Intrinsic approaches to prioritizing diagnoses in multi-context systems
Introduction
There is an increasing demand for interlinking knowledge contexts from heterogeneous sources in the community of knowledge representation and reasoning, especially with the advent of the world wide web. However, the diversity of representations and reasoning mechanisms in such contexts brings some important challenges to accessing each individual context as well as to interlinking these contexts [3].
Nonmonotonic multi-context systems (MCS) introduced by Brewka and Eiter [2], [3] leave different logics of contexts untouched and interlink contexts by modeling the inter-contextual information exchange in terms of so-called bridge rules uniformly. To be more precise, a bridge rule of a context describes a potential knowledge augment operation that adds information represented by the head of the bridge rule to the context by exchanging information with other contexts specified by the body of the bridge rule. Actually, unifying information exchange between contexts rather than logics of contexts makes MCS a promising starting point to interlink heterogeneous knowledge sources [3].
The semantics of multi-context systems is defined in terms of equilibria [2]. Under this semantics, we say that a multi-context system is inconsistent if there is no equilibrium for that system [2]. The inconsistency of a multi-context system makes the system useless in real world applications. However, inconsistency arises easily in such a system consisting of interlinked heterogeneous contexts [5]. This makes techniques for analyzing inconsistency more necessary in multi-context systems.
The notion of diagnosis for inconsistency has been proposed for analyzing the inconsistency in multi-context systems [2]. Informally speaking, given an inconsistent multi-context system, a diagnosis is a set of bridge rules that need to be changed (either deactivated or activated unconditionally) in order to restore the consistency of that multi-context system. It describes a possible proposal for resolving the inconsistency. Generally, diagnoses need to be prioritized to filter out undesired ones or to select the most desired ones from some specific perspectives [4], [5], [6], [7], [8], [19].
How can we prioritize diagnoses for an inconsistent multi-context system? Two strategies have been proposed so far. The first strategy is that we need to introduce some external information to prioritize diagnoses. For example, A. Weinzierl introduced categories of bridge rules as additional knowledge to assess and prioritize diagnoses [19]. Mu et al. considered the preference relation on contexts as useful information for filtering out undesired diagnoses [17]. Such external knowledge always depends on application scenarios. However, the nature of heterogeneity of the multi-context system makes finding appropriate external knowledge automatically more difficult in many cases. In the absence of external knowledge, it is not easy to identify the relative importance of a bridge rule in many cases. To illustrate this, consider an MCS with four contexts , , , and . In detail, describes Alice's hobbies, and is a set of new movies playing in theaters today; describes fruits and vegetables for lunch, while describes Alice's schedule for this afternoon. has only one bridge rule that allows to visit (i.e., access) in order to add durian to when says that Alice likes fruits unique to Southeast Asia. has two bridge rules and . Here allows to visit and to get a ticket of Alice's favorite movie in some theater on the afternoon. On the contrary, will add a reminder of no admittance to theaters to if durian is included in lunch by . Neither nor has bridge rules. Then the four contexts are interlinked as a whole system by the three bridge rules. When all the three bridge rules are activated, the inconsistency arises from the MCS. Note that we need to deactivate only one bridge rule to restore the consistency. If we knew that Alice prefers new movies to durian, we should deactivate rather than . However, just given , we have no idea about Alice's preference over her hobbies. Then in such cases, it is difficult to rank diagnoses by evaluating the relative importance of bridge rules.
The second strategy is that we may prioritize diagnoses based on some quantitative measurements for the inconsistency in a multi-context system [4], [19]. Roughly speaking, such inconsistency measurements aim to assess the contribution of each bridge rule to the inconsistency of the system. However, most are adapted from inconsistency measurements for knowledge bases (a set of logical formulas) in a straightforward manner, and then only focus on assessing the impact of deactivating a bridge rule on the occurrence of inconsistency. This implies that the impact of deactivating a bridge rule on the interconnection among contexts in the whole system has not yet been considered explicitly in this strategy. To illustrate this, consider the inconsistent MCS again. Note that deactivating any of the three bridge rules can make inconsistency disappear. Then there is no difference between these bridge rules from the perspective of inconsistency measuring. However, if Alice prefers new movies to durian, is more important than as mentioned earlier. Even we have no idea on the importance of bridge rules, we may find that deactivating makes being isolated from the three others, whilst deactivating any of and does not result in any isolated context. Such differences are not captured by inconsistency measurements.
