Learning Classifier Systems (LCSs), such as the accuracy-based XCS, evolve distributed problem solutions represented by a population of rules. During evolution, features are specialized, propagated, and recombined to provide increasingly accurate subsolutions. Recently, it was shown that, as in conventional genetic algorithms (GAs), some problems require efficient processing of subsets of features to find problem solutions efficiently. In such problems, standard variation operators of genetic and evolutionary algorithms used in LCSs suffer from potential disruption of groups of interacting features, resulting in poor performance. This paper introduces efficient crossover operators to XCS by incorporating techniques derived from competent GAs: the extended compact GA (ECGA) and the Bayesian optimization algorithm (BOA). Instead of simple crossover operators such as uniform crossover or one-point crossover, ECGA or BOA-derived mechanisms are used to build a probabilistic model of the global population and to generate offspring classifiers locally using the model. Several offspring generation variations are introduced and evaluated. The results show that it is possible to achieve performance similar to runs with an informed crossover operator that is specifically designed to yield ideal problem-dependent exploration, exploiting provided problem structure information. Thus, we create the first competent LCSs, XCS/ECGA and XCS/BOA, that detect dependency structures online and propagate corresponding lower-level dependency structures effectively without any information about these structures given in advance.