Lothar M. Schmitt

Theory of genetic algorithms

(i) We investigate spectral and geometric properties of the mutation-crossover operator in a genetic algorithm with general-size alphabet. By computing spectral estimates, we show how the crossover operator enhances the averaging procedure of the mutation operator in the random generator phase of the genetic algorithm. By mapping our model to the multi-set model often investigated in the literature, we compute corresponding spectral estimates for mutation-crossover in the multi-set model.(ii) Various types of unscaled or scaled fitness selection mechanisms are considered such as proportional fitness selection, rank selection, and tournament fitness selection. We allow fitness selection mechanisms where the fitness of an individual or creature depends upon the population it resides in. We investigate contracting properties of these fitness selection mechanisms and combine them with the crossover operator to obtain a model for genetic drift. This has applications to the study of genetic algorithms with zero or extremely low mutation rate.(iii) We discuss a variety of convergent simulated-annealing-type algorithms with mutation-crossover as generator matrix.(iv) The theory includes proof of strong ergodicity for various types of scaled genetic algorithms using common fitness selection methods. If the mutation rate converges to a positive value, and the other operators of the genetic algorithm converge, then the limit probability distribution over populations is fully positive at uniform populations whose members have not necessarily optimal fitness.(v) In what follows, suppose the mutation rate converges to zero sufficiently slow to assure weak ergodicity of the inhomogeneous Markov chain describing the genetic algorithm, unbounded power-law scaling for the fitness selection is used, mutation and crossover commute, and the fitness function is injective which is a minor restriction in regard to function optimization.(va) If a certain integrable convergence condition is satisfied such that the selection pressure increases fast, then there is essentially no other restriction on the crossover operation, and the algorithm asymptotically behaves as the following take-the-best search algorithm: (1) mutate in every step with rate decreasing to zero, and (2) map any population to the uniform population with the best creature. The take-the-best search algorithm is investigated, and its convergence is shown. Depending upon population-size, the take-the-best search algorithm does or does not necessarily converge to the optimal solution.(vb) If population size is larger than length of genome, and a certain logarithmic convergence condition is satisfied such that the selection pressure increases slowly but sufficiently fast, then the algorithm asymptotically converges to the optimal solution.

Theory of genetic algorithms II: models for genetic operators over the string-tensor representation of populations and convergence to global optima for arbitrary fitness function under scaling

We present a theoretical framework for an asymptotically converging, scaled genetic algorithm which uses an arbitrary-size alphabet and common scaled genetic operators. The alphabet can be interpreted as a set of equidistant real numbers and multiple-spot mutation performs a scalable compromise between pure random search and neighborhood-based change on the alphabet level. We discuss several versions of the crossover operator and their interplay with mutation. In particular, we consider uniform crossover and gene-lottery crossover which does not commute with mutation. The Vose-Liepins version of mutation-crossover is also integrated in our approach. In order to achieve convergence to global optima, the mutation rate and the crossover rate have to be annealed to zero in proper fashion, and unbounded, power-law scaled proportional fitness selection is used with logarithmic growth in the exponent. Our analysis shows that using certain types of crossover operators and large population size allows for particularly slow annealing schedules for the crossover rate. In our discussion, we focus on the following three major aspects based upon contraction properties of the mutation and fitness selection operators: (i) the drive towards uniform populations in a genetic algorithm using standard operations, (ii) weak ergodicity of the inhomogeneous Markov chain describing the probabilistic model for the scaled algorithm, (iii) convergence to globally optimal solutions. In particular, we remove two restrictions imposed in Theorem 8.6 and Remark 8.7 of (Theoret. Comput. Sci. 259 (2001) 1) where a similar type of algorithm is considered as described here: mutation need not commute with crossover and the fitness function (which may come from a coevolutionary single species setting) need not have a single maximum.

An ER-fMRI investigation of morphological inflection in German reveals that the brain makes a distinction between regular and irregular forms

The hypothesis that morphological processing is supported by a mental dictionary of stored entries plus a set of mental computations based on rules is examined using event-related fMRI. If a rules-plus-memory model () reflects the actual organization of the language faculty, two distinct patterns of brain activation should be observed for production of German irregular and regular noun and verb inflections. If a connectionist alternative to the rules-and-memory model (, and many others since), which seeks to explain the production of both irregular and regular forms within a single associative memory mechanism, is correct, there should be no neural differentiation between German regular and irregular inflection. The results we report support the existence of substantially differing patterns of activation for regulars vs. irregulars, an outcome that is consistent with the two-component rules-plus-memory account.

