Abstract
In this paper, we investigate the deductive inference for the interiors and exteriors of Horn knowledge bases, where the interiors and exteriors were introduced by Makino and Ibaraki [11] to study stability properties of knowledge bases. We present a linear time algorithm for the deduction for the interiors and show that it is co-NP-complete for the deduction for the exteriors. Under model-based representation, we show that the deduction problem for interiors is NP-complete while the one for exteriors is co-NP-complete. As for Horn envelopes of the exteriors, we show that it is linearly solvable under model-based representation, while it is co-NP-complete under formula-based representation. We also discuss the polynomially solvable cases for all the intractable problems.
This work is supported in part by the Grant-in-Aid of the Ministry of Education, Science, Sports and Culture of Japan and by the Asahi glass foundation.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Brachman, R.J., Levesque, H.J.: Knowledge Representation and Reasoning. Elsevier, Amsterdam (2004)
Dechter, R., Pearl, J.: Structure identification in relational data. Artificial Intelligence 58, 237–270 (1992)
Dowling, W., Galliear, J.H.: Linear-time algorithms for testing the satisfiability of propositional Horn theories. Journal of Logic Programming 3, 267–284 (1983)
Eiter, T., Ibaraki, T., Makino, K.: Computing intersections of Horn theories for reasoning with models. Artificial Intelligence 110, 57–101 (1999)
Eiter, T., Makino, K.: On computing all abductive explanations. In: Proc. AAAI 2002, pp. 62–67 (2002)
Kavvadias, D., Papadimitriou, C., Sideri, M.: On Horn Envelopes and Hypergraph Transversals. In: Ng, K.W., Balasubramanian, N.V., Raghavan, P., Chin, F.Y.L. (eds.) ISAAC 1993. LNCS, vol. 762, pp. 399–405. Springer, Heidelberg (1993)
Kautz, H., Kearns, M., Selman, B.: Reasoning with characteristic models. In: Proc. AAAI 1993, pp. 34–39 (1993)
Kautz, H., Kearns, M., Selman, B.: Horn approximations of empirical data. Artificial Intelligence 74, 129–245 (1995)
Khardon, R., Roth, D.: Reasoning with models. Artificial Intelligence 87, 187–213 (1996)
Khardon, R., Roth, D.: Defaults and relevance in model-based reasoning. Artificial Intelligence 97, 169–193 (1997)
Makino, K., Ibaraki, T.: Interior and exterior functions of Boolean functions. Discrete Applied Mathematics 69, 209–231 (1996)
Makino, K., Ono, H.: Deductive Inference for the Interiors and Exteriors of Horn Theories, Mathematical Engineering Technical Report, METR 2007-06, The University of Tokyo (February 2007)
Makino, K., Ono, H., Ibaraki, T.: Interior and exterior functions of positive Boolean functions. Discrete Applied Mathematics 130, 417–436 (2003)
Makowsky, J.: Why Horn formulas matter for computer science: Initial structures and generic examples. Journal of Computer and System Sciences 34, 266–292 (1987)
McKinsey, J.: The decision problem for some classes of sentences without quantifiers. Journal of Symbolic Logic 8, 61–76 (1943)
Selman, B., Kautz, H.: Knowledge compilation using Horn approximations. In: Proc. AAAI 1991, pp. 904–909 (1991)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Makino, K., Ono, H. (2008). Deductive Inference for the Interiors and Exteriors of Horn Theories. In: Hong, SH., Nagamochi, H., Fukunaga, T. (eds) Algorithms and Computation. ISAAC 2008. Lecture Notes in Computer Science, vol 5369. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-92182-0_36
Download citation
DOI: https://doi.org/10.1007/978-3-540-92182-0_36
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-92181-3
Online ISBN: 978-3-540-92182-0
eBook Packages: Computer ScienceComputer Science (R0)