FSM quasi-equivalence testing via reduction and observing absence

There has been significant interest in automatically generating test cases from a non-deterministic finite state machine (FSM). Most approaches check that the behaviours of the system under test (SUT) are allowed by the specification FSM; they therefore test for reduction. However, sometimes one wants all of the behaviours, and so features, of the specification to be implemented and then one is testing for equivalence. In this paper we first note that in order to test for equivalence one must effectively be able to observe the SUT not being able to produce an output $y$ in response to an input $x$ after trace $\sigma$; we model this as the absence of an output. We prove that the problem of testing for equivalence to FSM $M$ can be mapped to testing for reduction to an FSM $R(M)$ that extends $M$ with absences. Thus, one can use techniques developed for testing for reduction when testing for equivalence. We then consider the case where the specification is partial, generalising the result to quasi-equivalence. These results are proved for observable specifications and so we also show how a partial FSM can be mapped to an observable partial FSM from which we can test.