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Tauberian theorems by Norbert Wiener

By Norbert Wiener

Generalized Harmonic research and Tauberian Theorems

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Copyright 2014 Jeff Erickson. 0/). Free distribution is strongly encouraged; commercial distribution is expressly forbidden. edu/~jeffe/teaching/algorithms/ for the most recent revision.

Remove all states unreachable from q0 (via whatever first search). Then L(M ) is finite if and only if the reduced DFA is a dag; this condition can be checked by depth-first search. Alternatively, but less usefully, L(M ) is finite if and only if L(M ) contains no string w such that n ≤ |w| < 2n. • Is L(M) = Σ∗ ? Remove all states unreachable from q0 (via whatever first search). Then L(M ) = Σ∗ if and only if every state in M is an accepting state. • Is L(M) = L(M )? Build a DFA N such that L(N ) = L(M ) \ L(M ) using a standard product construction, and then check whether L(N ) = ∅.

From the previous example, we know that there is a three-state DFA M11 that accepts the set of strings with the substring 11 and a nearly identical DFA M00 that accepts the set of strings containing the substring 00. By identifying the accept state of M00 with the start state of M11 , we obtain a five-state DFA that accepts the set of strings with 00 before 11. Finally, by inverting which states are accepting, we obtain the DFA we want. 1 0 0,1 1 0 0 0 0,1 1 1 1 0 1 0 0 0,1 0 1 1 1 0 1 0 0 0,1 0 1 1 Building a DFA for the language of strings in which every 00 is after every 11.

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