Markov Chain Model and Poli Based Schema Model for Bentley's Genetic Algorithm

In this paper, I present the theoretical results obtained for Bentley’s genetic algorithm, which is used in CAD system to generate 3D-solids designs. The Vose-like-Markov-chain model for the Bentley’s GA is proposed. The transition matrices are found and the ergodity of the Markov chain and the asymptotic correctness in the probabilistic sense are shown by using the model. The microscopic Exact Poli GP Schema Theory for Subtree-Swapping Crossovers are applied for the Bentley’s GA to calculate the effective fitness and the total transmission probability for a fixedsize-and-shape schema under hierarchical crossover. Introduction The GA based on hierarchical structures like tree based genetic programming [3, 4], graph-based GA and hierarchical chromosome based GA [1, 5] are often used as the representation in evolutionary algorithms. There are not known effective methods of investigation of asymptotic properties or convergence for that class of algorithms. The paper deals with the two approaches and introduces two models for the Bentley’s GA (with hierarchical chromosome and hierarchical crossover) [1] – Vose-like Markov chain Model and Poli Schema Theory based one. Each approach models other aspects of genetic algorithms. The Markov chain model enables investigation on asymptotic properties and transition matrices. The second approach gives us possibility to calculate the total transmission probability for a fixed-size-and-shape schema under hierarchical crossover and the effective fitness. Markov chain model for Bentley’s GA In this section I will demonstrate a Markov chain Model for the Bentley’s GA under the assumptions, that crossover points are numbers of primitives and that only mutation of alleles is used. The phenotype space Fen is defined as a set of individuals consisting of n/3 primitives of each of three classes: base, seat, back. Each primitive is described by p genes (sequences of q bits). The search space X is defined as ( ) Z X 2 = . After defining the coding and decoding functions, the mutation and crossover are defined as genetic operators on the space X. The crossover operator with two similarity points can be replaced with crossover with one similarity point by the assumption, that the primitives are arbitrary located within the chromosome The probability distribution of the result of the crossing of the codes x, y is given by the following formula: [ ] ∑ = ⊗ ⊕ ⊗ + = ∈X t t t y x z y t t x z cross ) ( ) ( 2 }) ({ , η η where    = − − ≠ = 0 0 t type p p t type p