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Automates de coût borné sur un alphabet à une lettre. (French) Zbl 0578.68044

Summary: Let A be an automaton with n states on a single letter alphabet and having a cost function (called ”distance function” by K. Hashiguchi). If its cost c(A) is bounded, then \(c(A)<n^ 2\). Moreover, this bound is asymptotically optimal in the sense that there exists a family \(\{A_ n\}\) of automata, where \(A_ n\) has n states and such that \(\lim_{n\to +\infty}c(A_ n)/n^ 2=1\).

MSC:

68Q45 Formal languages and automata
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References:

[1] 1. S. EILENBERG, Automata, Languages and Machines, Academic Press, Vol. A, 1974. Zbl0317.94045 · Zbl 0317.94045
[2] 2. K. HASHIGUCHI, Limitedness Theorem on Finite Automata with Distance Functions, J. Comp. Syst Sc., Vol. 24, 1982, p. 233-244. Zbl0513.68051 MR661652 · Zbl 0513.68051
[3] 3. G. MARKOWSKY, Bounds on the Index and Period of a Binary Relation on a finite Set, Semigroup Forum, Vol. 13, 1977, p. 253-259. Zbl0353.20055 MR446762 · Zbl 0353.20055
[4] 4. E. SELMER, On the Linear Diophantine Problem of Frobenius, J. reine angew. Math., Vol. 293/294, 1977, p. 1-17. Zbl0349.10009 MR441855 · Zbl 0349.10009
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