We consider finite automata equipped with finitely many registers (or counters). Each transition can be fired only if the associated formula, the guard is satisfied by the current register values. Then the transition updates the registers and the control moves to a new state. In our framework, the guards are conjunctions of atomic constraints y i #y j + c i;j where y i is either x 0 i or x i , the values of the counter i respectively after and before the transition, and # is any relational symbol in f=; ; ; ?; !g. We show that the binary relation between the registers values, which is defined by a counter automaton whose control is flat (no nested loops), is definable in the additive theory of N (or Z or Q or R depending on the type of the counters). In particular, we show that adding a fixed point to the above guards, we stay within Presburger arithmetic (resp. additive theory of Z, Q, R). We actually prove a slightly more general result: the number of transition steps can be a ...

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