128 research outputs found
Bounded-depth Frege complexity of Tseitin formulas for all graphs
We prove that there is a constant K such that Tseitin formulas for a connected graph G requires proofs of size 2tw(G)javax.xml.bind.JAXBElement@531a834b in depth-d Frege systems for [Formula presented], where tw(G) is the treewidth of G. This extends HÃ¥stad's recent lower bound from grid graphs to any graph. Furthermore, we prove tightness of our bound up to a multiplicative constant in the top exponent. Namely, we show that if a Tseitin formula for a graph G has size s, then for all large enough d, it has a depth-d Frege proof of size 2tw(G)javax.xml.bind.JAXBElement@25a4b51fpoly(s). Through this result we settle the question posed by M. Alekhnovich and A. Razborov of showing that the class of Tseitin formulas is quasi-automatizable for resolution
Generic quantum walk using a coin-embedded shift operator
The study of quantum walk processes has been widely divided into two standard
variants, the discrete-time quantum walk (DTQW) and the continuous-time quantum
walk (CTQW). The connection between the two variants has been established by
considering the limiting value of the coin operation parameter in the DTQW, and
the coin degree of freedom was shown to be unnecessary [26]. But the coin
degree of freedom is an additional resource which can be exploited to control
the dynamics of the QW process. In this paper we present a generic quantum walk
model using a quantum coin-embedded unitary shift operation . The
standard version of the DTQW and the CTQW can be conveniently retrieved from
this generic model, retaining the features of the coin degree of freedom in
both variants.Comment: 5 pages, 1 figure, Publishe
Quantum lattice gases and their invariants
The one particle sector of the simplest one dimensional quantum lattice gas
automaton has been observed to simulate both the (relativistic) Dirac and
(nonrelativistic) Schroedinger equations, in different continuum limits. By
analyzing the discrete analogues of plane waves in this sector we find
conserved quantities corresponding to energy and momentum. We show that the
Klein paradox obtains so that in some regimes the model must be considered to
be relativistic and the negative energy modes interpreted as positive energy
modes of antiparticles. With a formally similar approach--the Bethe ansatz--we
find the evolution eigenfunctions in the two particle sector of the quantum
lattice gas automaton and conclude by discussing consequences of these
calculations and their extension to more particles, additional velocities, and
higher dimensions.Comment: 19 pages, plain TeX, 11 PostScript figures included with epsf.tex
(ignore the under/overfull \vbox error messages
Disordered quantum walk-induced localization of a Bose-Einstein condensate
We present an approach to induce localization of a Bose-Einstein condensate
in a one-dimensional lattice under the influence of unitary quantum walk
evolution using disordered quantum coin operation. We introduce a discrete-time
quantum walk model in which the interference effect is modified to diffuse or
strongly localize the probability distribution of the particle by assigning a
different set of coin parameters picked randomly for each step of the walk,
respectively. Spatial localization of the particle/state is explained by
comparing the variance of the probability distribution of the quantum walk in
position space using disordered coin operation to that of the walk using an
identical coin operation for each step. Due to the high degree of control over
quantum coin operation and most of the system parameters, ultracold atoms in an
optical lattice offer opportunities to implement a disordered quantum walk that
is unitary and induces localization. Here we present a scheme to use a
Bose-Einstein condensate that can be evolved to the superposition of its
internal states in an optical lattice and control the dynamics of atoms to
observe localization. This approach can be adopted to any other physical system
in which controlled disordered quantum walk can be implemented.Comment: 6 pages, 4 figures, published versio
Quantum phase transition using quantum walks in an optical lattice
We present an approach using quantum walks (QWs) to redistribute ultracold
atoms in an optical lattice. Different density profiles of atoms can be
obtained by exploiting the controllable properties of QWs, such as the variance
and the probability distribution in position space using quantum coin
parameters and engineered noise. The QW evolves the density profile of atoms in
a superposition of position space resulting in a quadratic speedup of the
process of quantum phase transition. We also discuss implementation in
presently available setups of ultracold atoms in optical lattices.Comment: 7 pages, 8 figure
Free Dirac evolution as a quantum random walk
Any positive-energy state of a free Dirac particle that is initially
highly-localized, evolves in time by spreading at speeds close to the speed of
light. This general phenomenon is explained by the fact that the Dirac
evolution can be approximated arbitrarily closely by a quantum random walk,
where the roles of coin and walker systems are naturally attributed to the spin
and position degrees of freedom of the particle. Initially entangled and
spatially localized spin-position states evolve with asymptotic two-horned
distributions of the position probability, familiar from earlier studies of
quantum walks. For the Dirac particle, the two horns travel apart at close to
the speed of light.Comment: 16 pages, 1 figure. Latex2e fil
On the absence of homogeneous scalar unitary cellular automata
Failure to find homogeneous scalar unitary cellular automata (CA) in one
dimension led to consideration of only ``approximately unitary'' CA---which
motivated our recent proof of a No-go Lemma in one dimension. In this note we
extend the one dimensional result to prove the absence of nontrivial
homogeneous scalar unitary CA on Euclidean lattices in any dimension.Comment: 7 pages, plain TeX, 3 PostScript figures included with epsf.tex
(ignore the under/overfull \vbox error messages); minor changes (including
title wording) in response to referee suggestions, also updated references;
to appear in Phys. Lett.
Abstract Canonical Inference
An abstract framework of canonical inference is used to explore how different
proof orderings induce different variants of saturation and completeness.
Notions like completion, paramodulation, saturation, redundancy elimination,
and rewrite-system reduction are connected to proof orderings. Fairness of
deductive mechanisms is defined in terms of proof orderings, distinguishing
between (ordinary) "fairness," which yields completeness, and "uniform
fairness," which yields saturation.Comment: 28 pages, no figures, to appear in ACM Trans. on Computational Logi
Decoherence on a two-dimensional quantum walk using four- and two-state particle
We study the decoherence effects originating from state flipping and
depolarization for two-dimensional discrete-time quantum walks using four-state
and two-state particles. By quantifying the quantum correlations between the
particle and position degree of freedom and between the two spatial ()
degrees of freedom using measurement induced disturbance (MID), we show that
the two schemes using a two-state particle are more robust against decoherence
than the Grover walk, which uses a four-state particle. We also show that the
symmetries which hold for two-state quantum walks breakdown for the Grover
walk, adding to the various other advantages of using two-state particles over
four-state particles.Comment: 12 pages, 16 figures, In Press, J. Phys. A: Math. Theor. (2013
Premise Selection for Mathematics by Corpus Analysis and Kernel Methods
Smart premise selection is essential when using automated reasoning as a tool
for large-theory formal proof development. A good method for premise selection
in complex mathematical libraries is the application of machine learning to
large corpora of proofs. This work develops learning-based premise selection in
two ways. First, a newly available minimal dependency analysis of existing
high-level formal mathematical proofs is used to build a large knowledge base
of proof dependencies, providing precise data for ATP-based re-verification and
for training premise selection algorithms. Second, a new machine learning
algorithm for premise selection based on kernel methods is proposed and
implemented. To evaluate the impact of both techniques, a benchmark consisting
of 2078 large-theory mathematical problems is constructed,extending the older
MPTP Challenge benchmark. The combined effect of the techniques results in a
50% improvement on the benchmark over the Vampire/SInE state-of-the-art system
for automated reasoning in large theories.Comment: 26 page
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