Abstract
Abduction is an important form of nonmonotonic reasoning allowing one to find explanations for certain symptoms or manifestations. When the application domain is described by a logical theory, we speak about logic- based abduction. Candidates for abductive explanations are usually subjected to minimality criteria such as subset-minimality, minimal cardinality, minimal weight, or minimality under prioritization of individual hypotheses. This paper presents a comprehensive complexity analysis of relevant problems related to abduction on propositional theories. They show that the different variations of abduction provide a rich collection of natural problems populating all major complexity classes between P and Σ P3 , Π P3 in the refined polynomial hierarchy. More precisely, besides polynomial, NP-complete and co-NP-complete abduction problems, abduction tasks that are complete for the classes Δ Pi , Δ Pi [O(logn), Σ Pi , and Π Pi , for i=2,3, are identified.
This paper is an overview of [10], which contains detailed proofs of all results.
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D. Allemang, M.C. Tanner, T. Bylander, and J. Josephson. Computational Complexity of Hypothesis Assembly. In Proc. IJCAI-87, 112–117, 1987.
G. Brewka. Nonmonotonic Reasoning: Logical Foundations of Commonsense. Cambridge Univ. Press, 1991.
T. Bylander. The Monotonic Abduction Problem: A Functional Characterization on the Edge of Tractability. In Proc. KR-91, 70–77, 1991.
T. Bylander, D. Allemang, M. Tanner, and J. Josephson. The computational complexity of abduction. Artificial Intelligence, 49:25–60, 1991.
M. Cadoli and M. Schaerf. A Survey on Complexity Results for Non-monotonic Logics. Technical report, Dipartimento di Informatica e Sistemistica, Università di Roma “La Sapienza”, 1992.
L. Console, D. Theseider Dupré, and P. Torasso. On the Relationship Between Abduction and Deduction. Journal of Logic and Computation, 1(5):661–690, 1991.
W. Dowling and J. H. Gallier. Linear-time Algorithms for Testing the Satisfiability of Propositional Horn Theories. Journal of Logic Programming, 3:267–284, 1984.
T. Eiter and G. Gottlob. Propositional Circumscription and Extended Closed World Reasoning are Π2/P-complete. Theoretical Computer Science, to appear.
T. Eiter and G. Gottlob. On the Complexity of Propositional Knowledge Base Revision, Updates, and Counterfactuals. Artificial Intelligence, 57:227–270, 1992.
T. Eiter and G. Gottlob. The Complexity of Logic-Based Abduction. Technical Report CD-TR 92/35, Christian Doppler Laboratory for Expert Systems, TU Vienna, 1992.
R. Fagin, J. D. Ullman, and M. Y. Vardi. On the Semantics of Updates in Data-bases. In Proc. PODS-83, 352–365, 1983.
G. Friedrich, G. Gottlob, and W. Nejdl. Hypothesis Classification, Abductive Diagnosis, and Therapy. In Proc. International Workshop on Expert Systems in Engineering, number 462 in LNAI, 69–78, September, 1990.
G. Gottlob. Complexity Results for Nonmonotonic Logics. Journal of Logic and Computation, 2(3):397–425, June 1992.
D. S. Johnson. A Catalog of Complexity Classes. volume A of Handbook of Theoretical Computer Science, chapter 2. 1990.
J. Josephson, B. Chandrasekaran, J. J. W. Smith, and M. Tanner. A Mechanism for Forming Composite Explanatory Hypotheses. IEEE Transactions on Systems, Man, and Cybernetics, SMC-17:445–454, 1987.
J. Kadin. P N P[O(log n)] and Sparse Turing-Complete Sets for N P. Journal of Computer and System Sciences, 39:282–298, 1989.
K. Konolige. Abduction versus closure in causal theories. Artificial Intelligence, 53:255–272, 1992.
R. Kowalski. Logic Programs in Artificial Intelligence. In Proc. IJCAI-91, 596–601, 1991.
M. Krentel. The Complexity of Optimization Problems. Journal of Computer and System Sciences, 36:490–509, 1988.
M. Krentel. Generalizations of OptP to the Polynomial Hierarchy. Theoretical Computer Science, 1992.
H. Levesque. A knowledge-level account for abduction. In Proc. IJCAI-89, 1061–1067, 1989.
V. Lifschitz. Computing Circumscription. In Proc. IJCAI-85, 121–127, 1985.
B. Nebel. Belief Revision and Default Reasoning: Syntax-Based Approaches. In Proc. KR-91, 417–428, 1991.
Y. Peng and J. Reggia. Abductive Inference Models for Diagnostic Problem Solving. Symbolic Computation — Artificial Intelligence. Springer, 1990.
D. Poole. A Logical Framework for Default Reasoning. Artificial Intelligence, 36:27–47, 1988.
R. Reiter. A Theory of Diagnosis From First Principles. Artificial Intelligence, 32:57–95, 1987.
V. Rutenburg. Complexity Classification in Truth Maintenance Systems. In Proc. STACS-91, 373–383, 1991.
G. Schwarz and M. Truszczyński. Nonmonotonic reasoning is sometimes easier. manuscript, August. 1992.
B. Selman and H. J. Levesque. Abductive and Default Reasoning: A Computational Core. In Proc. AAAI-90, 343–348, July 1990.
L. Stockmeyer and A. Meyer. Word Problems Requiring Exponential Time. In Proc. of the Fifth ACM STOC, 1–9, 1973.
K. Wagner. More Complicated Questions about Maxima and Minima, and Some Closures of NP. Theoretical Computer Science, 51:53–80, 1987.
K. Wagner. Bounded Query Computations. Technical Report 172, Universität Augsburg, March 1988.
K. Wagner. Bounded query classes. SIAM J. Comp., 19(5):833–846, 1990.
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Eiter, T., Gottlob, G. (1993). The complexity of logic-based abduction. In: Enjalbert, P., Finkel, A., Wagner, K.W. (eds) STACS 93. STACS 1993. Lecture Notes in Computer Science, vol 665. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-56503-5_10
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DOI: https://doi.org/10.1007/3-540-56503-5_10
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