Recall that a primary characteristic of the multi-context system is its capability of describing information exchange between heterogeneous contexts. The intention of introducing bridge rules is to interconnect contexts so as to make information flow along channels represented by bridge rules in the whole system. Correspondingly, the role of each context is characterized by its possible participation in information exchange in the system. For example, in the MCS mentioned above is the context with the maximal number of being visited by other contexts. This characteristics of reflects the influence of Alice's hobbies on knowledge updating of other contexts in the system. Then either deactivating or activating unconditionally some bridge rules of a context may bring changes on the information exchange between the context and others in the system, and then affect roles of the related contexts in the information exchange in the whole system. Moreover, changing different bridge rules may have different impacts on the information exchange between contexts. For example, each bridge rule in the MCS is a unique way to interlink the contexts and , then the channel for information exchange between and will disappear if we deactivate the bridge rule . However, all the four contexts are still interlinked by the rest bridge rules when we deactivate either or , that is, any context is still linked to at least one other context in the system. In contrast, when we deactivate , the unique bridge rule for linking to others, the system will split into two parts, namely and the others. Information about new movies will have nothing to do with Alice's hobbies, lunch and schedule in this instance. On the other hand, if there are at least two bridge rules involved in information exchange between two contexts, then there is still information exchange flow between two contexts if we deactivate only one bridge rule. But the strength of the connection of the two contexts due to information exchange may decrease in this case. Conversely, such impacts of deactivating a bridge rule on information exchange between contexts can be used to characterize the role of that bridge rule. From this point of view, considering the intrinsic role of bridge rules serving as channels for information exchange between contexts provides a practical perspective to evaluate diagnoses, especially in cases where no external knowledge is available.
In this paper, we leave the knowledge content involved in each bridge rule untouched and focus on the interconnection relation between contexts induced by potential information exchange by virtue of bridge rules. Roughly speaking, we construct a heterogeneous graph, termed information exchange network, to capture the communication over contexts via bridge rules for a multi-context system, in which each node is labeled by either a context or a bridge rule such that each edge is either a link from a bridge rule to a context that the bridge rule belongs to or that from a context to a bridge rule that involves the context in its body. Then we propose some notions to characterize the role of each context from a local perspective based on edges between bridge rules and the context. In detail, we use the term of authority of a context to describe how many bridge rules access that context in the information exchange network, and use the term of activity of a context to describe how active that context is in updating its knowledge base by virtue of its bridge rules in the information exchange network. We also introduce the betweenness centrality measure for a context with regard to the information exchange network, which reflects the extent to which the context is an intermediate in the information exchange over contexts in the whole information exchange network. It provides a characterization of the role of each context in information exchange from a global perspective. Besides these context-level intrinsic measures, we introduce a rule-level betweenness centrality measure for a set of bridge rules with regard to the information exchange network, which reflects the extent to which these bridge rules together are an intermediate in the information exchange over contexts via bridge rules. Based on these, we propose three kinds of approaches to prioritizing diagnoses for a multi-context system. The first two kinds of approaches prioritize diagnoses based on characterizations of roles of contexts. In detail, a series of approaches of the first kind focus on evaluating the impact of each diagnosis on each context's channels of information exchange by using local context-level measurements. They rank diagnoses of a multi-context system by comparing either authorities or activities or both of contexts in the original system and in its potential revisions under diagnoses. Approaches of the second type focus on the impact of a diagnosis on the global information flow over contexts. They rank diagnoses by comparing betweenness centralities of contexts w.r.t. the information exchange network of the original system and w.r.t. that of its potential revisions under diagnoses. In contrast, the last one uses the betweenness centralities of bridge rules in the information exchange network to prioritize diagnoses directly.
The main contributions of this paper are briefly summarized as follows:
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We propose the notion of information exchange network for a multi-context system, which allows us to represent communications of contexts by virtue of bridge rules in the whole system in a concise and intuitive way.
