Boolean Cellular Automata Research Articles

The Effect of Noise on the Density Classification Task for Various Cellular Automata Rules.

Cellular automata (CA) are discrete dynamical systems with a prominent place in the history and study of artificial life. Here, we focus on the density classification task (DCT) in which a 1-dimensional lattice of Boolean (on/off) automata must perform a form of rudimentary quorum sensing. Typically, the ring lattice consists of 149 cells (though we consider other sizes as well) that update their state according to their own state and its six nearest neighbors in the previous time step. The goal is obtaining Boolean CA rules whose dynamics converges to the majority state of the entire lattice for a given initial configuration of the lattice. This is a nontrivial task because cells have access only to local information, and thus need to integrate and coordinate information across the lattice to converge to the correct collective state. Because initial conditions are random, they have very similar proportions of on and off states, which makes the problem very difficult. This problem has hitherto been studied with the assumption that input to each cell is perfectly stable. Since biological systems that solve similar problems (e.g. bacterial quorum sensing) must operate in noisy environments, here we study the impact of noise on DCT accuracy for the 13 highest-accuracy CA rules from the literature. We use cubewalkers, a recently released GPU-accelerated Boolean simulator to conduct large-scale random experiments. We uncover a trade-off between maximum accuracy without noise and robustness to noise among these high-performance CAs. Moreover, there is no significant difference between rules that were human-designed or evolved computationally.

On Boolean Automata Networks (de)Composition1

Boolean automata networks (BANs) are a generalisation of Boolean cellular automata. In such, any theorem describing the way BANs compute information is a strong tool that can be applied to a wide range of models of computation. In this paper we explore a way of working with BANs which involves adding external inputs to the base model (via modules), and more importantly, a way to link networks together using the above mentioned inputs (via wirings). Our aim is to develop a powerful formalism for BAN (de)composition. We formulate three results: the first one shows that our modules/wirings definition is complete; the second one uses modules/wirings to prove simulation results amongst BANs; the final one expresses the complexity of the relation between modularity and the dynamics of modules.

Statistical Complexity of Boolean Cellular Automata with Short-Term Reaction-Diffusion Memory on a Square Lattice

Memory is a ubiquitous phenomenon in biological systems, in which the present system state is not entirely determined by the current conditions but also depends on the time evolutionary path of the system. Specifically, many phenomena related to memory are characterized by chemical memory reactions that may fire under particular system conditions. These conditional chemical reactions contradict the extant approaches for modeling chemical kinetics and have increasingly posed significant challenges to mathematical modeling and computer simulation. Along these lines, we can imagine a memory module contributing to cell therapy or the synthetic differentiation of certain cells in a certain fashion after experiencing a brief stimulus. We demonstrate that information processing properties of cellular automata (CAs) can be controlled by a signal composed of excitation pulses. We discuss how cellular memory can be incorporated into more complex systems like CAs to understand the controlling of information processing performed by a medium with the use of a pulse signal propagated from a number of control cells. In this paper, we also investigate the potential application of cellular computation for constructing pseudorandom number generators (PRNGs). Furthermore, the PRNG scheme based on CAs with reaction-diffusion memory is proposed for its capability of generating ultrahigh-quality random numbers. However, the quality bottleneck of a practical PRNG lies in the limited cycle of the generator. To close the gap between the pure randomness generation and the short period, we propose and implement a memory algorithm based on a reaction-diffusion process in a chemical system for Boolean CAs. This scheme is characterized by a tradeoff between, on one hand, the rate of generation of random bits and, on the other hand, the degree of randomness that the series can deliver. These successful applications of the memory modeling framework suggest that this innovative theory is an effective and powerful tool for studying memory processes and conditional chemical reactions in a wide range of complex biological systems. This result also opens a new perspective to apply CAs as a computational engine for the robust generation of pure random numbers, which has important applications in cryptography and other related areas. © 2019, Complex Systems Publications, Inc. All rights reserved.

A graph theory approach for regional controllability of Boolean cellular automata

ABSTRACT Controllability is one of the central concepts of modern control theory that allows a good understanding of a system's behaviour. It consists in constraining a system to reach the desired state from an initial state within a given time interval. When the desired objective affects only a sub-region of the domain, the control is said to be regional. The purpose of this paper is to study a particular case of regional control using cellular automata models since they are spatially extended systems where spatial properties can be easily defined thanks to their intrinsic locality. We investigate the case of boundary controls on the target region using an original approach based on graph theory. Necessary and sufficient conditions are given based on the Hamiltonian circuit and strongly connected component. The controls are obtained using a preimage approach.

