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.

Building Responsive Systems from Physically-correct Specifications

Predictability - the ability to foretell that an implementation will not violate a set of specified reliability and timeliness requirements - is a crucial, highly desirable property of responsive embedded systems. This paper overviews a development methodology for responsive systems, which enhances predictability by eliminating potential hazards resulting from physically-unsound specifications. The backbone of our methodology is the Time-constrained Reactive Automaton (TRA) formalism, which adopts a fundamental notion of space and time that restricts expressiveness in a way that allows the specification of only reactive, spontaneous, and causal computation. Using the TRA model, unrealistic systems - possessing properties such as clairvoyance, caprice, in finite capacity, or perfect timing - cannot even be specified. We argue that this "ounce of prevention" at the specification level is likely to spare a lot of time and energy in the development cycle of responsive systems - not to mention the elimination of potential hazards that would have gone, otherwise, unnoticed. The TRA model is presented to system developers through the CLEOPATRA programming language. CLEOPATRA features a C-like imperative syntax for the description of computation, which makes it easier to incorporate in applications already using C. It is event-driven, and thus appropriate for embedded process control applications. It is object-oriented and compositional, thus advocating modularity and reusability. CLEOPATRA is semantically sound; its objects can be transformed, mechanically and unambiguously, into formal TRA automata for verification purposes, which can be pursued using model-checking or theorem proving techniques. Since 1989, an ancestor of CLEOPATRA has been in use as a specification and simulation language for embedded time-critical robotic processes.

OS Support for Portable Bulk Synchronous Parallel Programs

Predictability -- the ability to foretell that an implementation will not violate a set of specified reliability and timeliness requirements -- is a crucial, highly desirable property of responsive embedded systems. This paper overviews a development methodology for responsive systems, which enhances predictability by eliminating potential hazards resulting from physically-unsound specifications. The backbone of our methodology is the Time-constrained Reactive Automaton (TRA) formalism, which adopts a fundamental notion of space and time that restricts expressiveness in a way that allows the specification of only reactive, spontaneous, and causal computation. Using the TRA model, unrealistic systems – possessing properties such as clairvoyance, caprice, infinite capacity, or perfect timing -- cannot even be specified. We argue that this "ounce of prevention" at the specification level is likely to spare a lot of time and energy in the development cycle of responsive systems -- not to mention the elimination of potential hazards that would have gone, otherwise, unnoticed. The TRA model is presented to system developers through the Cleopatra programming language. Cleopatra features a C-like imperative syntax for the description of computation, which makes it easier to incorporate in applications already using C. It is event-driven, and thus appropriate for embedded process control applications. It is object-oriented and compositional, thus advocating modularity and reusability. Cleopatra is semantically sound; its objects can be transformed, mechanically and unambiguously, into formal TRA automata for verification purposes, which can be pursued using model-checking or theorem proving techniques. Since 1989, an ancestor of Cleopatra has been in use as a specification and simulation language for embedded time-critical robotic processes.

Nearest-neighbor Queries in Probabilistic Graphs

Large probabilistic graphs arise in various domains spanning from social networks to biological and communication networks. An important query in these graphs is the k nearest-neighbor query, which involves finding and reporting the k closest nodes to a specific node. This query assumes the existence of a measure of the "proximity" or the "distance" between any two nodes in the graph. To that end, we propose various novel distance functions that extend well known notions of classical graph theory, such as shortest paths and random walks. We argue that many meaningful distance functions are computationally intractable to compute exactly. Thus, in order to process nearest-neighbor queries, we resort to Monte Carlo sampling and exploit novel graph-transformation ideas and pruning opportunities. In our extensive experimental analysis, we explore the trade-offs of our approximation algorithms and demonstrate that they scale well on real-world probabilistic graphs with tens of millions of edges.

A Local Circuit Model of Learned Straitial and Dopamine Cell Responses under Probabilistic Schedules of Reward

Before choosing, it helps to know both the expected value signaled by a predictive cue and the associated uncertainty that the reward will be forthcoming. Recently, Fiorillo et al. (2003) found the dopamine (DA) neurons of the SNc exhibit sustained responses related to the uncertainty that a cure will be followed by reward, in addition to phasic responses related to reward prediction errors (RPEs). This suggests that cue-dependent anticipations of the timing, magnitude, and uncertainty of rewards are learned and reflected in components of the DA signals broadcast by SNc neurons. What is the minimal local circuit model that can explain such multifaceted reward-related learning? A new computational model shows how learned uncertainty responses emerge robustly on single trial along with phasic RPE responses, such that both types of DA responses exhibit the empirically observed dependence on conditional probability, expected value of reward, and time since onset of the reward-predicting cue. The model includes three major pathways for computing: immediate expected values of cures, timed predictions of reward magnitudes (and RPEs), and the uncertainty associated with these predictions. The first two model pathways refine those previously modeled by Brown et al. (1999). A third, newly modeled, pathway is formed by medium spiny projection neurons (MSPNs) of the matrix compartment of the striatum, whose axons co-release GABA and a neuropeptide, substance P, both at synapses with GABAergic neurons in the SNr and with the dendrites (in SNr) of DA neurons whose somas are in ventral SNc. Co-release enables efficient computation of sustained DA uncertainty responses that are a non-monotonic function of the conditonal probability that a reward will follow the cue. The new model's incorporation of a striatal microcircuit allowed it to reveals that variability in striatal cholinergic transmission can explain observed difference, between monkeys, in the amplitutude of the non-monotonic uncertainty function. Involvement of matriceal MSPNs and striatal cholinergic transmission implpies a relation between uncertainty in the cue-reward contigency and action-selection functions of the basal ganglia. The model synthesizes anatomical, electrophysiological and behavioral data regarding the midbrain DA system in a novel way, by relating the ability to compute uncertainty, in parallel with other aspects of reward contingencies, to the unique distribution of SP inputs in ventral SN.