In the optimization of test sets for black-box conformance testing of objects specified and modelled as a finite state machine (FSM), a popular transformation is that under a certain precondition, a tail of a test is removed and appended to some other test. We propose a weaker precondition under which the transformation remains fault-coverage-preserving. Along with a weaker precondition, we propose some weaker sufficient conditions for its satisfaction. To demonstrate the usefulness of the relaxations, we employ them for generalizing the checking sequence (CS) construction method of Inan and Ural (1999), to incomplete FSMs and with additional dimensions for CS optimization. The method and its generalized version are exceptional in that they can handle also the case where the upper bound, call it m, assumed for the size of the state set of the FSM under test is not less than twice the size, call it n, of the state set of the specification FSM. We prove that for complete FSMs, the additional optimization dimensions facilitate that in the limit for increasingly large (m/n) and (a/m), with a the number of the defined inputs, the factor of CS length reduction is of the order

black-box conformance testing, model-based testing, finite state machine, quasi-equivalence, test set optimization, fault-coverage-preserving transformation, checking sequence