The Stable Roommates problem (SR) is characterized by the preferences of agents over other agents as roommates: each agent ranks all others in strict order of preference. A solution to SR is then a partition of the agents into pairs so that each pair shares a room, and there is no pair of agents that would block this matching (i.e., who prefers the other to their roommate in the matching). There are interesting variations of SR that are motivated by applications (e.g., the preference lists may be incomplete (SRI) and involve ties (SRTI)), and that try to find a more fair solution (e.g., Egalitarian SR). Unlike the Stable Marriage problem, every SR instance is not guaranteed to have a solution. For that reason, there are also variations of SR that try to find a good-enough solution (e.g., Almost SR). Most of these variations are NP-hard. We introduce a formal framework, called SRTI-ASP, utilizing the logic programming paradigm Answer Set Programming, that is provable and general enough to solve many of such variations of SR. Our empirical analysis shows that SRTI-ASP is also promising for applications.