By Azaria Paz
ISBN-10: 0125476507
ISBN-13: 9780125476508
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Extra info for Introduction to Probabilistic Automata
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I Assume now that two equivalent machines M and M* are in minimal-state form. 6]. This observation leads to the following corollary. 9: Let M be a minimal-state form machine. There exists a minimal state form machine M* which is equivalent but not isomorphic to M if and only if there exists a row ~(ylx), in a matrix A(ylx) of M, which is not simplicial. It follows from the above theorem and corollary that the uniqueness of the reduced or minimal-state form of a machine M is conditional on the nature of the points hM(~(ylx)), where ~(ylx) is a row in a matrix AM(ylx).
Find the matrix G(M,n) for the machine whose matrices are as in Exercise 1, with distribution 1t = (t, -h t, 0). 2. 1: Let 1t and p be two initial distributions for a given machine. 1t and p are called k-equivalent distributions if the functions pivlu) and pp(vlu) [see (7)] have the same values for all pairs (v, u) such that I(v, u) < k. 1t and p are called equivalent distributions if the functions Pn(vlu) and pp(vlu) have the same values for all pairs (v, u). 1: Two distributions 1t and p for a given machine are equivalent if and only if they are (n - I)-equivalent, where n is the number of states of the machine.
It is easily seen that from this viewpoint the first method is preferable, since by it. 1 is applied to a matrix A with IXI x lSI rows and lSI columns, so that IWI < (lSI - 1)(IX/ x lSI) + I, whereas by the second method the lemma is first applied to the matrices A(x); since there are at most