The Simple Genetic Algorithm and the Walsh Transform: Part I, Theory

Vose, Michael D.; Wright, Alden H.

doi:10.1162/evco.1998.6.3.253

Cited by 50 publications

(37 citation statements)

References 8 publications

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“…It is essential that for molecular-genetic systems operations of additions and subtractions can be organized simply by means of interconnections or disconnections of molecular elements which is easier than molecular organization of multiplication operations. It should be mentioned that some works about applications of Walsh functions for spectral analysis of genetic sequences and genetic algorithms are known (Geadah and Corinthios, 1977;Goldberg, 1989;Shiozaki, 1980;Vose and Wright, 1998;Waterman, 1999). These functions can be useful in researches in algebraic biology (Pellionisz et al, 2011;Petoukhov and He, 2010, etc).…”

Section: Discussionmentioning

confidence: 99%

Matrix Genetics and Algebraic Properties of the Multi-Level System of Genetic Alphabets

Petoukhov¹

2011

Neuroquantology

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The article is devoted to algebraic properties of the multi-level system of moleculargenetic alphabets. It leads to help solve the problem of algebraic unity of inherited information systems in living matter. These algebraic properties are revealed by means of Kronecker families of matrix forms of a presentation of molecular-genetic alphabets. A family of genetic (8x8)-matrices shows unexpected connections of the genetic system with Rademacher and Walsh functions and with special Hadamard matrices which are well-known in theory of noise-immunity coding and digital communication. Decompositions of such genetic (8x8)-matrices on the basis of the known principle of dyadic-shifts lead to sets of 8 sparse matrices. Each of these sets is closed in relation to multiplication and defines a special algebra of 8-dimensional hypercomplex numbers. Mathematical aspects of these 8-dimensional algebras are presented in connection with metric vector spaces, the sequency theory by Harmuth and some methods of spectral analysis. The diversity of known dialects of the genetic code can be analyzed from the viewpoint of these algebras. Our results are discussed taking into account the important role of dyadic shifts, hypercomplex numbers and Hadamard matrices in mathematics, informatics, theoretical physics, etc. These results testify that living matter has a profound algebraic essence which is interconnected with 8-dimensional vector spaces. In our opinion these results lead to a new way of knowledge of living matter in the field of algebraic biology and its mathematical modeling. The idea of a biological meaning of Kronecker multiplication of matrices is based on the structure of Punnett squares in the field of Mendelian genetics. The author believes that the Mendelian laws of independent inheritance of traits have revealed just the tip of an algebraic iceberg of informational structure of living matter and that matrix genetics has contributed to the next steps to disclose this important iceberg.

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Section: Discussionmentioning

confidence: 99%

Matrix Genetics and Algebraic Properties of the Multi-Level System of Genetic Alphabets

Petoukhov¹

2011

Neuroquantology

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“…But the fact that there is also a significant likelihood that the superior peak is not found means that the likelihood of solving all sub-functions by mutation alone becomes very small when the number of blocks is large. This point is important because although Walsh transform analysis [25] and schema disruption analysis [10][24], for example, will agree that our function, like other separable building-block functions, is GA-easy for moderate k, such analysis does not confirm that a problem is difficult for mutationonly algorithms.…”

Section: Discussionmentioning

confidence: 93%

A building-block royal road where crossover is provably essential

Watson

Jansen

2007

Proceedings of the 9th Annual Conference on Genetic and Evolutionary Computation

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One of the most controversial yet enduring hypotheses about what genetic algorithms (GAs) are good for concerns the idea that GAs process building-blocks. More specifically, it has been suggested that crossover in GAs can assemble short low-order schemata of above average fitness (building blocks) to create higher-order higher-fitness schemata. However, there has been considerable difficulty in demonstrating this rigorously and intuitively. Here we provide a simple building-block function that a GA with twopoint crossover can solve on average in polynomial time, whereas an asexual population or mutation hill-climber cannot. Categories and Subject Descriptors General TermsAlgorithms, Performance, Theory. KeywordsMutation, crossover, modularity, building block hypothesis, genetic algorithms theory, royal roads. , incorporates a wide fitness valley that must be overcome by a crossover event and all other fitness improvements are provided by mutation. These functions seem quite contrived: There is no intuitive explanation for why it should be the case that the positions of the global optimum and local optima should be just as they are (such that the global optimum is located at a genotype produced by crossover of the locally optimal genotypes). Although such a function does satisfy the essential property of distinguishing polynomial versus exponential expected time complexities of crossover and mutation-based algorithms respectively, their contrived structure makes it difficult to see what they have to offer with respect to our understanding of what GAs are good for in general. WHAT A GA IS GOOD FORSome of the other functions that show a principled distinction in the ability of crossover [26][27] do use a building-block structure but are too complex to facilitate rigorous proofs for their time complexities [13]. In particular, the HIFF function [27] illustrates building-block structure directly inspired by the BBH but the ability of a GA to outperform mutation-based algorithms on this function is dependent on its multi-level building-block structure, and indeed the requirement for a true GA rather than a 'crossover hill-climber ' [4][27] requires additional modifications to the function (specifically, the use of non-complementary global optima [27]). As yet, no simple single-level building-block function has been provided where crossover is provably essential.In this paper we return to a simple separable building-block function to see if we can rescue some intuition and yet maintain rigorous proofs that distinguish polynomial and non-polynomial time complexities for a GA with and without crossover respectively. In this paper we address only the original conception of a building-block that includes the assumption of tight linkage [6][8][10] [11][17] and ordinary one-or two-point crossover that could potentially exploit this structure as described in the BBH. Note that a large body of work on linkage learning [9] and modelbuilding methods, e.g. [19], has developed in the GA community that addresses the exploi...

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“…Geiringer's theorem ( [6], [10], [9]) gives the limiting population of a crossover-only infinitepopulation GA model on a flat fitness landscape. The limiting population has the same frequency of each allele as the initial population, and the frequency of a string is the product of the frequency of the alleles.…”

Section: Discussionmentioning

confidence: 99%