TRANSLATION FROM CLASSICAL TWO-WAYAUTOMATA TO PEBBLE TWO-WAY AUTOMATA

We study the relation between the standard two-way automataand more powerful devices, namely, two-way finite automataequipped with some _ additional “pebbles” that are movable along theinput tape, but their use is restricted (nested) in a stack-like fashion.Similarly as in the case of the classical two-way machines, it isnot known whether there exists a polynomial trade-off, in the numberof states, between the nondeterministic and deterministic two-wayautomata with _ nested pebbles. However, we show that these two machinemodels are not independent: if there exists a polynomial trade-offfor the classical two-way automata, then, for each _≥0, there must alsoexist a polynomial trade-off for the two-way automata with _ nestedpebbles. Thus, we have an upward collapse (or a downward separation)from the classical two-way automata to more powerful pebbleautomata, still staying within the class of regular languages. The sameupward collapse holds for complementation of nondeterministic twowaymachines. These results are obtained by showing that each pebblemachine can be, by using suitable inputs, simulated by a classical twowayautomaton (and vice versa), with only a linear number of states,despite the existing exponential blow-up between the classical and pebbletwo-way machines