Parallel dynamics and computational complexity of the Bak-Sneppen model

The parallel computational complexity of the Bak-Sneppen evolution model is studied. It is shown that Bak-Sneppen histories can be generated by a massively parallel computer in a time that is polylogarithmic in the length of the history. In this parallel dynamics, histories are built up via a nested hierarchy of avalanches. Stated in another way, the main result is that the logical depth of producing a Bak-Sneppen history is exponentially less than the length of the history. This finding is surprising because the self-organized critical state of the Bak-Sneppen model has long range correlations in time and space that appear to imply that the dynamics is sequential and history dependent. The parallel dynamics for generating Bak-Sneppen histories is contrasted to standard Bak-Sneppen dynamics. Standard dynamics and an alternate method for generating histories, conditional dynamics, are both shown to be related to P-complete natural decision problems implying that they cannot be efficiently implemented in parallel.Comment: 37 pages, 12 figure

Chaos in spin glasses revealed through thermal boundary conditions

We study the fragility of spin glasses to small temperature perturbations numerically using population annealing Monte Carlo. We apply thermal boundary conditions to a three-dimensional Edwards-Anderson Ising spin glass. In thermal boundary conditions all eight combinations of periodic versus antiperiodic boundary conditions in the three spatial directions are present, each appearing in the ensemble with its respective statistical weight determined by its free energy. We show that temperature chaos is revealed in the statistics of crossings in the free energy for different boundary conditions. By studying the energy difference between boundary conditions at free-energy crossings, we determine the domain-wall fractal dimension. Similarly, by studying the number of crossings, we determine the chaos exponent. Our results also show that computational hardness in spin glasses and the presence of chaos are closely related.Comment: 4 pages, 4 figure

Packing Squares in a Torus

The densest packings of N unit squares in a torus are studied using analytical methods as well as simulated annealing. A rich array of dense packing solutions are found: density-one packings when N is the sum of two square integers; a family of "gapped bricklayer" Bravais lattice solutions with density N/(N+1); and some surprising non-Bravais lattice configurations, including lattices of holes as well as a configuration for N=23 in which not all squares share the same orientation. The entropy of some of these configurations and the frequency and orientation of density-one solutions as N goes to infinity are discussed.Comment: 14 pages, 9 figures; v2 reflects minor changes in published versio

Parallel Complexity of Random Boolean Circuits

Random instances of feedforward Boolean circuits are studied both analytically and numerically. Evaluating these circuits is known to be a P-complete problem and thus, in the worst case, believed to be impossible to perform, even given a massively parallel computer, in time much less than the depth of the circuit. Nonetheless, it is found that for some ensembles of random circuits, saturation to a fixed truth value occurs rapidly so that evaluation of the circuit can be accomplished in much less parallel time than the depth of the circuit. For other ensembles saturation does not occur and circuit evaluation is apparently hard. In particular, for some random circuits composed of connectives with five or more inputs, the number of true outputs at each level is a chaotic sequence. Finally, while the average case complexity depends on the choice of ensemble, it is shown that for all ensembles it is possible to simultaneously construct a typical circuit together with its solution in polylogarithmic parallel time.Comment: 16 pages, 10 figures, matches published versio