A Characteristic Function Approach to Inconsistency Measures for Knowledge Bases

Abstract

Abstract. Knowledge is an important component in many intelligent systems. Since items of knowledge in a knowledge base can be conflicting, especially if there are multiple sources contributing to the knowledge in this base, significant research efforts have been made on developing inconsistency measures for knowledge bases and on developing merging approaches. Most of these efforts start with flat knowledge bases. However, in many real-world applications, items of knowledge are not perceived with equal importance, rather, weights (which can be used to indicate the importance or priority) are associated with items of knowledge. Therefore, measuring the inconsistency of a knowledge base with weighted formulae as well as their merging is an important but difficult task. In this paper, we derive a numerical characteristic function from each knowledge base with weighted formulae, based on the Dempster-Shafer theory of evidence. Using these functions, we are able to measure the inconsistency of the knowledge base in a convenient and rational way, and are able to merge multiple knowledge bases with weighted formulae, even if knowledge in these bases may be inconsistent. Furthermore, by examining whether multiple knowledge bases are dependent or independent, they can be combined in different ways using their characteristic functions, which cannot be handled (or at least have never been considered) in classic knowledge based merging approaches in the literature.

On the complexity of inconsistency measurement

“…Inconsistency measures, firstly mentioned in (Grant, 1978), can be used to analyse inconsistencies and to provide insights on how to repair them. An inconsistency measure I is a function on knowledge bases, such that the larger the value I(K) the more severe the inconsistency in K. A lot of different approaches of inconsistency measures have been proposed, mostly for classical propositional logic (Hunter and Konieczny, 2004Ma et al, 2010;Mu et al, 2011;Xiao and Ma, 2012;Grant and Hunter, 2011;Ma et al, 2012;Grant and Hunter, 2013;McAreavey et al, 2014;Jabbour et al, 2015Jabbour et al, , 2014bBesnard, 2016;Thimm, 2016b;Ammoura et al, 2015Ammoura et al, , 2017, but also for classical first-order logic (Grant and Hunter, 2008), description logics (Ma et al, 2007;Zhou et al, 2009), default logics (Doder et al, 2010), answer set programming (Ulbricht et al, 2016) probabilistic and other weighted logics (Thimm, 2013;Potyka, 2014;De Bona and Finger, 2015), and relational databases (Decker, 2011), see also (Thimm, 2017b(Thimm, , 2018 for some recent surveys.…”

“…it-has been conducted so far. The only complexity analyses on inconsistency measures we are aware of were presented in (Ma et al, 2010) and (Xiao and Ma, 2012) and each focused on a particular inconsistency measure. In (Ma et al, 2010) the complexity of a variant of the contension inconsistency measure I c (Grant and Hunter, 2011) and in (Xiao and Ma, 2012) the complexity of the measure I mv from (Xiao and Ma, 2012) themselves is investigated (we will recall the formal definitions of these measures in Section 3 and the corresponding results in Section 4, respectively).…”

“…The only complexity analyses on inconsistency measures we are aware of were presented in (Ma et al, 2010) and (Xiao and Ma, 2012) and each focused on a particular inconsistency measure. In (Ma et al, 2010) the complexity of a variant of the contension inconsistency measure I c (Grant and Hunter, 2011) and in (Xiao and Ma, 2012) the complexity of the measure I mv from (Xiao and Ma, 2012) themselves is investigated (we will recall the formal definitions of these measures in Section 3 and the corresponding results in Section 4, respectively). Recently, the algorithmic challenges in computing inconsistency measures have gained some attention (Ma et al, 2010;McAreavey et al, 2014;Jabbour et al, 2014aThimm, 2016b) and therefore calls for a theoretical investigation on the complexity of the involved computational problems.…”

Information Fusion of Conflicting Input Data

“…A measure based on vector distances between the data to be fused is introduced in [57]. Other works include [58,59,60]. …”

Plan Selection for Probabilistic BDI Agents

“…In the meantime, intention revision, which is closely related to plan selection w.r.t the revision/selection methods, is also investigated by recent works in BDI systems, e.g., [22], [21], [7], [5], [18]. While these works studies on plan selection or intention revision, they seldom consider uncertainty that can emerge in agent's beliefs (see [13], [9], [12]). However, in multiagent systems, especially in BDI systems, uncertainty has been discussed in many research articles (e.g., [14], [17], [8], [2]) in relation to both incomplete information about an environment and an agent's uncertain beliefs about some observations.…”