Proceedings, Fifteenth International Conference on Principles of Knowledge Representation and Reasoning (KR 2016) A MIS Partition Based Framework for Measuring Inconsistency Said Jabbour1, Yue Ma2,3, Badran Raddaoui4, Lakhdar Sa¨ıs1 and Yakoub Salhi1 1CRIL CNRS UMR 8188, University of Artois, France 2LRI, Univ. Paris-Sud, CNRS, Universite´ Paris-Saclay, France 3 Key Laboratory of Symbolic Computation and Knowledge Engineering, Jilin University, China 4LIAS - ENSMA, University of Poitiers, France {jabbour,sais,salhi}@cril.fr, [email protected], [email protected] Abstract this case, it is not desired to immediately conclude that both agents have a same conflict degree with A. Instead, some In this paper, we propose a general framework, both pa- proper inconsistency measures are necessary. rameterized and parameter-free, for defining a family of fine-grained inconsistency measures for propositional Among many possible ways to define an inconsistency knowledge bases. The parameterized approach allows measure (Hunter 2006; Grant and Hunter 2008; Hunter and to encompass several existing inconsistency measures Konieczny 2010; Ma, Qi, and Hitzler 2011; Jabbour et al. as specific cases, by properly setting its parameter. And 2014), minimal inconsistent subsets (MISes) are often used the parameter-free approach is defined to avoid the dif- because a MIS forms a direct representation of an incon- ficulty in choosing a suitable parameter in practice but sistency core in a KB. For example, a classical measure still keeps a desired ranking for knowledge bases by their IMI (Hunter and Konieczny 2010) is defined as the num- inconsistency degrees. The fine granularity of our frame- ber of MISes of a base K, i.e., IMI(K)=|MISes(K)|, work is based on the notion of MIS partition that con- I (A ∪ B )=I (A ∪ B )=2 siders the inner structure of all the minimal inconsistent by which we have MI 1 MI 2 subsets of a knowledge base. Moreover, MinCostSAT- for the previously mentioned example. Although useful based encodings are provided, which enable the use of for many scenarios (Hunter and Konieczny 2010; 2008; efficient SAT solvers for the computation of the pro- Mu, Liu, and Jin 2011), IMI measure fails to recommend posed measures. We implement these algorithms and B1 or B2 for A. We claim that B1 has more conflicts with test them on some real-world datasets. The preliminary A than B2 because there are two independent contradictory experimental results for a variety of inputs show that the topics between A and B1 (i.e. {t1, ¬t1}, {t2, ¬t2}), each of proposed framework gives a wide range of possibilities which should be modified to make an agreement between A for evaluating large knowledge bases. and B1. However, the disagreement between A and B2 (i.e. {t1, ¬t1}, {t1,t2, ¬t1 ∨¬t2}) can be handled by revising Introduction only one topic, for instance, if A deletes t1. The problem of no distinction between B1 and B2 for A is due to the fact that Reasoning about inconsistent knowledge bases (KBs) has IMI treats all MISes equally in terms of their contributions to been a long-standing challenge in the AI community. In re- the inconsistency degree. Recently, another measure, called cent years, measuring inconsistency has proved useful in di- ICC, has been developed as a lower bound of all standard in- verse scenarios, including software specifications (Barragans- consistency measures (Jabbour, Ma, and Raddaoui 2014), by Martinez, Arias, and Vilas 2004), belief merging (Qi, Liu, considering the most representative MISes. The ICC metric and Bell 2005), news reports (Hunter 2006), integrity con- can distinguish B1 and B2 for A because ICC(A ∪ B1)=2 straints (Grant and Hunter 2006; 2013), and multi-agents and ICC(A ∪ B2)=1. However, if another agent is avail- systems (Hunter, Parsons, and Wooldridge 2014; Jabbour, able, say B3 = {¬t1,t1 → t2},wegetICC(A ∪ B3)=1. Ma, and Raddaoui 2014). That is, the agent A can not distinguish between B2 and Inconsistencies are often unavoidable in real-world appli- B3, which is again insufficient in the following aspect: A cations. To achieve a certain goal, an agent may need to co- and B2 have more groups of topics in conflict (i.e. {t1, ¬t1}, operate with another agent, even in the presence of conflicts {t1,t2, ¬t1 ∨¬t2}) than A and B3 (i.e. {t1, ¬t1}). From the between them. In this case, the agent would prefer one that above discussion about IMI and ICC, a fine-grained analysis has the least disagreement with herself. For instance, suppose of the inner structures of MISes is clearly necessary. that an agent A with the support on two topics t1 and t2, writ- Studies on inner structures of MISes have been performed ten A = {t1,t2}, has to collaborate with one of the following in many different settings. For