Holland's Schema Theorem is widely taken to be the foundation for explanations of the power of genetic algorithms (GAs). Yet some dissent has been expressed as to its implications. Here, dissenting arguments are reviewed and elaborated upon, explaining why the Schema Theorem has no implications for how well a GA is performing. Interpretations of the Schema Theorem have implicitly assumed that a correlation exists between parent and offspring fitnesses, and this assumption is made explicit in results based on Price's Covariance and Selection Theorem. Schemata do not play a part in the performance theorems derived for representations and operators in general. However, schemata re-emerge when recombination operators are used. Using Geiringer's recombination distribution representation of recombination operators, a "missing" schema theorem is derived which makes explicit the intuition for when a GA should perform well. Finally, the method of "adaptive landscape" analysis is exa...

Fitness landscapes are an important concept in molecular evolution. Many important examples of landscapes in physics and combinatorial optimation, which are widely used as model landscapes in simulations of molecular evolution and adaptation, are "elementary", i.e., they are (up to an additive constant) eigenfuctions of a graph Laplacian. It is shown that elementary landscapes are characterized by their correlation functions. The correlation functions are in turn uniquely determined by the geometry of the underlying configuration space and the nearest neighbor correlation of the elementary landscape. Two types of correlation functions are investigated here: the correlation of a time series sampled along a random walk on the landscape and the correlation function with respect to a partition of the set of all vertex pairs.

In this paper we view the folding of polynucleotide (RNA) sequences as a map that assigns to each sequence a minimum free energy pattern of base pairings, known as secondary structure. Considering only the free energy leads to an energy landscape over the sequence space. Taking into account structure generates a less visualizable non-scalar "landscape", where a sequence space is mapped into a space of discrete "shapes". We investigate the statistical features of both types of landscapes by computing autocorrelation functions, as well as distributions of energy and structure distances, as a function of distance in sequence space. RNA folding is characterized by very short structure correlation lengths compared to the diameter of the sequence space. The correlation lengths depend strongly on the size and the pairing rules of the underlying nucleotide alphabet. Our data suggest that almost every minimum free energy structure is found within a small neighborhood of any random sequence. The...

Introduction The notion of fitness landscapes has been introduced to describe the dynamics of evolutionary adaptation in nature [40] and has become a powerful concept in evolutionary theory. Fitness landscapes are equally well suited to describe the behavior of heuristic search methods in optimization, since the process of evolution can be thought of as searching a collection of genotypes in order to find the genotype of an organism with highest fitness and thus highest chance of survival. Thinking of a heuristic search method as a strategy to "navigate" in the fitness landscape of a given optimization problem may help in predicting the performance of a heuristic search algorithm if the structure of the landscape is known in advance. Furthermore, the analysis of fitness landscapes may help in designing highly effective search algorithms. In the following we show how the analysis of fitness landscapes of combinatorial optimization problems can aid in designing the components of

We present and study the behavior of a simple kinetic model for the melting of RNA secondary structures, given that those structures are known. The model is then used as a map that assigns structure dependent overall rate constants of melting (or refolding) to a sequence. This induces a "landscape" of reaction rates, or activation energies, over the space of sequences with fixed length. We study the distribution and the correlation structure of these activation energies. 1. Introduction Single stranded RNA sequences fold into complex three-dimensional structures. A tractable, yet reasonable, model for the map from sequences to structures considers a more coarse grained level of resolution known as the secondary structure. The secondary structure is a list of base pairs such that no pairings occur between bases located in different loop regions. Algorithms based on empirical energy data have been developed to compute the minimum free energy secondary structure of an RNA sequence (Zuker...

The evolution of RNA molecules in replication assays, viroids and RNA viruses can be viewed as an adaptation process on a 'fitness' landscape. The dynamics of evolution is hence tightly linked to the structure of the underlying landscape. Global features of landscapes can be described by statistical measures like number of optima, lengths of walks, and correlation functions. The evolution of a quasispecies on such landscapes exhibits three dynamical regimes depending on the replication fidelity: Above the "localization threshold" the population is centered around a (local) optimum. Between localization and "dispersion threshold" the population is still centered around a consensus sequence, which, however, changes in time. For very large mutation rates the population spreads in sequence space like a gas. The critical mutation rates separating the three domains depend strongly on characteristics properties of the fitness landscapes. Statistical characteristics of RNA landscapes are acces...

In many cases fitness landscapes are obtained as particular instances of random fields by assigning a large number of random parameters. Models of this type are often characterized reasonably well by their covariance matrices. We characterize isotropic random fields on finite graphs in terms of their Fourier series expansions and investigate the relation between the covariance matrix of the random field model and the correlation structure of the individual landscapes constructed from this random field. Our formalism suggests to approximate landscape with known autocorrelation function by a random field model that has the same correlation structure.

Understanding complex behavior requires a multidisplinary effort from the neurosciences, psychology, behavioral biology, and computer science. This paper gives an overview of the current state of theoretical thinking in the field. The focus is on a behavior--oriented approach to cognition, i.e., not so much on the mental representations themselves, but on the behaviors that do require these representations. It is the intention of the paper to support the exchange between the different disciplines involved. Examples of different types of models and explanations are discussed, but no comprehensive review of all relevant work is attempted. In the second part, I collect a number of elements that in my view are essential to a future theory of cognitive behavior. Keywords: Cognition, Perception and Action, Brain Theory, Computational Theory, Artificial Life, Virtual Reality Mallot: Behavior--Oriented Approaches to Cognition Page 2 1 Introduction 1.1 Perception, Action, and Cognition The ...

The relationship of orthogonal functions associated with vertex transitive graphs and random walks on such graphs is investigated. We use this relations to characterize the exponentially decaying autocorrelation functions along random walks on isotropic random fields defined on vertex transitive graphs. The results are applied to a simple spin glass model. Motivation Recently "combinatory landscapes" --- maps from the vertex set of some graph into the real or complex numbers --- have received increasing attention. The classic example from physics is a Hamiltonian that assigns an energy value to a spin configuration (M'ezard et al. 1987). Combinatorial optimization problems, like the travelling salesman problem (Lawler et al. 1985), are of the same type. In evolutionary biology maps assigning free energies or "fitness values" to biomolecules --- encoded as strings over a finite alphabet --- are of particular interest (Eigen et al. 1989, Fontana et al. 1991, 1992). For each of these mo...

Numerical estimates for the number of local optima of p-spin Hamiltonians with p = 3; 4; 5 and 6 are reported. The data can be explained by means of a simple estimate in terms of the "correlation length" of the Hamiltonian.