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The Complexity of Abduction for Equality Constraint Languages

Authors: Johannes Schmidt and Michał Wrona

Published in: LIPIcs, Volume 23, Computer Science Logic 2013 (CSL 2013)

Abduction is a form of nonmonotonic reasoning that looks for an explanation for an observed manifestation according to some knowledge base. One form of the abduction problem studied in the literature is the propositional abduction problem parameterized by a structure \Gamma over the two-element domain. In that case, the knowledge base is a set of constraints over \Gamma, the manifestation and explanation are propositional formulas. In this paper, we follow a similar route. Yet, we consider abduction over infinite domain. We study the equality abduction problem parameterized by a relational first-order structure \Gamma over the natural numbers such that every relation in \Gamma is definable by a Boolean combination of equalities, a manifestation is a literal of the form (x = y) or (x != y), and an explanation is a set of such literals. Our main contribution is a complete complexity characterization of the equality abduction problem. We prove that depending on \Gamma, it is \Sigma^P_2-complete, or NP-complete, or in P.

Cite as

Johannes Schmidt and Michał Wrona. The Complexity of Abduction for Equality Constraint Languages. In Computer Science Logic 2013 (CSL 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 23, pp. 615-633, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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  author =	{Schmidt, Johannes and Wrona, Micha{\l}},
  title =	{{The Complexity of Abduction for Equality Constraint Languages}},
  booktitle =	{Computer Science Logic 2013 (CSL 2013)},
  pages =	{615--633},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-60-6},
  ISSN =	{1868-8969},
  year =	{2013},
  volume =	{23},
  editor =	{Ronchi Della Rocca, Simona},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-42229},
  doi =		{10.4230/LIPIcs.CSL.2013.615},
  annote =	{Keywords: Abduction, infinite structures, equality constraint languages, computational complexity, algebraic approach}
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