4,098 research outputs found
A Survey on Continuous Time Computations
We provide an overview of theories of continuous time computation. These
theories allow us to understand both the hardness of questions related to
continuous time dynamical systems and the computational power of continuous
time analog models. We survey the existing models, summarizing results, and
point to relevant references in the literature
Immunity and Simplicity for Exact Counting and Other Counting Classes
Ko [RAIRO 24, 1990] and Bruschi [TCS 102, 1992] showed that in some
relativized world, PSPACE (in fact, ParityP) contains a set that is immune to
the polynomial hierarchy (PH). In this paper, we study and settle the question
of (relativized) separations with immunity for PH and the counting classes PP,
C_{=}P, and ParityP in all possible pairwise combinations. Our main result is
that there is an oracle A relative to which C_{=}P contains a set that is
immune to BPP^{ParityP}. In particular, this C_{=}P^A set is immune to PH^{A}
and ParityP^{A}. Strengthening results of Tor\'{a}n [J.ACM 38, 1991] and Green
[IPL 37, 1991], we also show that, in suitable relativizations, NP contains a
C_{=}P-immune set, and ParityP contains a PP^{PH}-immune set. This implies the
existence of a C_{=}P^{B}-simple set for some oracle B, which extends results
of Balc\'{a}zar et al. [SIAM J.Comp. 14, 1985; RAIRO 22, 1988] and provides the
first example of a simple set in a class not known to be contained in PH. Our
proof technique requires a circuit lower bound for ``exact counting'' that is
derived from Razborov's [Mat. Zametki 41, 1987] lower bound for majority.Comment: 20 page
Different Approaches to Proof Systems
The classical approach to proof complexity perceives proof systems as deterministic, uniform, surjective, polynomial-time computable functions that map strings to (propositional) tautologies. This approach has been intensively studied since the late 70’s and a lot of progress has been made. During the last years research was started investigating alternative notions of proof systems. There are interesting results stemming from dropping the uniformity requirement, allowing oracle access, using quantum computations, or employing probabilism. These lead to different notions of proof systems for which we survey recent results in this paper
Inkdots as advice for finite automata
We examine inkdots placed on the input string as a way of providing advice to
finite automata, and establish the relations between this model and the
previously studied models of advised finite automata. The existence of an
infinite hierarchy of classes of languages that can be recognized with the help
of increasing numbers of inkdots as advice is shown. The effects of different
forms of advice on the succinctness of the advised machines are examined. We
also study randomly placed inkdots as advice to probabilistic finite automata,
and demonstrate the superiority of this model over its deterministic version.
Even very slowly growing amounts of space can become a resource of meaningful
use if the underlying advised model is extended with access to secondary
memory, while it is famously known that such small amounts of space are not
useful for unadvised one-way Turing machines.Comment: 14 page
The RAM equivalent of P vs. RP
One of the fundamental open questions in computational complexity is whether
the class of problems solvable by use of stochasticity under the Random
Polynomial time (RP) model is larger than the class of those solvable in
deterministic polynomial time (P). However, this question is only open for
Turing Machines, not for Random Access Machines (RAMs).
Simon (1981) was able to show that for a sufficiently equipped Random Access
Machine, the ability to switch states nondeterministically does not entail any
computational advantage. However, in the same paper, Simon describes a
different (and arguably more natural) scenario for stochasticity under the RAM
model. According to Simon's proposal, instead of receiving a new random bit at
each execution step, the RAM program is able to execute the pseudofunction
, which returns a uniformly distributed random integer in the
range . Whether the ability to allot a random integer in this fashion is
more powerful than the ability to allot a random bit remained an open question
for the last 30 years.
In this paper, we close Simon's open problem, by fully characterising the
class of languages recognisable in polynomial time by each of the RAMs
regarding which the question was posed. We show that for some of these,
stochasticity entails no advantage, but, more interestingly, we show that for
others it does.Comment: 23 page
Quantum counter automata
The question of whether quantum real-time one-counter automata (rtQ1CAs) can
outperform their probabilistic counterparts has been open for more than a
decade. We provide an affirmative answer to this question, by demonstrating a
non-context-free language that can be recognized with perfect soundness by a
rtQ1CA. This is the first demonstration of the superiority of a quantum model
to the corresponding classical one in the real-time case with an error bound
less than 1. We also introduce a generalization of the rtQ1CA, the quantum
one-way one-counter automaton (1Q1CA), and show that they too are superior to
the corresponding family of probabilistic machines. For this purpose, we
provide general definitions of these models that reflect the modern approach to
the definition of quantum finite automata, and point out some problems with
previous results. We identify several remaining open problems.Comment: A revised version. 16 pages. A preliminary version of this paper
appeared as A. C. Cem Say, Abuzer Yakary{\i}lmaz, and \c{S}efika
Y\"{u}zsever. Quantum one-way one-counter automata. In R\={u}si\c{n}\v{s}
Freivalds, editor, Randomized and quantum computation, pages 25--34, 2010
(Satellite workshop of MFCS and CSL 2010
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