Approximate Counting of Minimal Unsatisfiable Subsets

Abstract

Given an unsatisfiable formula F in CNF, i.e. a set of clauses, the problem of Minimal Unsatisfiable Subset (MUS) seeks to identify a minimal subset of clauses such that N is unsatisfiable. The emerging viewpoint of MUSes as the root causes of unsatisfiability has led MUSes to find applications in a wide variety of diagnostic approaches. Recent advances in identification and enumeration of MUSes have motivated researchers to discover applications that can benefit from rich information about the set of MUSes. One such extension is that of counting the number of MUSes. The current best approach for MUS counting is to employ a MUS enumeration algorithm, which often does not scale for the cases with a reasonably large number of MUSes. Motivated by the success of hashing-based techniques in the context of model counting, we design the first approximate MUS counting procedure with guarantees, called . Our approach avoids exhaustive MUS enumeration by combining the classical technique of universal hashing with advances in QBF solvers along with a novel usage of union and intersection of MUSes to achieve runtime efficiency. Our prototype implementation of is shown to scale to instances that were clearly beyond the realm of enumeration-based approaches.

Counting Minimal Unsatisfiable Subsets

“…The study of MUS counting without relying on exhaustive enumeration was initiated just recently by Bendík and Meel [13], who proposed an (ε, δ)approximation scheme called AMUSIC. AMUSIC extends a prior hashing-based model counting framework [15,18,63] to MUS counting.…”

“…It is worth remarking on a recent work by Bendík and Meel [14] that focuses on exact counting of maximal satisfiable subsets (MSSes). While MUSes and MSSes are closely related concepts, to the best of our knowledge, there does not exist any efficient reduction from MUS counting to MSS counting, or vice versa.…”

“…Subsequently, Durand, Hermann, and Koliatis [21] showed that the general problem of proj-#SAT is #NP-hard. A significant conceptual contribution of Durand et al was to show the importance of subtractive reductions for problems in #NP; this idea has been applied for reductions to projecting counting [14].…”

“…|F | and hence the complete enumeration is often practically intractable [9,10,39,69]. Inspired by the development of model counting techniques in the context of SAT, which in its nascent stages also depended on complete model enumeration while contemporary techniques often need to explicitly identify just a fraction of models, Bendík and Meel [13] recently initiated an investigation of counting MUSes without their explicit enumeration. In this context, they succeeded by developing a hashing-based approximate counter, AMUSIC [13], that provides the so-called PAC guarantees, also known as (ε, δ)-guarantees, wherein the computed answer is within the (1+ε)-factor of the exact count with confidence at least 1 − δ. AMUSIC reduces the problem of MUS counting to logarithmically many calls to a Σ P 3 oracle (3-QBF solver, in practice) wherein every Σ P 3 query is constructed over a CNF formula conjuncted with XORs.…”

“…Finally, our empirical analysis showcases that our wrappers W approximate the set of MUSes very tightly. Motivated by the tightness of our wrappers, we discuss several interesting applications of our framework: approximate MUS counting [13], MUS enumeration [5,40], MUS Sampling, estimation of minimum and maximum MUS cardinality [27,38], and MUS membership testing [31].…”