Pushdown and one-counter automata: Constant and non-constant memory usage

It cannot be decided whether a one-counter automaton accepts each string in its language using a counter whose value is bounded, with respect to the length of the input, by a constant. Furthermore, when the counter is bounded by a constant, its value cannot be limited by any recursive function in the size of the machine. By taking into account the costs of all computations (strong measure) or of all accepting computations (accept measure) instead of those of the least expensive accepting computations (weak measure), the above-mentioned problem becomes decidable for both pushdown automata and one-counter automata, while the bounds for the pushdown height or the value of the counter, when non constant, are recursive in the size of the machine. We also prove that, under the weak measure, if a one-counter automaton accepts with a counter that, with respect to the input length, is not bounded by any constants, then the counter grows at least as a logarithmic function. This is in contrast with the case of pushdown automata in which the bound is a double-logarithmic function. For the strong and accept measures these bounds are shown to be linear, for both pushdown and one-counter automata.