Summary. Traditional approaches to computing minimal conflicts and diagnoses use search technique. It is well known that search technique may cause combination explosion. Algebraic approach may be a way to solve the problem. In this paper we present an algebraic approach to model-based diagnosis. A system with an observation can be represented by a special Petri net PN, checking whether there is a conflict between the correct system behavior and the observation corresponds to checking whether there exists a marking M ∈ R(M0) such that M(p1) and M(p2) are not zero, where p1 and p2 are labeled with the output of the system and its negation respectively. Furthermore, we show that M = M0 +CX is such a marking, where M0 is the initial marking, C is the incidence matrix of PN, and X is the maximal vector in {V |V is a {0, 1}-vector and for each transition t, if V (t) = 1, then there is a firing sequence t1, t2,..., tm, t}. Then, we present an algorithm to compute the maximal vector X in V SE(PN) in polynomial time. Once the maximal vector in V SE(PN) is generated, we can check whether there is conflicts between the correct system behavior and the observation. We also present algorithms for computing minimal conflicts and diagnoses by using the above algorithm. Compared with related works, our algorithm terminates in polynomial time if the inputs of the each component in the system are not more than a given constant.
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
R. Reiter, A theory of diagnosis from first principles, Artificial Intelligence, vol. 32, pp. 57-95, 1987.
J. d. Kleer, A. K. Mackworth, and R. Reiter, Charactering diagnoses and sys-tems, Artificial Intelligence, vol. 56, pp. 197-222, 1992.
A. Darwiche, Model-based Diagnosis using causal networks, Proc. Int. Joint Conf. on Artificial Intelligence, pp. 211-217, Montreal, Canada, August 1995.
A. Darwiche, Model-based diagnosis using structured system descriptions, Jour-nal of Artificial Intelligence Research, vol.8, pp. 165-222, 1998.
A. Darwiche, and G. Provan, The Effect of observation on the complexity of model-based diagnosis, Proc. American National Conf. on Artificial Intelligence, pp. 94-99, Providence, Rhode Island, USA, July 1997.
R. Haenni, Generating diagnoses from conflict sets, Proc. Florida Artificial Intel-ligence Research Symposium, pp. 120-124, Florida, USA, May 1998.
P. Marquis, Consequence finding algorithms, in: (eds. S. Kholas, J. Moral) Handbook on Deafeasible Reasoning and Uncertainty Management Systems, pp. 41-145. Kluwer Academic, Boston, 2000.
R. Haenni, A query-driven anytime algorithm For argumentative and abductive reasoning, Proc. First Int. Conf. on SoftWare, pp. 114-127, Belfast, Northern Ireland, April 2002.
A. del Val, The complexity of restricted consequence finding and abduction, Proc. 17th American National Conf. on Artificial Intelligence, pp. 337-342, Texas, USA, July 2000.
L. Simon and A. del Val, Efficient consequence finding, Proc. Int. Joint Conf. on Artificial Intelligence, pp. 359-365, Seattle, Washington, USA, August 2001.
A. del Val, A new method for consequence finding and compilation in restricted languages, Proc. Sixteenth American National Conf. on Artificial Intelligence, pp. 259-264, Florida, USA, July 1999.
Y. E. Fattah, and R. Dechter, Diagnosing tree-decomposable circuits, Proc. Int. Joint Conf. on Artificial Intelligence, pp. 572-578, Montreal, Canada, August 1995.
M. Stumptner, and F. Wotawa, Diagnosing tree-structured systems, Artificial Intelligence, vol. 127, pp. 1-29, 2001.
Bartlomiej Gorny, Antoni Ligeza, Model-based diagnosis of dynamic Systems: systematic conflict generation, in: L. Magnani, N. J. Nersessian and C. Pizzi (eds.) Logical and Computational Aspects of Model-based Reasoning, pp. 273-291, Kluwer Academic Publisher, 2002.
I. Mozetic, A Polynomial-time algorithm for model-based diagnosis, Proc. 10th European Conf. on Artificial Intelligence, pp. 729-733, Vienna, Austria, August 1992.
R. L. Childress, and M. Valtorta, Polynomial-time model-based diagnosis with the critical set algorithm, Proc. Fourth Int. Fourth Int. Workshop on Principles of Diagnosis, pp. 166-177, Aberystwyth, Wales, UK, 1993.
S. Luan, L. Magnani, G. Dai, Algorithms for Computing Minimal Conflicts, Logic Journal of IGPL, Vol.14 No.2, 391-406, June 2006.
L. Magnani, Abduction, Reason, and Science: Processes of Discovery and Expla-nation, New York, Kluwer Academic/Plenum Publishers, 2001.
N. J. Nilsson, Artificial Intelligence: A New Synthesis, Morgan Kaufmann, San Fransisco, 1999.
P. Meseguer. A new method to checking rule bases for inconsistency: a Petri net approach. In Proceedings of the 9th European Conference on Artificial Intelli-gence (ECAI-90), Stockholm, August 1990, pp. 437-442.
Tadao Murata, Petri Nets: Properties, analysis and applications, Proceedings of the IEEE, Vol. 77, No 4, April, 1989, 541-580.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Luan, S., Magnani, L., Dai, G. (2007). An Algebraic Approach to Model-Based Diagnosis. In: Magnani, L., Li, P. (eds) Model-Based Reasoning in Science, Technology, and Medicine. Studies in Computational Intelligence, vol 64. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-71986-1_27
Download citation
DOI: https://doi.org/10.1007/978-3-540-71986-1_27
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-71985-4
Online ISBN: 978-3-540-71986-1
eBook Packages: EngineeringEngineering (R0)