On Computing Minimal Equivalent Subformulas

Abstract

Abstract. A propositional formula in Conjunctive Normal Form (CNF) may contain redundant clauses -clauses whose removal from the formula does not affect the set of its models. Identification of redundant clauses is important because redundancy often leads to unnecessary computation, wasted storage, and may obscure the structure of the problem. A formula obtained by the removal of all redundant clauses from a given CNF formula F is called a Minimal Equivalent Subformula (MES) of F. This paper proposes a number of efficient algorithms and optimization techniques for the computation of MESes. Previous work on MES computation proposes a simple algorithm based on iterative application of the definition of a redundant clause, similar to the well-known deletionbased approach for the computation of Minimal Unsatisfiable Subformulas (MUSes). This paper observes that, in fact, most of the existing algorithms for the computation of MUSes can be adapted to the computation of MESes. However, some of the optimization techniques that are crucial for the performance of the state-of-the-art MUS extractors cannot be applied in the context of MES computation, and thus the resulting algorithms are often not efficient in practice. To address the problem of efficient computation of MESes, the paper develops a new class of algorithms that are based on the iterative analysis of subsets of clauses. The experimental results, obtained on representative problem instances, confirm the effectiveness of the proposed algorithms. The experimental results also reveal that many CNF instances obtained from the practical applications of SAT exhibit a large degree of redundancy.

Quantified maximum satisfiability

“…More precisely, we are going to present the SMUS and SMES problems. In contrast to finding a smallest possible subset of clauses that are unsatisfiable or equivalent to the original CNF formula, the MUS and MES problems are well studied and the algorithms for solving them are deployed in practice [11,13,30,49,63,65,67].…”

“…The problem of computing MESes of a CNF formula [11] is tightly connected to the notions of a redundant clause and irredundant formula. A clause c ∈ ϕ is said to be redundant if ϕ \ {c} |= c. In other words, adding/removing c to/from ϕ does not change the set of models of ϕ.…”

“…11 It was originally proposed and applied to the Digger algorithm. To increase its performance, Liffiton et al [45] use a preprocessing technique -computing a set of disjoint MCSes.…”

“…The SMES problem, while being a generalization of SMUS, is tightly related to detection of CNF subformula redundancy. Therefore, SMES is of great importance in the context of CNF formula minimization [11,12,43,44]. Besides the SMUS/SMES problems studied in the paper, a number of other important practical applications (e.g.…”

“…The new algorithms for QMaxSAT are evaluated on the problems of computing the smallest minimally unsatisfiable subformula (SMUS) [29,45,49,53] and the smallest minimally equivalent subformula (SMES) [11,12,43,44]. The SMUS decision problem is well-known to be in the second level of the polynomial hierarchy (e.g.…”