Fault Tolerance in Cellular Automata at High Fault Rates

A commonly used model for fault-tolerant computation is that of cellular automata. The essential difficulty of fault-tolerant computation is present in the special case of simply remembering a bit in the presence of faults, and that is the case we treat in this paper. We are concerned with the degree (the number of neighboring cells on which the state transition function depends) needed to achieve fault tolerance when the fault rate is high (nearly 1/2). We consider both the traditional transient fault model (where faults occur independently in time and space) and a recently introduced combined fault model which also includes manufacturing faults (which occur independently in space, but which affect cells for all time). We also consider both a purely probabilistic fault model (in which the states of cells are perturbed at exactly the fault rate) and an adversarial model (in which the occurrence of a fault gives control of the state to an omniscient adversary). We show that there are cellular automata that can tolerate a fault rate $1/2 - \xi$ (with $\xi>0$) with degree $O((1/\xi^2)\log(1/\xi))$, even with adversarial combined faults. The simplest such automata are based on infinite regular trees, but our results also apply to other structures (such as hyperbolic tessellations) that contain infinite regular trees. We also obtain a lower bound of $\Omega(1/\xi^2)$, even with purely probabilistic transient faults only

Intrinsic Universality in Self-Assembly

We show that the Tile Assembly Model exhibits a strong notion of universality where the goal is to give a single tile assembly system that simulates the behavior of any other tile assembly system. We give a tile assembly system that is capable of simulating a very wide class of tile systems, including itself. Specifically, we give a tile set that simulates the assembly of any tile assembly system in a class of systems that we call \emph{locally consistent}: each tile binds with exactly the strength needed to stay attached, and that there are no glue mismatches between tiles in any produced assembly. Our construction is reminiscent of the studies of \emph{intrinsic universality} of cellular automata by Ollinger and others, in the sense that our simulation of a tile system $T$ by a tile system $U$ represents each tile in an assembly produced by $T$ by a $c \times c$ block of tiles in $U$, where $c$ is a constant depending on $T$ but not on the size of the assembly $T$ produces (which may in fact be infinite). Also, our construction improves on earlier simulations of tile assembly systems by other tile assembly systems (in particular, those of Soloveichik and Winfree, and of Demaine et al.) in that we simulate the actual process of self-assembly, not just the end result, as in Soloveichik and Winfree's construction, and we do not discriminate against infinite structures. Both previous results simulate only temperature 1 systems, whereas our construction simulates tile assembly systems operating at temperature 2

Computing by Temporal Order: Asynchronous Cellular Automata

Our concern is the behaviour of the elementary cellular automata with state set 0,1 over the cell set Z/nZ (one-dimensional finite wrap-around case), under all possible update rules (asynchronicity). Over the torus Z/nZ (n<= 11),we will see that the ECA with Wolfram rule 57 maps any v in F_2^n to any w in F_2^n, varying the update rule. We furthermore show that all even (element of the alternating group) bijective functions on the set F_2^n = 0,...,2^n-1, can be computed by ECA57, by iterating it a sufficient number of times with varying update rules, at least for n <= 10. We characterize the non-bijective functions computable by asynchronous rules.Comment: In Proceedings AUTOMATA&JAC 2012, arXiv:1208.249

Topology regulates pattern formation capacity of binary cellular automata on graphs

We study the effect of topology variation on the dynamic behavior of a system with local update rules. We implement one-dimensional binary cellular automata on graphs with various topologies by formulating two sets of degree-dependent rules, each containing a single parameter. We observe that changes in graph topology induce transitions between different dynamic domains (Wolfram classes) without a formal change in the update rule. Along with topological variations, we study the pattern formation capacities of regular, random, small-world and scale-free graphs. Pattern formation capacity is quantified in terms of two entropy measures, which for standard cellular automata allow a qualitative distinction between the four Wolfram classes. A mean-field model explains the dynamic behavior of random graphs. Implications for our understanding of information transport through complex, network-based systems are discussed.Comment: 16 text pages, 13 figures. To be published in Physica

Evolving cellular automata to generate nonlinear sequences with desirable properties

This paper presents a new chromosomal representation and associated genetic operators for the evolution of highly nonlinear cellular automata that generate pseudorandom number sequences with desirable properties ensured. This chromosomal representation reduces the computational complexity of genetic operators to evolve valid solutions while facilitating fitness evaluation based on the DIEHARD statistical tests