instance, a dependence relation two agents: B1 = {¬t1, ¬t2}, and B2 = {¬t1, ¬t1 ∨¬t2}. among MISes has been identified by the fact that resolving Clearly, A is in conflict with both, since B1 and B2 agree some MISes allows automatic resolution of others (Benferhat, upon the topic ¬t1 that contradicts with the topic t1 of A.In Dubois, and Prade 1995). In the context of ontology debug- Copyright c 2016, Association for the Advancement of Artificial ging, root or derived axioms for unsatisfiability have been Intelligence (www.aaai.org). All rights reserved. distinguished (Kalyanpur et al. 2005). A graphical represen- 84 tation of the relationships among justifications and axioms • Independence: I(K∪{α})=I(K) if α ∈ free(K∪{α}). is provided (Bail et al. 2011). In this paper, we propose the • MinInc: I(M)=1if M ∈ MISes(K). notion of MIS partition that results in a general framework • I(K ∪ ...∪K )=Σn I(K ) of inconsistency measures. Ind-decomposability: 1 n i=1 i if MISes(K ∪...∪K )=MISes(K ) ...MISes(K ) Our contribution can be summarized as follows: 1 n 1 n , where is the multi-set union over sets, and unfree(Ki) ∩ • Based on the MIS partition, we define a new family of unfree(Kj)=∅ for 1 ≤ i = j ≤ n. weighted inconsistency measures and show that these mea- The first four properties seem natural for an inconsistency sures satisfy some rational properties, and capture several measure. The idea behind the Ind-decomposability (Jabbour, existing inconsistency measures. Ma, and Raddaoui 2014) is that the inconsistency degrees • To overcome the difficulty in choosing a suitable parameter of several KBs should be additive if these bases are dis- in practice, we further provide a parameter-free inconsis- joint and have disjoint MISes. Notice that this revised prop- tency measure that can preserve the desired ranking (see erty is introduced to overcome the limitations of the De- below for details) generated from the MIS partition. composability property (Hunter and Konieczny 2010). An- I(K∪{α}) ≥ • We present Minimum-Cost Satisfiability (MinCostSAT) other property named Dominance says that I(K∪{β}) α β based encodings so that we can benefit from cutting-edge if , which is however criticized due to SAT solvers for computing the proposed fine-grained in- its unsuitability for characterizing inconsistency measures consistency measures. based on minimal inconsistent subsets (Mu et al. 2011; Besnard 2014). Therefore, in the rest of this paper, we only • An experimental study is conducted to show the relevance consider the five postulates given above which are widely of the proposed measures for finely quantifying the conflict accepted by the AI community. status of real-world KBs. Definition 2 An inconsistency measure is called a standard Preliminaries measure if it satisfies the Consistency, Monotonicity, Indepen- dence, MinInc, and Ind-decomposability properties. L A propositional language is built over a finite set of I propositional symbols PS using classical logical connec- It is easy to check that MI is a standard measure. {¬, ∧, ∨, →, ↔} ⊥ tives . The symbol denotes contradiction. I a, b, c, . represent atoms in PS. A literal is an atom a MIS-based Measure CC Revisited or its negation ¬a. A clause C is a disjunction of literals: In (Jabbour, Ma, and Raddaoui 2014), the authors proposed C = a1 ∨ ...∨ an. A formula α in conjunctive normal form a new inconsistency measure, denoted ICC, based on a subtle (CNF) is a conjunction of clauses. Let Var(α) denotes the analysis of the dependencies among the formulas of the KB. set of variables in α. An interpretation is a total function For example, in the ICC measure, overlap among MISes are from PS to {true, false}. An interpretation B is a model taken into account. Indeed, to characterize the inner struc- of α iff B(α)=true.AKBK is a finite set of propositional ture of MISes, each KB K is associated with an hypergraph formulas. For a set S, we denote by |S| its cardinality, and by GK whose vertices are associated with the formulas of K 2S its power set. K is inconsistent if K⊥, where is the and edges with the MISes of K. This hypergraph based rep- classical consequence relation. Minimal inconsistent subsets, resentation gives a better insight of the correlations among defined below, are often used to analyze inconsistency in a MISes so that it becomes useful for analyzing inconsistencies. KB. For example, the notion of strong-partition of a KB can be considered in the light of the connected components of GK, Definition 1 Let K be a KB and M ⊆K. M is a Minimal based on which the ICC measure can be defined. Inconsistent Subset (MIS)ofK iff M ⊥and ∀M M, M ⊥. The set of all minimal inconsistent subsets of K is Definition 3 (Jabbour, Ma, and Raddaoui 2014) Let K be denoted MISes(K).