33,350 research outputs found
Computing with and without arbitrary large numbers
In the study of random access machines (RAMs) it has been shown that the
availability of an extra input integer, having no special properties other than
being sufficiently large, is enough to reduce the computational complexity of
some problems. However, this has only been shown so far for specific problems.
We provide a characterization of the power of such extra inputs for general
problems. To do so, we first correct a classical result by Simon and Szegedy
(1992) as well as one by Simon (1981). In the former we show mistakes in the
proof and correct these by an entirely new construction, with no great change
to the results. In the latter, the original proof direction stands with only
minor modifications, but the new results are far stronger than those of Simon
(1981). In both cases, the new constructions provide the theoretical tools
required to characterize the power of arbitrary large numbers.Comment: 12 pages (main text) + 30 pages (appendices), 1 figure. Extended
abstract. The full paper was presented at TAMC 2013. (Reference given is for
the paper version, as it appears in the proceedings.
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
Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
A digital computer is generally believed to be an efficient universal
computing device; that is, it is believed able to simulate any physical
computing device with an increase in computation time of at most a polynomial
factor. This may not be true when quantum mechanics is taken into
consideration. This paper considers factoring integers and finding discrete
logarithms, two problems which are generally thought to be hard on a classical
computer and have been used as the basis of several proposed cryptosystems.
Efficient randomized algorithms are given for these two problems on a
hypothetical quantum computer. These algorithms take a number of steps
polynomial in the input size, e.g., the number of digits of the integer to be
factored.Comment: 28 pages, LaTeX. This is an expanded version of a paper that appeared
in the Proceedings of the 35th Annual Symposium on Foundations of Computer
Science, Santa Fe, NM, Nov. 20--22, 1994. Minor revisions made January, 199
- …