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Christopher R. Stephens
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Journal Articles
Publisher: Journals Gateway
Evolutionary Computation (2007) 15 (3): 291–320.
Published: 01 September 2007
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We present a covariant form for the dynamics of a canonical GA of arbitrary cardinality, showing how each genetic operator can be uniquely represented by a mathematical object — a tensor — that transforms simply under a general linear coordinate transformation. For mutation and recombination these tensors can be written as tensor products of the analogous tensors for one-bit strings thus giving a greatly simplified formulation of the dynamics. We analyze the three most well known coordinate systems — string, Walsh and Building Block — discussing their relative advantages and disadvantages with respect to the different operators, showing how one may transform from one to the other, and that the associated coordinate transformation matrices can be written as a tensor product of the corresponding one-bit matrices. We also show that in the Building Block basis the dynamical equations for all Building Blocks can be generated from the equation for the most fine-grained block (string) by a certain projection (“zapping”).
Journal Articles
Publisher: Journals Gateway
Evolutionary Computation (2007) 15 (1): 95–131.
Published: 01 March 2007
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This is the second part of a two-part paper where we propose, model theoretically and study a general notion of recombination for fixed-length strings where homologous recombination, inversion, gene duplication, gene deletion, diploidy and more are just special cases. In Part I, we derived both microscopic and coarse-grained evolution equations for strings and schemata for a selecto-recombinative GA using generalised recombination, and we explained the hierarchical nature of the schema evolution equations. In this part, we provide a variety of fixed points for evolution in the case where recombination is used alone, thereby generalising Geiringer's theorem. In addition, we numerically integrate the infinite-population schema equations for some interesting problems, where selection and recombination are used together to illustrate how these operators interact. Finally, to assess by how much genetic drift can make a system deviate from the infinite-population-model predictions we discuss the results of real GA runs for the same model problems with generalised recombination, selection and finite populations of different sizes.
Journal Articles
Publisher: Journals Gateway
Evolutionary Computation (2006) 14 (4): 411–432.
Published: 01 December 2006
Abstract
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This is the first part of a two-part paper where we propose, model theoretically and study a general notion of recombination for fixed-length strings, where homologous recombination, inversion, gene duplication, gene deletion, diploidy and more are just special cases. The analysis of the model reveals that the notion of schema emerges naturally from the model's equations. In Part I, after describing and characterising the notion of generalised recombination, we derive both microscopic and coarse-grained evolution equations for strings and schemata and illustrate their features with simple examples. Also, we explain the hierarchical nature of the schema evolution equations and show how the theory presented here generalises past work in evolutionary computation. In Part II, the study provides a variety of fixed points for evolution in the case where recombination is used alone, which generalise Geiringer's theorem. In addition, we numerically integrate the infinite-population schema equations for some interesting problems, where selection and recombination are used together to illustrate how these operators interact. Finally, to assess by how much genetic drift can make a system deviate from the infinite-population-model predictions we discuss the results of real GA runs for the same model problems with generalised recombination, selection and finite populations of different sizes.