Minimal automaton of a rational cover
A cover is the set of finite factors of a bi-infinite word. It is quite clear that a rational cover is recognized by a trimmed automaton in the following sense : in every state q, there exists a transition beginning in q and a transition ending in q; furthermore every state is an initial and a final state.
We prove here that every rational cover is recognized by a minimal deterministic trimmed automaton Q in the sense of Eilenberg  : if B is a deterministic trimmed automaton which recognizes C, there exists a morphism from B to Q (of course Q is unique save on an isomorphism.
KeywordsRational Cover Finite Automaton Elementary Cycle Deterministic Automaton Elementary Path
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- Beauquier D. (1986) Ensembles homogènes minces.Google Scholar
- Beauquier D., Nivat M. (1985) About rational sets of factors of a bi-infinite word. Lecture Notes in Computer Science, 194, 33–42 ICALP 85.Google Scholar
- Blanchard F., Hansel G. (1984) Languages and subshifts, Lecture Notes in Computer Science, 192, 138–146.Google Scholar
- Eilenberg S. (1974), Automata, Languages and Machines, Vol. A, Academic Press.Google Scholar
- Fisher R. (1975), Sofic systems and graphs. Monants. Math. 80, 179–186.Google Scholar
- Lallement G. (1979) Semigroup and combinatorial applications Wiley — Interscience.Google Scholar
- Nivat M., Perrin D. (1982), Ensembles reconnaissables de mots bi-infinis, Proc. 14th A.C.M. Symp. on Theory of Computing, 47–59.Google Scholar
- Pin J.-E. (1984), Variétés de langages formels — Masson.Google Scholar