Two-Way Finite Automata: Old and Recent Results

TL;DR

This paper reviews the history and recent developments in two-way finite automata, focusing on their computational power, state complexity, and open problems like the Sakoda-Sipser conjecture, especially in unary cases and space complexity.

Contribution

It summarizes recent progress on the state complexity of simulating nondeterministic automata deterministically, highlighting connections to space complexity and open conjectures.

Findings

Two-way automata have the same language recognition power as one-way automata.

Recent results provide insights into the state complexity of automata simulation.

Open problems like the Sakoda-Sipser conjecture remain unresolved.

Abstract

The notion of two-way automata was introduced at the very beginning of automata theory. In 1959, Rabin and Scott and, independently, Shepherdson, proved that these models, both in the deterministic and in the nondeterministic versions, have the same power of one-way automata, namely, they characterize the class of regular languages. In 1978, Sakoda and Sipser posed the question of the cost, in the number of the states, of the simulation of one-way and two-way nondeterministic automata by two-way deterministic automata. They conjectured that these costs are exponential. In spite of all attempts to solve it, this question is still open. In the last ten years the problem of Sakoda and Sipser was widely reconsidered and many new results related to it have been obtained. In this work we discuss some of them. In particular, we focus on the restriction to the unary case and on the connections with open questions in space complexity.