Limited Automata: Properties, Complexity and Variants

Abstract

Limited automata are single-tape Turing machines with severe rewriting restrictions. They have been introduced in 1967 by Thomas Hibbard, who proved that they have the same computational power as pushdown automata. Hence, they provide an alternative characterization of the class of context-free languages in terms of recognizing devices. After that paper, these models have been almost forgotten for many years. Only recently limited automata were reconsidered in a series of papers, where several properties of them and of their variants have been investigated. In this work we present an overview of the most important results obtained in these researches. We also discuss some related models and possible lines for future investigations.

A Linear-Time Simulation of Deterministic d-Limited Automata

“…We agree with the remark from [17] that T. Hibbard claimed in [11] that deterministic d-limited automata do not recognize palindromes for any d, while the formal proof is missing and would be interesting.…”

“…d-LAs were abandoned for decades but then the formal language's community returned to their study. G. Pighizzini, who actively worked on this topic, made a survey of results on d-LA automata and related models [17], focusing in part on state complexity.…”

“…We define the Hibbard's model, introduced in [11], as it is defined today [17]. An equivalent definition in a more formal style could be found in [18].…”

Once-Marking and Always-Marking 1-Limited Automata

“…The investigation of these models has been reconsidered in the last decade, mainly from a descriptional point of view. Starting with [8,9], several works investigating properties of limited automata and their relationships with other computational models appeared in the literature (for a recent survey see [7]).…”

Performing Regular Operations with 1-Limited Automata

“…Limited automata are single-tape Turing machines with rewriting restrictions, introduced by Hibbard in 1967 [10] and recently reconsidered and deeply investigated (see, e.g., [11][12][13][14][15][16][17]). These devices are two-way finite automata with the extra capability of overwriting the contents of each tape cell only in the first d visits, for a fixed constant d ≥ 0 (we use the name d-limited automaton to explicitly mention the constant d).…”