Wim Hordijk

A measure of landscapes

This paper introduces a statistical fitness landscape analysis, based on Weinbergers random walk method and on a time series analysis known as the Box-Jenkins approach, to measure and express the correlation structure of fitness landscapes. The analysis has some additions to and advantages over previous methods for measuring this structure. The analysis is demonstrated on fitness landscapes constructed with Kauffmans NK-model, using two operators (point mutation and a form of crossover) and a combination of the two. Furthermore, the predictive value of the method is shown.

Mechanisms of Emergent Computation in Cellular Automata

We introduce a class of embedded-particle models for describing the emergent computational strategies observed in cellular automata (CAs) that were evolved for performing certain computational tasks. The models are evaluated by comparing their estimated performances with the actual performances of the CAs they model. The results show, via a close quantitative agreement, that the embedded-particle framework captures the main information processing mechanisms of the emergent computation that arise in these evolved CAs.

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

Material Representations: From the Genetic Code to the Evolution of Cellular Automata

We present a new definition of the concept of representation for cognitive science that is based on a study of the origin of structures that are used to store memory in evolving systems. This study consists of novel computer experiments in the evolution of cellular automata to perform nontrivial tasks as well as evidence from biology concerning genetic memory. Our key observation is that representations require inert structures to encode information used to construct appropriate dynamic configurations for the evolving system. We propose criteria to decide if a given structure is a representation by unpacking the idea of inert structures that can be used as memory for arbitrary dynamic configurations. Using a genetic algorithm, we evolved cellular automata rules that can perform nontrivial tasks related to the density task (or majority classification problem) commonly used in the literature. We present the particle catalogs of the new rules following the computational mechanics framework. We discuss if the evolved cellular automata particles may be seen as representations according to our criteria. We show that while they capture some of the essential characteristics of representations, they lack an essential one. Our goal is to show that artificial life can be used to shed new light on the computation-versus-dynamics debate in cognitive science, and indeed function as a constructive bridge between the two camps. Our definitions of representation and cellular automata experiments are proposed as a complementary approach, with both dynamics and informational modes of explanation.

Amplitude Spectra of Fitness Landscapes

Fitness landscapes can be decomposed into elementary landscapes using a Fourier transform that is determined by the structure of the underlying configuration space. The amplitude spectrum obtained from the Fourier transform contains information about the ruggedness of the landscape. It can be used for classification and comparison purposes. We consider here three very different types of landscapes using both mutation and recombination to define the topological structure of the configuration spaces. A reliable procedure for estimating the amplitude spectra is presented. The method is based on certain correlation functions that are easily obtained from empirical studies of the landscapes.

Correlation analysis of the synchronizing-CA landscape

A correlation analysis will be applied to subspaces of the fitness landscape generated by the synchronization task for one- dimensional cellular automata. This results in a stochastic model that can be used to characterize the correlation structure of those subspaces. The results show that both subspaces can be characterized by an AR(2) model, both for point mutation and crossover.

The Usefulness of Recombination

In this paper, we examine the usefulness of recombination from two points of view. First, the problem of crossover disruption is investigated. This is done by comparing two Genetic Algorithms with different crossover operators (one-point and uniform) to each other on NK-landscapes with different values of K relative to N, and with different epistatic interactions (random and nearest neighbor). Second, the usefulness of recombination in relation to the location of local optima in the fitness landscape is investigated.

Phase transition and landscape statistics of the number partitioning problem

The phase transition in the number partitioning problem (NPP), i.e., the transition from a region in the space of control parameters in which almost all instances have many solutions to a region in which almost all instances have no solution, is investigated by examining the energy landscape of this classic optimization problem. This is achieved by coding the information about the minimum energy paths connecting pairs of minima into a tree structure, termed a barrier tree, the leaves and internal nodes of which represent, respectively, the minima and the lowest energy saddles connecting those minima. Here we apply several measures of shape (balance and symmetry) as well as of branch lengths (barrier heights) to the barrier trees that result from the landscape of the NPP, aiming at identifying traces of the easy-hard transition. We find that it is not possible to tell the easy regime from the hard one by visual inspection of the trees or by measuring the barrier heights. Only the difficulty measure, given by the maximum value of the ratio between the barrier height and the energy surplus of local minima, succeeded in detecting traces of the phase transition in the tree. In addition, we show that the barrier trees associated with the NPP are very similar to random trees, contrasting dramatically with trees associated with the p spin-glass and random energy models. We also examine critically a recent conjecture on the equivalence between the NPP and a truncated random energy model.