Isotopic Inheritance: A Topological Approach to Genotype Transfer

摘要

Classic genetic algorithms (GA) incorporate a concept of variability in genotype to phenotype mapping only by the notion of generative encodings. In such cases, the phenotype depends not only on the pure values of genes but also on their interaction and influence of other factors like the environment. In case of generative encoding GA not exact phenotype of an individual is inherited but the isomorphic function to form the individual. The same information may lead to the formation of different phenotypes due to alterations in conditions or permutations in the gene sequence. We propose a mathematical definition of the abstract notion of genetic variability and a new approach to the problem of the genotype transition to a new generation called "isotopic inheritance". To this extent, we apply the notion of topological isotopy and use the branch of low-dimensional topology, known as braid theory to encode the values of a gene of arbitrary length into the genome and to propagate the isomorphic genotypes among the offspring. We illustrate the propositional variations of such encoding in different applications. Multiple knapsack problem was solved using the proposed approach.