24,688 research outputs found
QIP = PSPACE
We prove that the complexity class QIP, which consists of all problems having
quantum interactive proof systems, is contained in PSPACE. This containment is
proved by applying a parallelized form of the matrix multiplicative weights
update method to a class of semidefinite programs that captures the
computational power of quantum interactive proofs. As the containment of PSPACE
in QIP follows immediately from the well-known equality IP = PSPACE, the
equality QIP = PSPACE follows.Comment: 21 pages; v2 includes corrections and minor revision
Some applications of logic to feasibility in higher types
In this paper we demonstrate that the class of basic feasible functionals has
recursion theoretic properties which naturally generalize the corresponding
properties of the class of feasible functions. We also improve the Kapron -
Cook result on mashine representation of basic feasible functionals. Our proofs
are based on essential applications of logic. We introduce a weak fragment of
second order arithmetic with second order variables ranging over functions from
N into N which suitably characterizes basic feasible functionals, and show that
it is a useful tool for investigating the properties of basic feasible
functionals. In particular, we provide an example how one can extract feasible
"programs" from mathematical proofs which use non-feasible functionals (like
second order polynomials)
On monotone circuits with local oracles and clique lower bounds
We investigate monotone circuits with local oracles [K., 2016], i.e.,
circuits containing additional inputs that can perform
unstructured computations on the input string . Let be
the locality of the circuit, a parameter that bounds the combined strength of
the oracle functions , and
be the set of -cliques and the set of complete -partite graphs,
respectively (similarly to [Razborov, 1985]). Our results can be informally
stated as follows.
1. For an appropriate extension of depth- monotone circuits with local
oracles, we show that the size of the smallest circuits separating
(triangles) and (complete bipartite graphs) undergoes two phase
transitions according to .
2. For , arbitrary depth, and , we
prove that the monotone circuit size complexity of separating the sets
and is , under a certain restrictive
assumption on the local oracle gates.
The second result, which concerns monotone circuits with restricted oracles,
extends and provides a matching upper bound for the exponential lower bounds on
the monotone circuit size complexity of -clique obtained by Alon and Boppana
(1987).Comment: Updated acknowledgements and funding informatio
Efficient computation of exact solutions for quantitative model checking
Quantitative model checkers for Markov Decision Processes typically use
finite-precision arithmetic. If all the coefficients in the process are
rational numbers, then the model checking results are rational, and so they can
be computed exactly. However, exact techniques are generally too expensive or
limited in scalability. In this paper we propose a method for obtaining exact
results starting from an approximated solution in finite-precision arithmetic.
The input of the method is a description of a scheduler, which can be obtained
by a model checker using finite precision. Given a scheduler, we show how to
obtain a corresponding basis in a linear-programming problem, in such a way
that the basis is optimal whenever the scheduler attains the worst-case
probability. This correspondence is already known for discounted MDPs, we show
how to apply it in the undiscounted case provided that some preprocessing is
done. Using the correspondence, the linear-programming problem can be solved in
exact arithmetic starting from the basis obtained. As a consequence, the method
finds the worst-case probability even if the scheduler provided by the model
checker was not optimal. In our experiments, the calculation of exact solutions
from a candidate scheduler is significantly faster than the calculation using
the simplex method under exact arithmetic starting from a default basis.Comment: In Proceedings QAPL 2012, arXiv:1207.055
Chasing diagrams in cryptography
Cryptography is a theory of secret functions. Category theory is a general
theory of functions. Cryptography has reached a stage where its structures
often take several pages to define, and its formulas sometimes run from page to
page. Category theory has some complicated definitions as well, but one of its
specialties is taming the flood of structure. Cryptography seems to be in need
of high level methods, whereas category theory always needs concrete
applications. So why is there no categorical cryptography? One reason may be
that the foundations of modern cryptography are built from probabilistic
polynomial-time Turing machines, and category theory does not have a good
handle on such things. On the other hand, such foundational problems might be
the very reason why cryptographic constructions often resemble low level
machine programming. I present some preliminary explorations towards
categorical cryptography. It turns out that some of the main security concepts
are easily characterized through the categorical technique of *diagram
chasing*, which was first used Lambek's seminal `Lecture Notes on Rings and
Modules'.Comment: 17 pages, 4 figures; to appear in: 'Categories in Logic, Language and
Physics. Festschrift on the occasion of Jim Lambek's 90th birthday', Claudia
Casadio, Bob Coecke, Michael Moortgat, and Philip Scott (editors); this
version: fixed typos found by kind reader
Privacy-Preserving Outsourcing of Large-Scale Nonlinear Programming to the Cloud
The increasing massive data generated by various sources has given birth to
big data analytics. Solving large-scale nonlinear programming problems (NLPs)
is one important big data analytics task that has applications in many domains
such as transport and logistics. However, NLPs are usually too computationally
expensive for resource-constrained users. Fortunately, cloud computing provides
an alternative and economical service for resource-constrained users to
outsource their computation tasks to the cloud. However, one major concern with
outsourcing NLPs is the leakage of user's private information contained in NLP
formulations and results. Although much work has been done on
privacy-preserving outsourcing of computation tasks, little attention has been
paid to NLPs. In this paper, we for the first time investigate secure
outsourcing of general large-scale NLPs with nonlinear constraints. A secure
and efficient transformation scheme at the user side is proposed to protect
user's private information; at the cloud side, generalized reduced gradient
method is applied to effectively solve the transformed large-scale NLPs. The
proposed protocol is implemented on a cloud computing testbed. Experimental
evaluations demonstrate that significant time can be saved for users and the
proposed mechanism has the potential for practical use.Comment: Ang Li and Wei Du equally contributed to this work. This work was
done when Wei Du was at the University of Arkansas. 2018 EAI International
Conference on Security and Privacy in Communication Networks (SecureComm
Classes of representable disjoint NP-pairs
For a propositional proof system P we introduce the complexity class of all disjoint -pairs for which the disjointness of the pair is efficiently provable in the proof system P. We exhibit structural properties of proof systems which make canonical -pairs associated with these proof systems hard or complete for . Moreover, we demonstrate that non-equivalent proof systems can have equivalent canonical pairs and that depending on the properties of the proof systems different scenarios for and the reductions between the canonical pairs exist
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