MUST: Provide a Finer-Grained Explanation of Unsatisfiability

Abstract

In this paper, a new form of explanation and recovery technique for the unsatisfiability of discrete CSPs is introduced. Whereas most approaches amount to providing users with a minimal number of constraints that should be dropped in order to recover satisfiability, a finer-grained alternative technique is introduced. It allows the user to reason both at the constraints and tuples levels by exhibiting both problematic constraints and tuples of values that would allow satisfiability to be recovered if they were not forbidden. To this end, the Minimal Set of Unsatisfiable Tuples (MUST) concept is introduced. Its formal relationships with Minimal Unsatisfiable Cores (MUCs) are investigated. Interestingly, a concept of shared forbidden tuples is derived. Allowing any such tuple makes the corresponding MUC become satisfiable. From a practical point of view, a two-step approach to the explanation and recovery of unsatisfiable CSPs is proposed. First, a recent approach proposed by Hemery et al.'s is used to locate a MUC. Second, a specific SAT encoding of a MUC allows MUSTs to be computed by taking advantage of the best current technique to locate Minimally Unsatisfiable Sub-formulas (MUSes) of Boolean formulas. Interestingly enough, shared tuples coincide with protected clauses, which are one of the keys to the efficiency of this SAT-related technique. Finally, the feasibility of the approach is illustrated through extensive experimental results.