Limited automata and unary languages

Abstract

Limited automata are one-tape Turing machines that are allowed to rewrite the content of any tape cell only in the first d visits, for a fixed constant d. When d = 1 these models characterize regular languages. An exponential gap between the size of limited automata accepting unary languages and the size of equivalent finite automata is proved. Since a similar gap was already known from unary contextfree grammars to finite automata, also the conversion of such grammars into limited automata is investigated. It is proved that from each unary context-free grammar it is possible to obtain an equivalent 1-limited automaton whose description has a size which is polynomial in the size of the grammar. Furthermore, despite the exponential gap between the sizes of limited automata and of equivalent unary finite automata, there are unary regular languages for which d-limited automata cannot be significantly smaller than equivalent finite automata, for any arbitrarily large d.

Limited Automata: Properties, Complexity and Variants

“…Kutrib and Wendlandt proved state lower bounds for the simulation of unary dlimited automata by different variants of finite automata [15,14]. More recently, the following result has been proved: Theorem 10 ( [30]). For each integer n > 0, the singleton language {a 2 n } is accepted by a deterministic 1-limited automaton of size O(n), while each one-way nondeterministic finite automaton accepting it needs 2 n + 1 states.…”

“…With a small modification, it can be shown that for each n > 0 even the language {a 2 n } * can be accepted by a deterministic 1-limited automaton of size O(n), while each two-way nondeterministic finite automaton accepting it requires 2 n states [30].…”

Converting Nondeterministic Two-Way Automata into Small Deterministic Linear-Time Machines

“…By allowing nondeterministic transitions (1nfas) or/and movements of the head in both directions on the input tape, so obtaining two-way deterministic and nondeterministic finite automata (2dfas/2nfas), the computational power does not increase [21,24]. Other extensions of finite automata have been proved to capture the same class of languages, such as constant-height pushdown automata [2,4], straight-line programs [2], 1-limited automata [28,18,19], or, as will be of interest for this work, linear-time one-tape Turing machines [8,20,5,6]. 1 A natural question concerning models that share the same computational power is the comparison of the sizes of their descriptions.…”

Linear-time limited automata

“…This cost reduces to a single exponential when starting from a deterministic 1-limited automaton. Also, an exponential lower bound, using a single-letter input alphabet, has been obtained in [11], for the simulation of deterministic 1-limited automata by nondeterministic two-way nite automata.…”