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To characterize the role of each context in information exchange over contexts from different perspectives, three notions are proposed based on the information exchange network, namely authority, activity, and betweenness centrality. The authority of a context describes how frequently that context is visited by other contexts, while the activity of a context describes how active that context is in knowledge updating based on information exchange. The last one describes the extent to which a context is an intermediate in the information exchange over contexts.
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We propose three kinds of approaches to prioritizing diagnoses for a multi-context system. The first two kinds of approaches focus on ranking diagnoses by comparing the original system and the revised systems under diagnosis at context level, while the last one uses the betweenness centrality of diagnoses to rank diagnoses directly. The main difference between the first two kinds of approaches lies in characterizations of contexts used in these approaches. In brief, approaches of the first kind use one or both of the authority and the activity to characterize contexts, while ones of the second kind use the betweenness centrality of contexts. We also introduce a regularization method to combine difference approaches (or perspectives) for prioritizing diagnoses.
The rest of this paper is organized as follows: we introduce some necessary notions about inconsistency analysis in multi-context systems in Section 2. In section 3, we propose the information exchange network for a multi-context system, as well as essential concepts that describe the roles of bridge rules and contexts in the information exchange network, respectively. In section 4, we propose three types of approaches to prioritizing diagnoses for an inconsistent multi-context system. In Section 5, we introduce a regularization method to combine some specific perspectives for prioritizing diagnoses. In section 6, we compare our approaches with some closely related works. Some potential extensions of our approaches are also discussed in this section. Finally, we conclude this paper in Section 7.
Section snippets
Preliminaries
In this section, we give a brief introduction to multi-context systems and inconsistency analysis in multi-context systems. The material here is largely taken from [2], [4] and [17].
The multi-context system presented by Brewka and Eiter aims to combine arbitrary monotonic and nonmonotonic logics [2]. Here an abstract logic (or just logic) L is defined as a triple , where is the set of well-formed knowledge bases of L, which characterizes the syntax of L; is the set of
Information exchange network
In this section, we use a heterogeneous graph, termed information exchange network, to describe the information exchange over contexts via bridge rules in a multi-context system. Then we introduce the notion of authority to describe how frequently a context is accessed by other contexts in the information exchange network of a multi-context system, as well as the notion of activity to describe how active a context is in its knowledge updating based on information exchange over contexts. We
Prioritizing diagnoses
In this section, we consider approaches to prioritizing diagnoses based on information exchange network. We use to denote that D is preferred to under some given articulation of ⪯. In particular, we use as an abbreviation of and in this paper. Also, we use to denote that but . We say a diagnosis D is the most preferred one w.r.t. ⪯ if there is no diagnosis such that . We use to denote the most preferred diagnoses w.r.t. ⪯.
In the first two
Regularization
As summarized earlier, context-level approaches to evaluating a diagnosis D fully depend on comparisons between and from their own respective perspectives. However, we may use the following regularization method to combine any two perspectives to compare and : where is a regularization parameter used to balance the first perspective against the second one . For example, we may use different values of λ to combine the perspective of
Comparison and discussion
Our approaches to prioritizing diagnoses for an MCS focus on scenarios where external knowledge on beliefs of bridge rules is not available. Then the structure of information exchange over contexts via bridge rules instead of the content of information exchange over contexts plays a central role in these approaches. The structure of information exchange over contexts in an MCS is represented by the information exchange network, which is a heterogeneous graph with two kinds of vertices (contexts
Conclusion
The problem of selecting desirable diagnoses for resolving inconsistency is still a challenge in multi-context systems. Most approaches to prioritizing diagnoses often depend on some external knowledge, which is often difficult to be obtained in many applications.
We have proposed three types of approaches to prioritizing diagnoses for a multi-context system based on its intrinsic structure of information exchange over contexts via bridge rules in the whole system. The approaches of the first
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
The author is grateful to anonymous reviewers for their valuable comments. This work was partly supported by the National Natural Science Foundation of China under Grant No. 61572002, No. 61690201, and No. 61732001.
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