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Extra info for Developments in Language Theory: 12th International Conference, DLT 2008, Kyoto, Japan, September 16-19, 2008. Proceedings
The method of proving this begins with embedding the Post Correspondence Problem  into integer matrices (a method introduced by Paterson ), and continues by a procedure introduced by P. Turakainen  to convert any set of matrices into doubly stochastic ones, still preserving some essential properties of the original matrix set. It is possible to show that the emptiness problem for cut point and strict cut point languages is in fact undecidable for 25-state probabilistic automata over a binary alphabet.
Then this accepting computation accepts all words of L corresponding to words on the intersections of the rows α1 , α2 , . . , αi , . . and the columns β1 , β2 , . . , βi , . .. In Fig. 3 one immediately sees that this intersection of rows and columns determines unambiguously a 1-monochromatic submatrix of ML . To solve the combinatorial problem of covering all 1’s of a matrix by the minimal number of potentially overlapping 1-monochromatic submatrices is not easy. One possibility to use this fact is to restrict ML to a ﬁnite submatrix ML and then to estimate the largest 1-monochromatic submatrix S of ML .
And Rj = (rj1 , rj2 , . ) be diﬀerent rows of ML . , rik = rjk . If C1 sends the same message m to C2 for its inputs αi and αj corresponding to the rows Ri and Rj , then C2 either has to accept both αi βk and αj βk or to reject both words αi βk and αj βk (The arguments of C2 are the message m and the word βk and these arguments are the same for inputs αi βk and αj βk ). Since the rows Ri and Rj diﬀer in column k, the number of messages used is at least the number of diﬀerent rows of ML . Now, one can easily observe that the number of diﬀerent rows of ML is nothing else than the number of the equivalence classes of the Nerode relation for L and we are done (for a detailed argumentation see ).