Specifications are an essential component of software development, and getting specifications right, especially \emph{formal specifications}, can be very challenging. While the use of tools such as model finders and model checkers can be very effective for specification analysis through property checking, researchers have also realized that by the explicit provision of wanted and unwanted specification scenarios, in the style of testing in programs, specification assessment can be significantly enhanced. Thus, various testing and test generation techniques have been recently proposed for assessing formal specifications.

In this paper, we present such a specification testing approach, in the form of a novel combinatorial testing technique for Alloy specifications, called COMBA. COMBA implements an automated partitioning of the state space of Alloy specifications solely based on elements of the specification (thus not requiring user intervention), and defines a family of test criteria, that indicate how such partitions are to be covered. The coverage of the partitions is defined by a family of combinatorial criteria that, given a positive integer $t$, require to cover through test cases all feasible $t$-uples of elements from different partitions. Finally, COMBA introduces an efficient algorithm to generate test cases that satisfy the combinatorial criteria. By leveraging on incremental SAT solving techniques, COMBA achieves significantly better performance in test generation.

We experimentally assess COMBA against existing test generation approaches for Alloy, using a large number of Alloy specifications with known errors. The results show COMBA runs faster, produces smaller test suites, and finds a significantly larger
number of real bugs than related approaches.