Querying inconsistent knowledge bases is a problem that has attracted a great deal of interest over the last decades. While several semantics of query answering have been proposed, and their complexity is rather well-understood, little attention has been paid to the problem of explaining query answers. Explainability has recently become a prominent problem in different areas of AI. In particular, explaining query answers allows users to understand not only what is entailed by an inconsistent knowledge base, but also why. In this paper, we address the problem of explaining query answers for existential rules under three popular inconsistency-tolerant semantics, namely, the ABox repair, the intersection of repairs, and the intersection of closed repairs semantics. We provide a thorough complexity analysis for a wide range of existential rule languages and for different complexity measures.

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There has been extensive work in many different fields on how phenomena of interest (e.g., diseases, innovation, product adoption) "diffuse" through a social network. As social networks increasingly become a fabric of society, there is a need to make "optimal" decisions with respect to an observed model of diffusion. For example, in epidemiology, officials want to find a set of k individuals in a social network which, if treated, would minimize spread of a disease. In marketing, campaign managers try to identify a set of k customers that, if given a free sample, would generate maximal "buzz" about the product. In this article, we first show that the well-known Generalized Annotated Program (GAP) paradigm can be used to express many existing diffusion models. We then define a class of problems called Social Network Diffusion Optimization Problems (SNDOPs). SNDOPs have four parts: (i) a diffusion model expressed as a GAP, (ii) an objective function we want to optimize with respect to a given diffusion model, (iii) an integer k > 0 describing resources (e.g., medication) that can be placed at nodes, (iv) a logical condition VC that governs which nodes can have a resource (e.g., only children above the age of 5 can be treated with a given medication). We study the computational complexity of SNDOPs and show both NP-completeness results as well as results on complexity of approximation. We then develop an exact and a heuristic algorithm to solve a large class of SNDOPproblems and show that our GREEDY-SNDOP algorithm achieves the best possible approximation ratio that a polynomial algorithm can achieve (unless P = NP). We conclude with a prototype experimental implementation to solve SNDOPs that looks at a real-world Wikipedia dataset consisting of over 103,000 edges. ACM Reference Format:Shakarian, P., Broecheler, M., Subrahmanian, V. S. and Molinaro, C. 2013. Using generalized annotated programs to solve social network diffusion optimization problems.

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