The testing of a state-based system involves the application of sequences of
inputs and the observation of the resultant input/output sequences (traces).
These traces can result from preset input sequences or adaptive test cases in
which the choice of the next input depends on the trace that has observed
up to that input. Adaptive test cases are used in a number of areas including
protocol conformance testing and adaptivity forms
the basis of the standardised test language TTCN.
Suppose that we apply adaptive test case ° to the system under test (SUT)
and observe the trace ¯¾. If the SUT is deterministic and we apply ° again, after
resetting the SUT, then we will observe ¯¾ again. Further, if we have another
adaptive test case °0 where a prefix ¯¾0 of ¯¾ is a possible response to °0 then we
know that the application of °0 must lead to ¯¾0. Thus, for a deterministic SUT
the response of the SUT to an adaptive test case °0 might be deduced from
the response of the SUT to another adaptive test case. This observation
can be used to reduce the cost of testing: we only apply adaptive test case °0
if we cannot deduce the response to °0 from the set of observations.
While many systems are deterministic, nondeterminism is becoming increasingly
common. Nondeterminism in the SUT is typically a consequence of limits
in the ability to observe the SUT. For example, it could be a result of information
hiding, real time properties, or of different possible interleavings in a
concurrent system (see, for example. This paper investigates the case
where the SUT is nondeterministic. We consider the situation in which a set
O of traces has been observed in testing and we are considering applying an adaptive test case °. In general we cannot expect to be able to deduce the
response of a nondeterministic SUT to an adaptive test case ° since there may
be more than one possible response. Instead we consider the question of how
we can decide whether the application of ° could lead to a trace that has not
been observed. A solution to this would allow us to reduce the cost of testing:
if all possible responses of the SUT to ° have already been observed then we
do not have to apply ° in testing and thus reduce the cost of test execution.
This paper considers three cases. Section 3 considers the case where we can
apply a fairness assumption. Section 4 weakens this assumption to us having
a lower bound p on the probability of observing alternative responses of the
SUT to any input and in any state. Section 5 then considers the general case
