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Abstractions of Finite-State Machines and Immediately-Detectable Output Faults
March 1992 (vol. 41 no. 3)
pp. 325-338

A general way to make a smaller model of a large system, or to represent the fact that the observations possible on it are limited, is to apply an abstraction A to it. If the system is modeled by a finite-state machine M, the abstraction consists of three partitions, one for each of the state, input, and output sets. States, inputs, or outputs lumped together in one block by the partition are indistinguishable from each other, resulting in a nondeterministic machine M/sub A/. An observer of M/sub A/, whose task is to detect erroneous behavior in M, is prevented by the abstraction from seeing some of the faults. The authors investigate the choice of an abstraction that is optimal with respect to immediately detectable faults in the output map. It is shown that this requires solving an NP-complete 'set-partitioning' problem. A polynomial-time algorithm for finding an approximately optimal partition of either the states or the inputs of M, together with a way to check the goodness of the approximation is given. This algorithm also solves the undetectable fault minimization problem exactly, and in polynomial time.

[1] M. A. Arbib,Theories of Abstract Automata. Englewood Cliffs, NJ: Prentice-Hall, 1969.
[2] A. Avizienis, "Fault-tolerance by means of external monitoring of computer systems," inProc. Nat. Comput. Conf., 1981.
[3] R. E. Bellman, "On a routing problem,"Quarterly Appl. Mathemat., vol. 16, 1958.
[4] A. K. Chakravarty, J. B. Orlin, and U. G. Rothblum, "A partitioning problem with additive objective with an application to optimal inventory groupings for joint replenishment,"Oper. Res., vol. 30, no. 5, Sept.-Oct. 1982.
[5] M. R. Garey and D. S. Johnson,Computers and Intractability: A Guide to Theory of NP-Completeness. San Francisco, CA: Freeman, 1979.
[6] M. R. Garey, D. S. Johnson, and L. Stockmeyer, "Some simplified NP-complete graph problems,"Theoret. Comput. Sci., vol. 1, 1976.
[7] F. K. Hwang, J. Sun, and E. Y. Yao, "Optimal set partitioning,"SIAM J. Algebraic and Discrete Methods, vol. 6, no. 1, Jan. 1985.
[8] D. E. Knuth,The Art of Computer Programming, Vol. 1. Reading, MA: Addison-Wesley, 1973.
[9] K. N. Oikonomou and R. Y. Kain, "Abstractions for node-level passive fault detection in distributed systems,"IEEE Trans. Comput., vol. C-32, June 1983.
[10] K. N. Oikonomou, "Abstractions of finite-state machines optimal with respect to single undetectable output faults,"IEEE Trans. Comput., vol. C-36, Feb. 1987.
[11] A. Mahmood and E. J. McCluskey, "Concurrent error detection using watchdog processors--A survey,"IEEE Trans. Comput., vol. C-37, Feb. 1988.

Index Terms:
set partitioning; finite-state machines; immediately-detectable output faults; abstraction; nondeterministic machine; NP-complete; polynomial-time algorithm; approximately optimal partition; computational complexity; data structures; fault tolerant computing; finite automata.
K.N. Oikonomou, "Abstractions of Finite-State Machines and Immediately-Detectable Output Faults," IEEE Transactions on Computers, vol. 41, no. 3, pp. 325-338, March 1992, doi:10.1109/12.127444
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