Recursive Online Enumeration of All Minimal Unsatisfiable Subsets

Abstract

In various areas of computer science, we deal with a set of constraints to be satisfied. If the constraints cannot be satisfied simultaneously, it is desirable to identify the core problems among them. Such cores are called minimal unsatisfiable subsets (MUSes). The more MUSes are identified, the more information about the conflicts among the constraints is obtained. However, a full enumeration of all MUSes is in general intractable due to the large number (even exponential) of possible conflicts. Moreover, to identify MUSes, algorithms have to test sets of constraints for their simultaneous satisfiability. The type of the test depends on the application domains. The complexity of the tests can be extremely high especially for domains like temporal logics, model checking, or SMT. In this paper, we propose a recursive algorithm that identifies MUSes in an online manner (i.e., one by one) and can be terminated at any time. The key feature of our algorithm is that it minimises the number of satisfiability tests and thus speeds up the computation. The algorithm is applicable to an arbitrary constraint domain and its effectiveness demonstrates itself especially in domains with expensive satisfiability checks. We benchmark our algorithm against the state-of-the-art algorithm Marco on the Boolean and SMT constraint domains and demonstrate that our algorithm really requires less satisfiability tests and consequently finds more MUSes in the given time limits.

MUST: Minimal Unsatisfiable Subsets Enumeration Tool

“…Since the list of constraint domains where MUSes find an application is quite long and new applications still arise, there have been proposed several domain agnostic MUS enumeration algorithms (e.g. [3,22,9,7,10]). Such algorithms can be used in an arbitrary constraint domain, and thus theoretically serve as ready-to-use solutions for any constraint domain where MUSes might eventually find an application.…”

Evaluation of Domain Agnostic Approaches for Enumeration of Minimal Unsatisfiable Subsets

“…ReMUS Recursive algorithm ReMUS [12] also employs black-box shrinking procedures and uses a symbolic representation of unexplored subsets. Similarly as TOME, ReMUS exploits the observation that the larger the set being shrunk is the harder is to shrink it.…”

“…Benchmarks As experimental data in the SAT domain, we use a collection of 291 Boolean formulae in conjunctive normal form that comes from the MUS track of the SAT 2011 competition 1 . These benchmarks are used in several recent papers that focus on MUS enumera- tion [29,27,11,12] as well. The benchmarks range in their size from 70 to 16 million constraints and use from 26 to 4.4 million variables.…”

“…ReMUS [12] tends to find relatively small seeds for shrinking by recursively reducing the search space in which the seeds are searched for. The reduction is based on previously found MUSes and in order to perform deep recursive calls, the input instance has to contain many similar MUSes [12]. Also, the bigger is the number of constraint in the input instance, the more significant reduction is possible.…”

“…The domain agnostic algorithms were extensively studied in the recent years and several algorithms were proposed [4,27,29,34,12,11,13,10]. However, the relevant papers usually present experimental evaluation of algorithms only in one constraint domain, typically in the domain of Boolean (SAT) constraints.…”