The paper presents a theoretical foundation for test sequence generation based on an input specification. The set of possible test sequences is first partitioned according to a generic “triaging” function, which can be created from a state machine specification in various ways. The notion of coverage metric is then expressed in terms of the categories produced by this function. Many existing test generation problems, such as t-way state or transition coverage, become particular cases of this generic framework. We then present algorithms for generating sets of test sequences providing guaranteed full coverage with respect to a metric, by building and processing a special type of graph called a Cayley graph. An implementation of these concepts is then experimentally evaluated against existing techniques, and shows it provides better performance in terms of running time and test suite size.