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Abduction and the Dualization Problem

  • Thomas Eiter
  • Kazuhisa Makino
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2843)

Abstract

Computing abductive explanations is an important problem, which has been studied extensively in Artificial Intelligence (AI) and related disciplines. While computing some abductive explanation for a literal χ with respect to a set of abducibles A from a Horn propositional theory Σ is intractable under the traditional representation of Σ by a set of Horn clauses, the problem is polynomial under model-based theory representation, where Σ is represented by its characteristic models. Furthermore, computing all the (possibly exponentially) many explanations is polynomial-time equivalent to the problem of dualizing a positive CNF, which is a well-known problem whose precise complexity in terms of the theory of NP-completeness is not known yet. In this paper, we first review the monotone dualization problem and its connection to computing all abductive explanations for a query literal and some related problems in knowledge discovery. We then investigate possible generalizations of this connection to abductive queries beyond literals. Among other results, we find that the equivalence for generating all explanations for a clause query (resp., term query) χ to the monotone dualization problem holds if χ contains at most k positive (resp., negative) literals for constant k, while the problem is not solvable in polynomial total-time, i.e., in time polynomial in the combined size of the input and the output, unless P=NP for general clause resp. term queries. Our results shed new light on the computational nature of abduction and Horn theories in particular, and might be interesting also for related problems, which remains to be explored.

Keywords

Abduction monotone dualization hypergraph transversals Horn functions model-based reasoning polynomial total-time computation NP-hardness 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Thomas Eiter
    • 1
  • Kazuhisa Makino
    • 2
  1. 1.Institut für InformationssystemeTechnische Universität WienWienAustria
  2. 2.Division of Mathematical Science for Social Systems, Graduate School of Engineering ScienceOsaka UniversityToyonaka, OsakaJapan

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