Ergodicity and Mixing Rate of One-Dimensional Cellular Automata

One-and two-dimensional cellular automata which are known to be fault-tolerant are very complex. On the other hand, only very simple cellular automata have actually been proven to lack fault-tolerance, i.e., to be mixing. The latter either have large noise probability ε or belong to the small family of two-state nearest-neighbor monotonic rules which includes local majority voting. For a certain simple automaton L called the soldiers rule, this problem has intrigued researchers for the last two decades since L is clearly more robust than local voting: in the absence of noise, L eliminates any finite island of perturbation from an initial configuration of all 0's or all 1's. The same holds for a 4-state monotonic variant of L, K, called two-line voting. We will prove that the probabilistic cellular automata Kε and Lε asymptotically lose all information about their initial state when subject to small, strongly biased noise. The mixing property trivially implies that the systems are ergodic. The finite-time information-retaining quality of a mixing system can be represented by its relaxation time Relax(⋅), which measures the time before the onset of significant information loss. This is known to grow as (1/ε)^c for noisy local voting. The impressive error-correction ability of L has prompted some researchers to conjecture that Relax(Lε) = 2^(c/ε). We prove the tight bound 2^(c1log^21/ε) ＜ Relax(Lε) ＜ 2^(c2log^21/ε) for a biased error model. The same holds for Kε. Moreover, the lower bound is independent of the bias assumption. The strong bias assumption makes it possible to apply sparsity/renormalization techniques, the main tools of our investigation, used earlier in the opposite context of proving fault-tolerance.

Reliable Cellular Automata with Self-Organization

In a probabilistic cellular automaton in which all local transitions have positive probability, the problem of keeping a bit of information for more than a constant number of steps is nontrivial, even in an infinite automaton. Still, there is a solution in 2 dimensions, and this solution can be used to construct a simple 3-dimensional discrete-time universal fault-tolerant cellular automaton. This technique does not help much to solve the following problems: remembering a bit of information in 1 dimension; computing in dimensions lower than 3; computing in any dimension with non-synchronized transitions. Our more complex technique organizes the cells in blocks that perform a reliable simulation of a second (generalized) cellular automaton. The cells of the latter automaton are also organized in blocks, simulating even more reliably a third automaton, etc. Since all this (a possibly infinite hierarchy) is organized in "software", it must be under repair all the time from damage caused by errors. A large part of the problem is essentially self-stabilization recovering from a mess of arbitrary-size and content caused by the faults. The present paper constructs an asynchronous one-dimensional fault-tolerant cellular automaton, with the further feature of "self-organization". The latter means that unless a large amount of input information must be given, the initial configuration can be chosen to be periodical with a small period.

The Hippocampus and Cerebellum in Adaptively Timed Learning, Recognition, and Movement

The concepts of declarative memory and procedural memory have been used to distinguish two basic types of learning. A neural network model suggests how such memory processes work together as recognition learning, reinforcement learning, and sensory-motor learning take place during adaptive behaviors. To coordinate these processes, the hippocampal formation and cerebellum each contain circuits that learn to adaptively time their outputs. Within the model, hippocampal timing helps to maintain attention on motivationally salient goal objects during variable task-related delays, and cerebellar timing controls the release of conditioned responses. This property is part of the model's description of how cognitive-emotional interactions focus attention on motivationally valued cues, and how this process breaks down due to hippocampal ablation. The model suggests that the hippocampal mechanisms that help to rapidly draw attention to salient cues could prematurely release motor commands were not the release of these commands adaptively timed by the cerebellum. The model hippocampal system modulates cortical recognition learning without actually encoding the representational information that the cortex encodes. These properties avoid the difficulties faced by several models that propose a direct hippocampal role in recognition learning. Learning within the model hippocampal system controls adaptive timing and spatial orientation. Model properties hereby clarify how hippocampal ablations cause amnesic symptoms and difficulties with tasks which combine task delays, novelty detection, and attention towards goal objects amid distractions. When these model recognition, reinforcement, sensory-motor, and timing processes work together, they suggest how the brain can accomplish conditioning of multiple sensory events to delayed rewards, as during serial compound conditioning.

A Neural Network of Adaptively Timed Reinforcement Learning and Hippocampal Dynamics

A neural model is described of how adaptively timed reinforcement learning occurs. The adaptive timing circuit is suggested to exist in the hippocampus, and to involve convergence of dentate granule cells on CA3 pyramidal cells, and NMDA receptors. This circuit forms part of a model neural system for the coordinated control of recognition learning, reinforcement learning, and motor learning, whose properties clarify how an animal can learn to acquire a delayed reward. Behavioral and neural data are summarized in support of each processing stage of the system. The relevant anatomical sites are in thalamus, neocortex, hippocampus, hypothalamus, amygdala, and cerebellum. Cerebellar influences on motor learning are distinguished from hippocampal influences on adaptive timing of reinforcement learning. The model simulates how damage to the hippocampal formation disrupts adaptive timing, eliminates attentional blocking, and causes symptoms of medial temporal amnesia. It suggests how normal acquisition of subcortical emotional conditioning can occur after cortical ablation, even though extinction of emotional conditioning is retarded by cortical ablation. The model simulates how increasing the duration of an unconditioned stimulus increases the amplitude of emotional conditioning, but does not change adaptive timing; and how an increase in the intensity of a conditioned stimulus "speeds up the clock", but an increase in the intensity of an unconditioned stimulus does not. Computer simulations of the model fit parametric conditioning data, including a Weber law property and an inverted U property. Both primary and secondary adaptively timed conditioning are simulated, as are data concerning conditioning using multiple interstimulus intervals (ISIs), gradually or abruptly changing ISis, partial reinforcement, and multiple stimuli that lead to time-averaging of responses. Neurobiologically testable predictions are made to facilitate further tests of the model.

