Agreement between conformance and composition

Abstract

In our previous paper [1], a new model of a Labeled Transition System (LTS)-type implementation was proposed. In ordinary LTSs, transitions are labeled by actions; therefore, they can be called LTSs of actions. The new model is an LTS of observations; in this model, observations and test actions (buttons) are used instead of actions. This model generalizes many testing semantics that are based on the LTS of actions but use additional observations (refusals, ready sets, etc.). Moreover, systems with priority, which are not described by the LTS of actions, are simulated uniformly. In the present paper, we develop this approach by focusing on the composition of systems. The point is that, on observation traces, one cannot define a composition with respect to which a composition of LTSs would possess the property of additivity: the set of traces of a composition of LTSs coincides with the set of all pairwise compositions of traces of LTS operands. This is explained by the fact that an observation in a composition state is not calculated based on observations in states-operands. In this paper, we propose an approach that eliminates this drawback. To this end, we label the transitions of LTSs by symbols (events) that, on the one hand, can be composed to guarantee the property of additivity, and, on the other hand, can be used to generate observations under testing: a transition by an event gives rise to an observation related to this event. This model is called an LTS of events. In this paper, we define (1) a transformation of an LTS of events into an LTS of observations to conform with the principles of our previous paper [1]; (2) a composition of LTSs of events; (3) a composition of specifications that preserves conformance: a composition of conformal implementations is conformal to a composition of specifications; and (4) a uniform simulation of LTSs of actions in terms of the LTSs of events, which allows one to consider an implementation in any interaction semantics admissible for LTSs of actions. In this case, a composition of LTSs of events obtained as a result of simulation of the original LTSs of actions is equivalent to the LTS of events obtained as a result of simulation of the composition of these LTSs of actions.