Toward a boundary regional control problem for Boolean cellular automata

An important question to be addressed regarding system control on a time interval [0, T] is whether some particular target state in the configuration space is reachable from a given initial state. When the target of interest refers only to a portion of the spatial domain, we speak about regional analysis. Cellular automata approach have been recently promoted for the study of control problems on spatially extended systems for which the classical approaches cannot be used. An interesting problem concerns the situation where the subregion of interest is not interior to the domain but a portion of its boundary . In this paper we address the problem of regional controllability of cellular automata via boundary actions, i.e., we investigate the characteristics of a cellular automaton so that it can be controlled inside a given region only acting on the value of sites at its boundaries.

On the relationship between fuzzy and Boolean cellular automata

Fuzzy cellular automata (FCA) are continuous cellular automata where the local rule is defined as the “fuzzification” of the local rule of a corresponding Boolean cellular automaton in disjunctive normal form. In this paper, we are interested in the relationship between Boolean and fuzzy models and, for the first time, we analytically show the existence of a strong connection between them by focusing on two properties: density conservation and additivity. We begin by showing that the density conservation property, extensively studied in the Boolean domain, is preserved in the fuzzy domain: a Boolean CA is density conserving if and only if the corresponding FCA is sum preserving. A similar result is established for another novel “spatial” density conservation property. Second, we prove an interesting parallel between the additivity of Boolean CA and oscillations of the corresponding fuzzy CA around its fixed point. In fact, we show that a Boolean CA is additive if and only if the behaviour of the corresponding fuzzy CA around its fixed point coincides with the Boolean behaviour. Finally, we give a probabilistic interpretation of our fuzzification which establishes an equivalence between convergent fuzzy CA and the mean field approximation on Boolean CA, an estimation of their asymptotic density. These connections between the Boolean and the fuzzy models are the first formal proofs of a relationship between them.

On the Asymptotic Behavior of Fuzzy Cellular Automata

Fuzzy cellular automata (FCA) are continuous cellular automata where the local rule is defined as the “fuzzification” of the local rule of a corresponding Boolean cellular automaton in disjunctive normal form. In this paper we consider circular FCA; their asymptotic behavior has been observed through simulations and FCA have been empirically classified accordingly. No analytical study exists so far to support those observations. We now start the analytical study of circular FCA's dynamics by considering a particular set of FCA ( Weighted Average rules) which includes rules displaying most of the observed dynamics, and we precisely derive their behavior. We confirm the empirical observations proving that all weighted average rules are periodic in time and space, and we derive their periods.

On the Relationship Between Boolean and Fuzzy Cellular Automata

Fuzzy cellular automata (FCA) are continuous cellular automata where the local rule is defined as the ''fuzzification'' of the local rule of a corresponding Boolean cellular automaton in disjunctive normal form. In this paper we are interested in the relationship between Boolean and fuzzy models and we analytically show, for the first time, the existence of a strong connection between them by focusing on two properties: density conservation and additivity. We begin by giving a probabilistic interpretation of our fuzzification which leads to two important results. First, it establishes an equivalence between convergent fuzzy CA and the mean field approximation on Boolean CA, an estimation of their asymptotic density. Second, we show that the density conservation property, extensively studied in the Boolean domain, is preserved in the fuzzy domain: a Boolean CA is density conserving if and only if the corresponding FCA is sum preserving. A similar result is established for another novel ''spatial'' density conservation property. Finally, we prove an interesting parallel between additivity of Boolean CA and oscillation of the corresponding fuzzy CA around its fixed point. In fact, we show that a Boolean CA has a certain form of additivity if and only if the behavior of the corresponding fuzzy CA around its fixed point coincides with the Boolean behavior. These connections between the Boolean and the fuzzy models are the first formal proofs of a relationship between them.

A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM'S NEW KIND OF SCIENCE PART VI: FROM TIME-REVERSIBLE ATTRACTORS TO THE ARROW OF TIME

