We define new families of dpda's by restricting the appearance of equivalent configurations in certain computations. These families capture the notion of a necessary usage of the stack. We discuss the equivalence and containment problems for these families and show how to maintain on-line regularity indication (i.e., how to know at each time in the computation whether the language accepted from here on is regular). As a consequence we get a linearity test for these families. We also give polynomial time algorithms to decide regularity and real time strictness for some of these families.