A Universe of Sorts

Statement coverage is different from branch coverage, since an if (cond) { s1; } s2 will say that s1 and s2 were executed when cond=True, so we have full statement coverage. On the other hand, this does not guarantee full branch coverage, since we have not exectuted the branch where cond=False. We can't tell that we haven't covered this branch since there is no statement to record that we have taken the else branch!

Branch distances: for conditions a == b, a != b, a < b, a <= b, define the "distance true/distance false" to be the number that is to be added/subtracted to a to make the condition true/false (for a fixed b). So, for example, the "distance true" for a == b is abs(b - a), while "distance false" is 1 - int(a == b).

What are we missing in coverage? The problem here is that coverage is unable to evaluate the quality of our assertions. Indeed, coverage does not care about assertions at all. However, as we saw above, assertions are an extremely important part of test suite effectiveness. Hence, what we need is a way to evaluate the quality of assertions.

Competent Programmer Hypothesis / Finite Nbhd Hypothesis : Mutation Analysis provides an alternative to a curated set of faults. The key insight is that, if one assumes that the programmer understands the program in question, the majority of errors made are very likely small transcription errors (a small number of tokens). A compiler will likely catch most of these errors. Hence, the majority of residual faults in a program is likely to be due to small (single token) variations at certain points in the structure of the program from the correct program (This particular assumption is called the Competent Programmer Hypothesis or the Finite Neighborhood Hypothesis).

Equivalent mutants : However, if the number of mutants are sufficiently large (say > 1000), one may choose a smaller number of mutants from the alive mutants randomly and manually evaluate them to see whether they represent faults. We choose the sample size by sampling theory of binomial distributions .
Chao's estimator : way to estimate the number of true mutants (and hence the number of equivalent mutants) is by means of Chao's estimator:

hat M \equiv \begin{cases} M(n) + k_1^2 / (2 k_2) & \text{if } k_2 > 0 \\ M(n) + k_1(k_1 - 1)/2 & \text{otherwise} \\ \end{cases}

$k_1$ is the number of mutants that were killed exactly once, $k_2$ is the number of mutants that were killed exactly twice. $\hat M$ estimates the the true numbe of mutants.
If $T$ is the total mutants generated, then $T - M(n)$ represents immortal mutants.
$\hat M$ is the is the mutants that the testset can detect given an infinite amount of time.

§ Fuzzing Book