Translation from Classical Two-Way Automata to Pebble Two-Way Automata

TL;DR

This paper explores the relationship between classical two-way automata and pebble two-way automata, demonstrating that polynomial trade-offs in classical models imply similar trade-offs in pebble models, revealing a fundamental connection.

Contribution

It establishes a polynomial trade-off relationship between classical and pebble two-way automata, showing their computational equivalence under certain conditions.

Findings

Polynomial trade-offs in classical automata imply similar trade-offs in pebble automata.

Pebble automata can be simulated by classical automata with linear state increase.

Results hold for both nondeterministic and deterministic models.

Abstract

We study the relation between the standard two-way automata and more powerful devices, namely, two-way finite automata with an additional "pebble" movable along the input tape. Similarly as in the case of the classical two-way machines, it is not known whether there exists a polynomial trade-off, in the number of states, between the nondeterministic and deterministic pebble two-way automata. However, we show that these two machine models are not independent: if there exists a polynomial trade-off for the classical two-way automata, then there must also exist a polynomial trade-off for the pebble two-way automata. Thus, we have an upward collapse (or a downward separation) from the classical two-way automata to more powerful pebble automata, still staying within the class of regular languages. The same upward collapse holds for complementation of nondeterministic two-way machines. These results are obtained by showing that each pebble machine can be, by using suitable inputs, simulated by a classical two-way automaton with a linear number of states (and vice versa), despite the existing exponential blow-up between the classical and pebble two-way machines.