A subclass of the class of all pseudopolynomial problems is defined as a family of sets acceptable by some automaton operating with simultaneous time and space bounds. That the class is large enough can be seen in that it contains many (if not all) of the pseudopolynomial problems described in the literature. We study structure preserving reductions within this class and give intuitive reasons (borrowed from our knowledge about space bounded automata) that there exist at least four well known problems which are pairwise not equivalent under these reductions.