A decomposition theorem for probabilistic transition systems
In this paper we prove that every finite Markov chain can be decomposed into a cascade product of a Bernoulli process and several simple permutation-reset deterministic automata. The original chain is a statehomomorphic image of the product. By doing so we give a positive answer to an open question stated in [Paz71] concerning the decomposability of probabilistic systems. Our result is based on the surprisingly-original observation that in probabilistic transition systems, “randomness” and “memory” can be separated in such a way that allows the non-random part to be treated using common deterministic automata-theoretic techniques. The same separation technique can be applied as well to other kinds of non-determinism.
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- [Arb68]M.A. Arbib, Theories of Abstract Automata, Prentice-Hall, Englewood Cliffs, 1968.Google Scholar
- [Eil76]S. Eilenberg, Automata, Languages and Machines, Vol. B, Academic Press, New York, 1976.Google Scholar
- [Gin68]A. Ginzburg, Algebraic Theory of Automata, Academic Press, New York, 1968.Google Scholar
- [KS60]J.G. Kemeny and J.L. Snell, Finite Markov Chains, Van Nostrand, New York, 1960.Google Scholar
- [KR65]K. Krohn and J.L. Rhodes, Algebraic Theory of Machines, I Principles of Finite Semigroups and Machines, Transactions of the American Mathematical Society 116, 450–464, 1965.Google Scholar
- [Lal79]G. Lallement, Semigroups and Combinatorial Applications, Wiley, New York, 1979.Google Scholar
- [MP90]O. Maler and A. Pnueli, Tight Bounds on the Complexity of Cascaded Decomposition of Automata, Proc. 31st FOCS, 672–682, 1990.Google Scholar
- [Paz70]A. Paz, Introduction to Probabilistic Automata, Academic Press, New York, 1970.Google Scholar
- [Pin86]J.-E. Pin, Varieties of Formal Languages, Plenum, New York, 1986.Google Scholar
- [Sta72]P.H. Starke, Abstract Automata, North-Holland, Amsterdam, 1972.Google Scholar