This paper proves, via an analytical approach, that 170 (out of 256) Boolean CA rules in a one-dimensional cellular automata (CA) are time-reversible in a generalized sense. The dynamics on each attractor of a time-reversible rule N is exactly mirrored, in both space and time, by its bilateral twin ruleN†. In particular, all 69 period-1 rules, 17 (out of 25) period-2 rules, and 84 (out of 112) Bernoulli rules are time-reversible. The remaining 86 CA rules are time-irreversible in the sense that N and N† mirror their dynamics only in space, but not in time. In this case, each attractor of N defines a unique arrow of time. A simple "time-reversal test" is given for testing whether an attractor of a CA rule is time-reversible or time-irreversible. For a time-reversible attractor of a CA rule N the past can be uniquely recovered from the future of N†, and vice versa. This remarkable property provides 170 concrete examples of CA time machines where time travel can be routinely achieved by merely hopping from one attractor to its bilateral twin attractor, and vice versa. Moreover, the time-reversal property of some local rules can be programmed to mimic the matter–antimatter "annihilation" or "pair-production" phenomenon from high-energy physics, as well as to mimic the "contraction" or "expansion" scenarios associated with the Big Bang from cosmology. Unlike the conventional laws of physics, which are based on a unique universe, most CA rules have multiple universes (i.e. attractors), each blessed with its own laws. Moreover, some CA rules are endowed with both time-reversible attractors and time-irreversible attractors. Using an analytical approach, the time-τ return map of each Bernoulli στ-shift attractor of all 112 Bernoulli rules are shown to obey an ultra-compact formula in closed form, namely,. [Formula: see text] or its inverse map. These maps completely characterize the time-asymptotic (steady state) behavior of the nonlinear dynamics on the attractors. In-depth analysis of all but 18 global equivalence classes of CA rules have been derived, along with their basins of attraction, which characterize their transient regimes. Above all, this paper provides a rigorous nonlinear dynamics foundation for a paradigm shift from an empirical-based approach à la Wolfram to an attractor-based analytical theory of cellular automata.

Fuzzy automata and life

AbstractBoolean cellular automata may be generalized to fuzzy automata in a consistent manner. Several fuzzy logics are used to create 1‐dimensional automata and also 2‐dimensional automata that generalize the game of life. These generalized automata are investigated and compared to their Boolean counterparts empirically and using rule entropy and repeated input response functions. Fuzzy automata offer new mechanisms for classification of classical automata and can be used for insight into their qualitative behavior. © 2002 Wiley Periodicals, Inc.

CHAOS IN A SIMPLE BOOLEAN NETWORK

We study the complex dynamics of a simple stochastic Boolean network. The investigated system is equivalent to a randomly connected Boolean cellular automaton. The dynamical evolution of the cellular automaton is exactly described by a polynomial map with binomial coefficients. We show that the map is chaotic and the route to chaos is period-doubling bifurcations.

One-Dimensional Lattice Dynamics of the Diffusion-Limited Reaction A +A A +S: Transient Behavior

We use a Boolean cellular automaton model to describe the diffusion-limited dynamics of the irreversible reaction A+A→A+S on a 1D lattice. We derive a set of equations for the dynamics of the empty interval probabilities from which explicit expressions for the particle concentration and the two-point correlation can be obtained. It is shown that the long-time dynamics is in agreement with the off-lattice solution. The early-time behavior, however, predicts a slower decay of the concentration.

Control parameters in Boolean networks and cellular automata revisited from a logical and a sociological point of view

Control parameters in Boolean networks and cellular automata revisited from a logical and a sociological point of view

Control parameters in Boolean networks and cellular automata revisited from a logical and a sociological point of view

Control parameters in Boolean networks and cellular automata revisited from a logical and a sociological point of view

On the threshold of chaos in random boolean cellular automata

AbstractA random boolean cellular automaton is a network of boolean gates where the inputs, the boolean function, and the initial state of each gate are chosen randomly. In this article, each gate has two inputs. Let a (respectively c) be the probability that the gate is assigned a constant function (respectively a noncanalyzing function, i.e., EQUIVALENCE or EXCLUSIVE OR). Previous work has shown that when a>c, with probability asymptotic to 1, the random automaton exhibits very stable behavior: Almost all of the gates stabilize, almost all of them are weak, i.e., they can be perturbed without affecting the state cycle that is entered, and the state cycle is bounded in size. This article gives evidence that the condition a = c is a threshold of chaotic behavior: with probability asymptotic to 1, almost all of the gates are still stable and weak, but the state cycle size is unbounded. In fact, the average state cycle size is superpolynomial in the number of gates.

Towards the classification of all Boolean cellular automata

A classification scheme for all Boolean cellular automata models involving an arbitrary number of inputs is proposed. The scheme is based upon counting the number of stable points for a given rule, i.e., points whose time development is stable with respect to small perturbations. When applied to all cellular automata with 2, 3, 4, or 5 Boolean inputs (the latter involving more than 4000 million cases), it yields, in each case, a nontrivial classification (i.e., there is no single class in which nearly all the rules lie). In addition, it is shown how rules satisfying a given Boolean differential equation can easily be obtained.

Stability of vertices in random boolean cellular automata

AbstractBased on computer simulations, Kauffman (Physica D, 10, 145‐156, 1984) made several generalizations about a random Boolean cellular automaton which he invented as a model of cellular metabolism. Here we give the first rigorous proofs of two of Kauffman's generalizations: a large fraction of vertices stabilize quickly, consequently the length of cycles in the automaton's behavior is small compared to that of a random mapping with the same number of states; and reversal of the states of a large fraction of the vertices does not affect the cycle to which the automaton moves.