Linear analysis of genetic algorithms

Original article can be found at: http://www.sciencedirect.com/science/journal/03043975 Copyright Elsevier B.V. DOI: 10.1016/S0304-3975(98)00004-8 [Full text of this article is not available in the UHRA]

Reconstructing occlusal surfaces of teeth using a genetic algorithm with simulated annealing type selection

In this paper, we present an application of numerical optimization for surface reconstruction (more precisely: reconstruction of missing parts of a real geometric object represented by volume data) by employing a specially designed genetic algorithm to solve a problem concerning computer-aided design in dentistry. Using a space mapping technique the surface of a given model tooth is fitted by a shape transformation to extrapolate (or reconstruct) the remaining surface of a patients tooth with occurring damage such as a “drill hole.” Thereby, the genetic algorithm minimizes the error of the approximation by optimizing a set of control points that determine the coefficients for spline functions, which in turn define a space transformation. The fitness function to be minimized by the genetic algorithm is the error between the transformed occlusal surface of the model tooth and the remaining occlusal surface of the damaged (drilled) tooth. The algorithm, that is used, is based upon a proposal by Mahfoud and Goldberg. It uses a simulated-annealing type selection scheme, which is applied sequentially (pair-wise, or one-by-one) to the members in the parent generation and their respective offspring generated by mutation-crossover. We outline a proof of convergence for this algorithm. The algorithm is parallel in regard to computing the fitness-values of creatures.

Theory of coevolutionary genetic algorithms

We discuss stochastic modeling of scaled coevolutionary genetic algorithms (coevGA) which converge asymptotically to global optima. In our setting, populations contain several types of interacting creatures such that for some types (appropriately defined) globally maximal creatures exist. These algorithms particularly demand parallel processing in view of the nature of the fitness function. It is shown that coevolutionary arms races yielding global optima can be implemented in a procedure similar to simulated annealing.

Pedagogical aspects of a UNIX-based network management system for English instruction

Abstract We have developed a unix -based management system (named uneim ) which supports the instructor in teaching English as a second language using a network of workstations. The present implementation is aimed at teaching English composition to Japanese students at The University of Aizu. uneim has a convenient set-up mechanism designed to assist, in particular, the computer novice. While running, uneim takes care of the following tasks using the cron mechanism of unix : assignments are sent out via e-mail on preset dates; if necessary, students are reminded of missing homework; homework sent back by students via e-mail is sorted in regard to course, section and assignment; submission deadlines and required length of the homework are enforced; homework is partially evaluated in regard to mechanical mistakes such as spelling or punctuation; results of the evaluations by the machine are sent back to the students automatically (to trigger resubmission); the writings of students are reformatted to make human correction easier; the use of global or specialized vocabulary can be measured for individual students as well as classes; authentic, interesting or critical examples of grammatical patterns can be identified and collected for presentation in class or research purposes; desired statistical evidence is generated; and graphical display of data is generated.

Exploiting quotients of markov chains to derive properties of the stationary distribution of the markov chain associated to an evolutionary algorithm

In this work, a method is presented for analysis of Markov chains modeling evolutionary algorithms through use of a suitable quotient construction. Such a notion of quotient of a Markov chain is frequently referred to as “coarse graining” in the evolutionary computation literature. We shall discuss the construction of a quotient of an irreducible Markov chain with respect to an arbitrary equivalence relation on the state space. The stationary distribution of the quotient chain is “coherent” with the stationary distribution of the original chain. Although the transition probabilities of the quotient chain depend on the stationary distribution of the original chain, we can still exploit the quotient construction to deduce some relevant properties of the stationary distribution of the original chain. As one application, we shall establish inequalities that describe how fast the stationary distribution of Markov chains modelling evolutionary algorithms concentrates on the uniform populations as the mutation rate converges to 0. Further applications are discussed.

Coevolutionary convergence to global optima

We discuss a theory for a realistic, applicable scaled genetic algorithm (GA) which converges asymptoticly to global optima in a coevolutionary setting involving two species. It is shown for the first time that coevolutionary arms races yielding global optima can be implemented successfully in a procedure similar to simulated annealing.

Convergence to global optima for genetic programming systems with dynamically scaled operators

This work shows asymptotic convergence to global optima for a family of dynamically scaled genetic programming systems where the underlying population consists of a fixed number of creatures (individuals) each of arbitrary size. The genetic programming systems use common mutation and crossover operators as well as fitness-proportional selection. In addition, the mutation and crossover rates are annealed to zero in predefined fashion over the course of the algorithm, and power-law scaling is used for the (possibly population-dependent) initial fitness function with (unbounded) logarithmic growth in the exponent.We assume that a set of globally optimal creatures for the optimization problem instance exists. In addition, it is assumed that the ratio of the best fitness of globally optimal creatures vs the fitness of other creatures is greater or equal a constant ρ>1 in any population they jointly reside in. We discuss how both conditions can usually be satisfied in application settings. Under the above conditions, a selected, traceable sequence of probability distributions over the possible states of the properly scaled genetic programming system converge in time towards the convex set of probability distributions over uniform populations that contain only globally optimal creatures.