By Julius T. Tou (Eds.)
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Additional info for Applied Automata Theory
Any relation that is reflexive, symmetric, and transitive is an equivalence relation. We have had examples of both of these in preceding sections. 6 Since relations are actually sets, a number of operations on relations are at once seen to be natural: (a) The complement R of the relation R on a set A is the set of all pairs (a, b) in A X A which are not in R. (b) The transpose RT of a relation fiona set A is the set of all pairs (6, a) where (a, b) G R. (c) The union of two relations R\ and R2 on A is simply the set of all pairs (a, b) of A X A which G Ri or G Ä2 or both.
It is stationery in the sense that the transition function F and the output function G do not depend upon the sampling instant under consideration. I t is memoryless in the sense that the present output state and the next automaton state do not depend upon the past inputs, outputs, or automaton states. The transition function F and the output function G of an automaton are usually characterized by three basic representations: the transition table, the transition diagram, and the transition matrix.
To illustrate a more involved application, it will now be used to show that (0*1)* = λ VJ (0 W 1)*1. Note that this equation is an instance of law (7). It is easy to see that the proof below is adequate to establish the more general law, but the more specific instance is more easily discussed. To show first that (0*1)* = λ U (0 VJ 1)*1, consider an arbitrary word We (0*1)*. Then W = UiU*"-Un,n > 0. If n = 0, then W = λ, and then W 6 λ VJ (0 VJ 1)*1. If n > 1, then for each i, 1 < i < n, Ui € 0*1.
Applied Automata Theory by Julius T. Tou (Eds.)