4,662 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
Quantum, Stochastic, and Pseudo Stochastic Languages with Few States
Stochastic languages are the languages recognized by probabilistic finite
automata (PFAs) with cutpoint over the field of real numbers. More general
computational models over the same field such as generalized finite automata
(GFAs) and quantum finite automata (QFAs) define the same class. In 1963, Rabin
proved the set of stochastic languages to be uncountable presenting a single
2-state PFA over the binary alphabet recognizing uncountably many languages
depending on the cutpoint. In this paper, we show the same result for unary
stochastic languages. Namely, we exhibit a 2-state unary GFA, a 2-state unary
QFA, and a family of 3-state unary PFAs recognizing uncountably many languages;
all these numbers of states are optimal. After this, we completely characterize
the class of languages recognized by 1-state GFAs, which is the only nontrivial
class of languages recognized by 1-state automata. Finally, we consider the
variations of PFAs, QFAs, and GFAs based on the notion of inclusive/exclusive
cutpoint, and present some results on their expressive power.Comment: A new version with new results. Previous version: Arseny M. Shur,
Abuzer Yakaryilmaz: Quantum, Stochastic, and Pseudo Stochastic Languages with
Few States. UCNC 2014: 327-33
Partially Ordered Two-way B\"uchi Automata
We introduce partially ordered two-way B\"uchi automata and characterize
their expressive power in terms of fragments of first-order logic FO[<].
Partially ordered two-way B\"uchi automata are B\"uchi automata which can
change the direction in which the input is processed with the constraint that
whenever a state is left, it is never re-entered again. Nondeterministic
partially ordered two-way B\"uchi automata coincide with the first-order
fragment Sigma2. Our main contribution is that deterministic partially ordered
two-way B\"uchi automata are expressively complete for the first-order fragment
Delta2. As an intermediate step, we show that deterministic partially ordered
two-way B\"uchi automata are effectively closed under Boolean operations.
A small model property yields coNP-completeness of the emptiness problem and
the inclusion problem for deterministic partially ordered two-way B\"uchi
automata.Comment: The results of this paper were presented at CIAA 2010; University of
Stuttgart, Computer Scienc
Implications of quantum automata for contextuality
We construct zero-error quantum finite automata (QFAs) for promise problems
which cannot be solved by bounded-error probabilistic finite automata (PFAs).
Here is a summary of our results:
- There is a promise problem solvable by an exact two-way QFA in exponential
expected time, but not by any bounded-error sublogarithmic space probabilistic
Turing machine (PTM).
- There is a promise problem solvable by an exact two-way QFA in quadratic
expected time, but not by any bounded-error -space PTMs in
polynomial expected time. The same problem can be solvable by a one-way Las
Vegas (or exact two-way) QFA with quantum head in linear (expected) time.
- There is a promise problem solvable by a Las Vegas realtime QFA, but not by
any bounded-error realtime PFA. The same problem can be solvable by an exact
two-way QFA in linear expected time but not by any exact two-way PFA.
- There is a family of promise problems such that each promise problem can be
solvable by a two-state exact realtime QFAs, but, there is no such bound on the
number of states of realtime bounded-error PFAs solving the members this
family.
Our results imply that there exist zero-error quantum computational devices
with a \emph{single qubit} of memory that cannot be simulated by any finite
memory classical computational model. This provides a computational perspective
on results regarding ontological theories of quantum mechanics \cite{Hardy04},
\cite{Montina08}. As a consequence we find that classical automata based
simulation models \cite{Kleinmann11}, \cite{Blasiak13} are not sufficiently
powerful to simulate quantum contextuality. We conclude by highlighting the
interplay between results from automata models and their application to
developing a general framework for quantum contextuality.Comment: 22 page
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