17,000 research outputs found
On probabilistic analog automata
We consider probabilistic automata on a general state space and study their
computational power. The model is based on the concept of language recognition
by probabilistic automata due to Rabin and models of analog computation in a
noisy environment suggested by Maass and Orponen, and Maass and Sontag. Our
main result is a generalization of Rabin's reduction theorem that implies that
under very mild conditions, the computational power of the automaton is limited
to regular languages
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
Context-dependent nondeterminism for pushdown automata
AbstractPushdown automata using a limited and unlimited amount of nondeterminism are investigated. Moreover, nondeterministic steps are allowed only within certain contexts, i.e., in configurations that meet particular conditions. The relationships of the accepted language families with closures of the deterministic context-free languages (DCFL) under regular operations are studied. For example, automata with unbounded nondeterminism that have to empty their pushdown store up to the initial symbol in order to make a guess are characterized by the regular closure of DCFL. Automata that additionally have to reenter the initial state are (almost) characterized by the Kleene star closure of the union closure of the prefix-free deterministic context-free languages. Pushdown automata with bounded nondeterminism are characterized by the union closure of DCFL in any of the considered contexts. Proper inclusions between all language classes discussed are shown. Finally, closure properties of these families under AFL operations are investigated
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
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
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
From Finite Automata to Regular Expressions and Back--A Summary on Descriptional Complexity
The equivalence of finite automata and regular expressions dates back to the
seminal paper of Kleene on events in nerve nets and finite automata from 1956.
In the present paper we tour a fragment of the literature and summarize results
on upper and lower bounds on the conversion of finite automata to regular
expressions and vice versa. We also briefly recall the known bounds for the
removal of spontaneous transitions (epsilon-transitions) on non-epsilon-free
nondeterministic devices. Moreover, we report on recent results on the average
case descriptional complexity bounds for the conversion of regular expressions
to finite automata and brand new developments on the state elimination
algorithm that converts finite automata to regular expressions.Comment: In Proceedings AFL 2014, arXiv:1405.527
Formal Languages in Dynamical Systems
We treat here the interrelation between formal languages and those dynamical
systems that can be described by cellular automata (CA). There is a well-known
injective map which identifies any CA-invariant subshift with a central formal
language. However, in the special case of a symbolic dynamics, i.e. where the
CA is just the shift map, one gets a stronger result: the identification map
can be extended to a functor between the categories of symbolic dynamics and
formal languages. This functor additionally maps topological conjugacies
between subshifts to empty-string-limited generalized sequential machines
between languages. If the periodic points form a dense set, a case which arises
in a commonly used notion of chaotic dynamics, then an even more natural map to
assign a formal language to a subshift is offered. This map extends to a
functor, too. The Chomsky hierarchy measuring the complexity of formal
languages can be transferred via either of these functors from formal languages
to symbolic dynamics and proves to be a conjugacy invariant there. In this way
it acquires a dynamical meaning. After reviewing some results of the complexity
of CA-invariant subshifts, special attention is given to a new kind of
invariant subshift: the trapped set, which originates from the theory of
chaotic scattering and for which one can study complexity transitions.Comment: 23 pages, LaTe
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