Abstract
We prove that two automata with multiplicity in \({\mathbb Z}\) are equivalent, i.e. define the same rational series, if and only if there is a sequence of \({\mathbb Z}\)-coverings, co- \({\mathbb Z}\)-coverings, and circulations of –1, which transforms one automaton into the other. Moreover, the construction of these transformations is effective.
This is obtained by combining two results: the first one relates coverings to conjugacy of automata, and is modeled after a theorem from symbolic dynamics; the second one is an adaptation of Schützenberger’s reduction algorithm of representations in a field to representations in an Euclidean domain (and thus in \({\mathbb Z}\)).
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Béal, MP., Lombardy, S., Sakarovitch, J. (2005). On the Equivalence of \({\mathbb Z}\)-Automata. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds) Automata, Languages and Programming. ICALP 2005. Lecture Notes in Computer Science, vol 3580. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11523468_33
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DOI: https://doi.org/10.1007/11523468_33
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