Horizontal and Vertical Structuring Techniques for Statechartsa

In this paper we present an algebraic approach to statecharts as they are used in the Statemate tool in the style of "Petri-Nets are Monoids" for place-transition nets developed by Meseguer and Montanari. We apply the framework of high-level-replacement systems, a categorical generalization of graph transformation systems, in order to deene union as horizontal as well as transformation and reenement as vertical structuring techniques for statecharts. The rst main result shows compatibility of union and transformation in a suitable category of statecharts. We present an algorithm for the computation of all transitions enabled within one step. The second main result shows the correctness of this algorithm. We deene reenement morphisms for statecharts, which allow reenement of arbitrary states, in contrast to concepts in the literature where only basic and root states are subject of reenement. The third main result shows that reenement mor-phisms are compatible with the behavior of statecharts as deened in the formal semantics.