8 research outputs found
One-Tape Turing Machine Variants and Language Recognition
We present two restricted versions of one-tape Turing machines. Both
characterize the class of context-free languages. In the first version,
proposed by Hibbard in 1967 and called limited automata, each tape cell can be
rewritten only in the first visits, for a fixed constant .
Furthermore, for deterministic limited automata are equivalent to
deterministic pushdown automata, namely they characterize deterministic
context-free languages. Further restricting the possible operations, we
consider strongly limited automata. These models still characterize
context-free languages. However, the deterministic version is less powerful
than the deterministic version of limited automata. In fact, there exist
deterministic context-free languages that are not accepted by any deterministic
strongly limited automaton.Comment: 20 pages. This article will appear in the Complexity Theory Column of
the September 2015 issue of SIGACT New
Once-Marking and Always-Marking 1-Limited Automata
Single-tape nondeterministic Turing machines that are allowed to replace the
symbol in each tape cell only when it is scanned for the first time are also
known as 1-limited automata. These devices characterize, exactly as finite
automata, the class of regular languages. However, they can be extremely more
succinct. Indeed, in the worst case the size gap from 1-limited automata to
one-way deterministic finite automata is double exponential.
Here we introduce two restricted versions of 1-limited automata, once-marking
1-limited automata and always-marking 1-limited automata, and study their
descriptional complexity. We prove that once-marking 1-limited automata still
exhibit a double exponential size gap to one-way deterministic finite automata.
However, their deterministic restriction is polynomially related in size to
two-way deterministic finite automata, in contrast to deterministic 1-limited
automata, whose equivalent two-way deterministic finite automata in the worst
case are exponentially larger. For always-marking 1-limited automata, we prove
that the size gap to one-way deterministic finite automata is only a single
exponential. The gap remains exponential even in the case the given machine is
deterministic.
We obtain other size relationships between different variants of these
machines and finite automata and we present some problems that deserve
investigation.Comment: In Proceedings AFL 2023, arXiv:2309.0112
Forgetting 1-Limited Automata
We introduce and investigate forgetting 1-limited automata, which are
single-tape Turing machines that, when visiting a cell for the first time,
replace the input symbol in it by a fixed symbol, so forgetting the original
contents. These devices have the same computational power as finite automata,
namely they characterize the class of regular languages. We study the cost in
size of the conversions of forgetting 1-limited automata, in both
nondeterministic and deterministic cases, into equivalent one-way
nondeterministic and deterministic automata, providing optimal bounds in terms
of exponential or superpolynomial functions. We also discuss the size
relationships with two-way finite automata. In this respect, we prove the
existence of a language for which forgetting 1-limited automata are
exponentially larger than equivalent minimal deterministic two-way automata.Comment: In Proceedings NCMA 2023, arXiv:2309.0733
Limited automata and unary languages
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
REGULAR LANGUAGES: TO FINITE AUTOMATA AND BEYOND - SUCCINCT DESCRIPTIONS AND OPTIMAL SIMULATIONS
\uc8 noto che i linguaggi regolari \u2014 o di tipo 3 \u2014 sono equivalenti agli automi a stati finiti. Tuttavia, in letteratura sono presenti altre caratterizzazioni di questa classe di linguaggi, in termini di modelli riconoscitori e grammatiche. Per esempio, limitando le risorse computazionali di modelli pi\uf9 generali, quali grammatiche context-free, automi a pila e macchine di Turing, che caratterizzano classi di linguaggi pi\uf9 ampie, \ue8 possibile ottenere modelli che generano o riconoscono solamente i linguaggi regolari. I dispositivi risultanti forniscono delle rappresentazioni alternative dei linguaggi di tipo 3, che, in alcuni casi, risultano significativamente pi\uf9 compatte rispetto a quelle dei modelli che caratterizzano la stessa classe di linguaggi. Il presente lavoro ha l\u2019obiettivo di studiare questi modelli formali dal punto di vista della complessit\ue0 descrizionale, o, in altre parole, di analizzare le relazioni tra le loro dimensioni, ossia il numero di simboli utilizzati per specificare la loro descrizione. Sono presentati, inoltre, alcuni risultati connessi allo studio della famosa domanda tuttora aperta posta da Sakoda e Sipser nel 1978, inerente al costo, in termini di numero di stati, per l\u2019eliminazione del nondeterminismo dagli automi stati finiti sfruttando la capacit\ue0 degli automi two-way deterministici di muovere la testina avanti e indietro sul nastro di input.It is well known that regular \u2014 or type 3 \u2014 languages are equivalent to finite automata. Nevertheless, many other characterizations of this class of languages in terms of computational devices and generative models are present in the literature. For example, by suitably restricting more general models such as context-free grammars, pushdown automata, and Turing machines, that characterize wider classes of languages, it is possible to obtain formal models that generate or recognize regular languages only. The resulting formalisms provide alternative representations of type 3 languages that may be significantly more concise than other models that share the same expressing power. The goal of this work is to investigate these formal systems from a descriptional complexity perspective, or, in other words, to study the relationships between their sizes, namely the number of symbols used to write down their descriptions. We also present some results related to the investigation of the famous question posed by Sakoda and Sipser in 1978, concerning the size blowups from nondeterministic finite automata to two-way deterministic finite automata
Limited automata and context-free languages
Limited automata are one-tape Turing machines which are allowed to rewrite each tape cell only in the first d visits, for a given constant d. For each d 65 2, these devices characterize the class of context-free languages. We investigate the equivalence between 2-limited automata and pushdown automata, comparing the relative sizes of their descriptions. We prove exponential upper and lower bounds for the sizes of pushdown automata simulating 2-limited automata. In the case of the conversion of deterministic 2-limited automata into deterministic pushdown automata the upper bound is double exponential and we conjecture that it cannot be reduced. On the other hand, from pushdown automata we can obtain equivalent 2-limited automata of polynomial size, also preserving determinism. From our results, it follows that the class of languages accepted by deterministic 2-limited automata coincides with the class of deterministic context-free languages