Exploratory analysis of spatiotemporal patterns of cellular automata by clustering compressibility

In this paper we study the classification of spatiotemporal pattern of one-dimensional cellular automata (CA) whereas the classification comprises CA rules including their initial conditions. We propose an exploratory analysis method based on the normalized compression distance (NCD) of spatiotemporal patterns which is used as dissimilarity measure for a hierarchical clustering. Our approach is different with respect to the following points. First, the classification of spatiotemporal pattern is comparative because the NCD evaluates explicitly the difference of compressibility among two objects, e.g., strings corresponding to spatiotemporal patterns. This is in contrast to all other measures applied so far in a similar context because they are essentially univariate. Second, Kolmogorov complexity, which underlies the NCD, was used in the classification of CA with respect to their spatiotemporal pattern. Third, our method is semiautomatic allowing us to investigate hundreds or thousands of CA rules or initial conditions simultaneously to gain insights into their organizational structure. Our numerical results are not only plausible confirming previous classification attempts but also shed light on the intricate influence of random initial conditions on the classification results.

Design of quantum-dot cellular automata circuits using cut-set retiming

As a potential alternative to CMOS technology, QCA provides an interesting paradigm in both communication and computation. However, QCAs unique four-phase clocking scheme and timing constraints present serious timing issues for interconnection and feedback. In this work, a cut-set retiming design procedure is proposed to resolve these QCA timing issues. The proposed design procedure can accommodate QCAs unique characteristics by performing delay-transfer and time-scaling to reallocate the existing delays so as to achieve efficient clocking zone assignment. Cut-set retiming makes it possible to effectively design relatively complex QCA circuits that include feedback. It utilizes the similar characteristics of synchronization, deep pipelines and local interconnections common to both QCA and systolic architectures. As a case study, a systolic Montgomery modular multiplier is designed to illustrate the procedure. Furthermore, a nonsystolic architecture, an S27 benchmark circuit, is designed and compared with previous designs. The comparison shows that the cut-set retiming method achieves a more efficient design, with a reduction of 22%, 44%, and 46% in terms of cell count, area, and latency, respectively.

Bit erasure analysis of binary adders in quantum-dot cellular automata

As a post-CMOS technology, the incipient Quantum-dot Cellular Automata technology has various advantages. A key aspect which makes it highly desirable is low power dissipation. One method that is used to analyse power dissipation in QCA circuits is bit erasure analysis. This method has been applied to analyse previously proposed QCA binary adders. However, a number of improved QCA adders have been proposed more recently that have only been evaluated in terms of area and speed. As the three key performance metrics for QCA circuits are speed, area and power, in this paper, a bit erasure analysis of these adders will be presented to determine their power dissipation. The adders to be analysed are the Carry Flow Adder (CFA), Brent-Kung Adder (B-K), Ladner-Fischer Adder (L-F) and a more recently developed area-delay efficient adder. This research will allow for a more comprehensive comparison between the different QCA adder proposals. To the best of the authors' knowledge, this is the first time power dissipation analysis has been carried out on these adders.

A First Step Towards Cost Functions for Quantum-dot Cellular Automata Designs

Quantum-dot cellular automata (QCA) is potentially a very attractive alternative to CMOS for future digital designs. Circuit designs in QCA have been extensively studied. However, how to properly evaluate the QCA circuits has not been carefully considered. To date, metrics and area-delay cost functions directly mapped from CMOS technology have been used to compare QCA designs, which is inappropriate due to the differences between these two technologies. In this paper, several cost metrics specifically aimed at QCA circuits are studied. It is found that delay, the number of QCA logic gates, and the number and type of crossovers, are important metrics that should be considered when comparing QCA designs. A family of new cost functions for QCA circuits is proposed. As fundamental components in QCA computing arithmetic, QCA adders are reviewed and evaluated with the proposed cost functions. By taking the new cost metrics into account, previous best adders become unattractive and it has been shown that different optimization goals lead to different “best” adders.

Cost-Efficient Decimal Adder Design in Quantum-dot Cellular Automata

Applications that cannot tolerate the loss of accuracy that results from binary arithmetic demand hardware decimal arithmetic designs. Binary arithmetic in Quantum-dot cellular automata (QCA) technology has been extensively investigated in recent years. However, only limited attention has been paid to QCA decimal arithmetic. In this paper, two cost-efficient binary-coded decimal (BCD) adders are presented. One is based on the carry flow adder (CFA) using a conventional correction method. The other uses the carry look ahead (CLA) algorithm which is the first QCA CLA decimal adder proposed to date. Compared with previous designs, both decimal adders achieve better performance in terms of latency and overall cost. The proposed CFA-based BCD adder has the smallest area with the least number of cells. The proposed CLA-based BCD adder is the fastest with an increase in speed of over 60% when compared with the previous fastest decimal QCA adder. It also has the lowest overall cost with a reduction of over 90% when compared with the previous most cost-efficient design.