Enumerating Minimally Revised Specifications Using Dualization

Abstract

We consider the problem of enumerating minimally revised specifications in software engineering in the situation where a new specification is added to the current specification and causes a conflict. We assume that a specification is expressed as a set of ground Horn clauses which is divided into two sets Tpst and Ttmp that are the unchangeable and changeable parts of the specification, respectively. Since a minimal revision is obtained by removing a minimal set of clauses from Ttmp so that the remaining set is consistent, our task can be restated as enumerating maximal consistent subsets of a given set of Horn clauses. Moreover, consistency property is monotone, that is, if a set of Horn clauses is consistent then every subset of the set is also consistent. Then, we can apply our previous method of enumerating maximal frequent sets in data mining which can be used for any enumeration for maximal subsets w.r.t. a monotone property. We show that our algorithm performs a dualization only once for an enumeration of maximal subsets, and the number of consistency checks is at most \(|{MinI_{T_{pst}}({T_{tmp}})}|+|{MaxC_{T_{pst}}({T_{tmp}})}|\cdot|T_{tmp}|\) and the necessary space is \(\mbox{O}(\Sigma_{S\in {MaxC_{T_{pst}}({T_{tmp}})}}|S|)\) where \(|{MinI_{T_{pst}}({T_{tmp}})}|\) is the number of minimal subsets of Ttmp that are inconsistent with Tpst, and \(|{MaxC_{T_{pst}}({T_{tmp}})}|\) is the number of maximal subsets of Ttmp that are consistent with Tpst.