For partial, nondeterministic, finite state machines, a new conformance relation called strong reduction is presented. It complements other existing conformance relations in the sense that the new relation is well-suited for model-based testing of systems whose inputs are enabled or disabled, depending on the actual system state. Examples of such systems are graphical user interfaces and systems with interfaces that can be enabled or disabled in a mechanical way. We present a new test generation algorithm producing complete test suites for strong reduction. The suites are executed according to the grey-box testing paradigm: it is assumed that the state-dependent sets of enabled inputs can be identified during test execution, while the implementation states remain hidden, as in black-box testing. It is shown that this grey-box information is exploited by the generation algorithm in such a way that the resulting best-case test suite size is only linear in the state space size of the reference model. Moreover, examples show that this may lead to significant reductions of test suite size in comparison to true black-box testing for strong reduction.