A new quantifying crossover with ternary representation

As a fundamental operator in genetic algorithms (GAs), crossover may not only make an existing schema survive, but also construct a new one from other existing schemata. Unfortunately, the existing schema theorems do not exactly quantify the positive effects of schema construction by a crossover. Consequently, they cannot well characterize the evolution of a schema. In this paper, a new ternary representation is proposed through which the survival and construction of a schema for any crossover operator can be distinguished easily. Subsequently, the probabilities that a schema will survive and be constructed from other schemata can be estimated for an arbitrary crossover. As a result, we present four new schema theorems, through which the survival and construction abilities of different crossover operations are easier to be compared. Moreover, these results generalize the existing schema theorems.