To compare the performance of different testing strategies, P-measure and E-measure are two effectiveness measures used in previous analytical studies. P-measure, which is defined as the probability of detecting at least one failure, is a measure of how likely it is that failure-causing inputs are selected at least once as test cases. E-measure, which is defined as the expected number of failures detected, is a measure of how frequently failure-causing inputs are selected as test cases. However, we have no a priori knowledge of how many failure-causing inputs there are, or where they may lie. In this paper, we study P-measure and E-measure in terms of how much attention an arbitrary input receives. In the context of P-measure, the attention received by an arbitrary input is the probability that the input is selected at least once as a test case, while in the context of E-measure, the attention is the expected number of times that an input is selected as a test case. The attentions received by an input using the proportional sampling strategy and random testing are then compared. The attention received by an input is found to be the same under the two testing strategies for E-measure, whereas for P-measure the attention is always higher for the proportional sampling strategy than for random testing. This new perspective allows us to provide simpler proofs of some known results. Furthermore, we are able to show that the difference in the expected number of distinct test cases considered by the proportional sampling strategy is larger than that of random testing by at most 2k, where k is the number of partitions and is independent of the number of test cases selected.

We resolve an open problem raised by Ravikumar and Ibarra [SIAM J. Comput., 18 (1989), pp. 1263--1282] on the succinctness of representations relating to the types of ambiguity of finite automata. We show that there exists a family of nondeterministic finite automata {A n } over a two-letter alphabet such that, for any positive integer n, A n is exponentially ambiguous and has n states, whereas the smallest equivalent deterministic finite automaton has 2 n states, and any smallest equivalent polynomially ambiguous finite automaton has 2 n -1 states.

Weyuker and Jeng (1991) have investigated the conditions that affect the performance of partition testing and have compared analytically the fault-detecting ability of partition testing and random testing. Chen and Yu (1994) have generalized some of Weyuker and Jeng's results. We extend the analysis to subdomain testing in which subdomains may overlap. We derive several results for a special case and demonstrate a technique to extend some of our results to more general cases. We believe that this technique should be very useful in further investigating the behaviour of subdomain testing.