Geodesic
Finite automata and numbers
Diskretnaya Matematika, Tome 27 (2015) no. 4, pp. 3-20

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We study finite automata representations of numerical rings. Such representations correspond to the class of linear $p$-adic automata that compute homogeneous linear functions with rational coefficients in the ring of $p$-adic integers. Finite automata act both as ring elements and as operations. We also study properties of transition diagrams of automata that compute a function $f(x)=cx$ of one variable. In particular we obtain precise values for the number of states of such automata and show that for $c>0$ transition diagrams are self-dual (this property generalises self-duality of Boolean functions). We also obtain the criterion for an automaton computing a function $f(x)=cx$ to be a permutation automaton, and fully describe groups that are transition semigroups of such automata.
Keywords: linear automata, $p$-adic numbers, automata structure, transition diagrams, transition semigroups. 
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     author = {S. V. Aleshin and P. A. Panteleev},
     title = {Finite automata and numbers},
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     url = {http://geodesic.mathdoc.fr/item/DM_2015_27_4_a0/}
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S. V. Aleshin; P. A. Panteleev. Finite automata and numbers. Diskretnaya Matematika, Tome 27 (2015) no. 4, pp. 3-20. http://geodesic.mathdoc.fr/item/DM_2015_27_4_a0/