By Julius T. Tou (Eds.)

ISBN-10: 1483200167

ISBN-13: 9781483200163

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**Extra resources for Applied Automata Theory**

**Example text**

1. ■ FIG. & w 1. The transition matrix of an n-state automaton is an n X n matrix whose ijth entry is the label of the branch ba from the ith state Xi to the j t h state Xj if the branch bij exists, and is equal to zero if the branch does not exist. The transition matrix for the three-state automaton shown in Fig. 1 is ~n/c2 r2/ci 0 0 r2/c2 n/ci n/c2 0 \_r2/ci A finite automaton can also be described by the state matrix PoO/e) = [>ii(r*)], i, j = 1, 2, · · ·, n; k = 1, 2, · · ·, a, (6) where Pij(rk) is conditional probability for the automaton to go from state Xi to state Xj when the input is rk.

Washington, D. C , 1967. ACKNOWLEDGEMENT We are grateful to the editors and publishers of reference [la] for permission to re-use here the material written for that article. STOCHASTIC AUTOMATA AND DISCRETE SYSTEMS THEORY JULIUS T. TOU UNIVERSITY OF FLORIDA GAINESVILLE, FLORIDA 1. 2. 3. 4. 5. 6. 7. 8. Introduction Finite Automata Characterization of Stochastic Automata State Probabilities and «-Transform Analysis Multistage Decision Process Entropy of a Stochastic Automaton Unreliable Finite Automaton Eigenvalues of a Stochastic Automaton Bibliography 55 56 59 62 67 69 71 78 80 1.

0 U 4 U ( 0 U 2 ) ( 0 U 4 ) * ( 2 U 4 ) ] * can be reduced to Σ*04Σ* U ( \ U Σ*02) (22 U 4)*(λ U 0Σ* U 24Σ*). But the argument, explained at the end of the author's work [13], is remote and seems to be peculiar to this example. At present, it would appear that there is no unified method of reducing the star height of those regular expressions that can be so reduced (see the last section of the author's work [13]). 54 ROBERT MCNAUGHTON REFERENCES 1. Brzozowski, J. , A survey of regular expressions and their applications.

### Applied Automata Theory by Julius T. Tou (Eds.)

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