duplicated. For signals <11> and <101>, the CC covers approximately 230 cells. CC size is proportional to the number and length of inputs and outputs. The RTCO is a square organ, comprising 64 cells, as shown in figure 2. It has two inputs and two outputs, arranged in orthogonal input/output pairs. Signals are duplicated at input, routed along a pair of internal paths, and joined into a single signal at output. There are four different signal paths internal to the RTCO, all of identical length. The RTCO has five clocks, each of identical structure and emitting a period six signal <101010>, which drive inputs to outputs and operate in-phase with one-another. Four of these clocks are positioned at the four corners of the RTCO, with the fifth clock located directly in the middle of the RTCO. The four internal signal paths of the RTCO CDDD CD CD C CDDD CDDD CD CD CD C E E F F E E F F A in DDDD CD CD C DDDDDDDD CD CD CD CD CD C DDDDD B out E F CDDD CDDD CD CD CD C E F CDDD CD CD C E E F F E F E E F F B in DD CD CD CD CD C DDDDDD E DD CD CD C DDDDDDD A out Fig. 1. The minimal CC is a configuration that crosses two signals, <11> and <101>. Input Ain is accepted by a decoder/pulser pair, the result being then injected into the internal channel, where an identical decoder/pulser pair again accepts the signal. A single ordinary transmission state separates the decoder from the pulser, each organ being constructed of confluent and ordinary transmission states. The decoder of input Ain , outlined in this figure with dashed lines, is an organ of dimension five cells by three cells