Cosma Rohilla Shalizi

Computational Mechanics: Pattern and Prediction, Structure and Simplicity

Computational mechanics, an approach to structural complexity, defines a processs causal states and gives a procedure for finding them. We show that the causal-state representation—an ∈-machine—is the minimal one consistent with accurate prediction. We establish several results on ∈-machine optimality and uniqueness and on how ∈-machines compare to alternative representations. Further results relate measures of randomness and structural complexity obtained from ∈-machines to those from ergodic and information theories.

Upper bound on the products of particle interactions in cellular automata

Particle-like objects are observed to propagate and interact in many spatially extended dynamical systems. For one of the simplest classes of such systems, one-dimensional cellular automata, we establish a rigorous upper bound on the number of distinct products that these interactions can generate. The upper bound is controlled by the structural complexity of the interacting particles -- a quantity which is defined here and which measures the amount of spatio-temporal information that a particle stores. Along the way we establish a number of properties of domains and particles that follow from the computational mechanics analysis of cellular automata; thereby eludicating why that approach is of general utility. The upper bound is tested against several relatively complex domain-particle cellular automata and found to be tight. PACS: 45.70.Qj, 05.45, 05.65+b

INFORMATION BOTTLENECKS, CAUSAL STATES, AND STATISTICAL RELEVANCE BASES: HOW TO REPRESENT RELEVANT INFORMATION IN MEMORYLESS TRANSDUCTION

Discovering relevant, but possibly hidden, variables is a key step in constructing useful and predictive theories about the natural world. This brief note explains the connections between three approaches to this problem: the recently introduced information-bottleneck method, the computational mechanics approach to inferring optimal models, and Salmons statistical relevance basis.