Convergence analysis of rule-generality on the XCS classifier system

The XCS classifier system adaptively controls a rule-generality of a rule-condition through a rule-discovery process. However, there is no proof that the rule-generality can eventually converge to its optimum value even under some ideal assumptions. This paper conducts a convergence analysis of the rule-generality on the rule-discovery process with the ternary alphabet coding. Our analysis provides the first proof that an average rule-generality of rules in a population can converge to its optimum value under some assumptions. This proof can be used to mathematically conclude that the XCS framework has a natural pressure to explore rules toward optimum rules if XCS satisfies our derived conditions. In addition, our theoretical result returns a rough setting-up guideline for the maximum population size, the mutation rate, and the GA threshold, improving the convergence speed of the rule-generality and the XCS performance.