Abstract
Abstract 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.
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