A new schema theorem for uniform crossover based on ternary representation

Abstract

Crossover is a fundamental operator in genetic algorithms, through which not only an existing schema may be either eliminated or survived, but also a new schema is constructed via other existing schemata. Unfortunately the traditional schema theorem (Holland, J.H., 1975) does not take into account the positive effects of a schema construction through crossover operation. Recently, some works have been done by considering the schema construction, but they could not well characterize the evolution of a schema via crossover. We propose a new representation of a schema, called ternary representation, through which the survival and construction probabilities of a schema are given out. Eventually, we present a new improved schema theorem that considers both schema survival and construction in a uniform crossover.