Consider a DFA \(A\) together with \(d\) pairs off counters \((x_1,y_1),\dots,(x_d,y_d)\). Each transition of \(A\) is labelled with an update function for all counters. The update functions for the x-counters are \(id:n\rightarrow n\) and \(++ : n\rightarrow n+1\). The update function for the y-counters are \(id:n\rightarrow n\), \(\times 2:n\rightarrow 2n\) and \(\times 2+1:n\rightarrow 2n+1\). An accepting configuration is a configuration in which we are in an accepting state and all the counter pairs have the same value, i.e. \(x_1=y_1,\dots,x_d=y_d\).

*Question:* is the emptiness problem for this class of automata decidable?