Quantum and classical strong direct product theorems and optimal time-space tradeoffs

A strong direct product theorem says that if we want to compute $k$ independent instances of a function, using less than $k$ times the resources needed for one instance, then our overall success probability will be exponentially small in $k$. We establish such theorems for the classical as well as quantum query complexity of the OR-function. This implies slightly weaker direct product results for all total functions. We prove a similar result for quantum communication protocols computing $k$ instances of the disjointness function. Our direct product theorems imply a time-space tradeoff T^2S=\Om{N^3} for sorting $N$ items on a quantum computer, which is optimal up to polylog factors. They also give several tight time-space and communication-space tradeoffs for the problems of Boolean matrix-vector multiplication and matrix multiplication

Unbounded-Error Classical and Quantum Communication Complexity

Since the seminal work of Paturi and Simon \cite[FOCS'84 & JCSS'86]{PS86}, the unbounded-error classical communication complexity of a Boolean function has been studied based on the arrangement of points and hyperplanes. Recently, \cite[ICALP'07]{INRY07} found that the unbounded-error {\em quantum} communication complexity in the {\em one-way communication} model can also be investigated using the arrangement, and showed that it is exactly (without a difference of even one qubit) half of the classical one-way communication complexity. In this paper, we extend the arrangement argument to the {\em two-way} and {\em simultaneous message passing} (SMP) models. As a result, we show similarly tight bounds of the unbounded-error two-way/one-way/SMP quantum/classical communication complexities for {\em any} partial/total Boolean function, implying that all of them are equivalent up to a multiplicative constant of four. Moreover, the arrangement argument is also used to show that the gap between {\em weakly} unbounded-error quantum and classical communication complexities is at most a factor of three.Comment: 11 pages. To appear at Proc. ISAAC 200

Asymmetric exclusion process with next-nearest-neighbor interaction: some comments on traffic flow and a nonequilibrium reentrance transition

We study the steady-state behavior of a driven non-equilibrium lattice gas of hard-core particles with next-nearest-neighbor interaction. We calculate the exact stationary distribution of the periodic system and for a particular line in the phase diagram of the system with open boundaries where particles can enter and leave the system. For repulsive interactions the dynamics can be interpreted as a two-speed model for traffic flow. The exact stationary distribution of the periodic continuous-time system turns out to coincide with that of the asymmetric exclusion process (ASEP) with discrete-time parallel update. However, unlike in the (single-speed) ASEP, the exact flow diagram for the two-speed model resembles in some important features the flow diagram of real traffic. The stationary phase diagram of the open system obtained from Monte Carlo simulations can be understood in terms of a shock moving through the system and an overfeeding effect at the boundaries, thus confirming theoretical predictions of a recently developed general theory of boundary-induced phase transitions. In the case of attractive interaction we observe an unexpected reentrance transition due to boundary effects.Comment: 12 pages, Revtex, 7 figure

Automata for true concurrency properties

We present an automata-theoretic framework for the model checking of true concurrency properties. These are specified in a fixpoint logic, corresponding to history-preserving bisimilarity, capable of describing events in computations and their dependencies. The models of the logic are event structures or any formalism which can be given a causal semantics, like Petri nets. Given a formula and an event structure satisfying suitable regularity conditions we show how to construct a parity tree automaton whose language is non-empty if and only if the event structure satisfies the formula. The automaton, due to the nature of event structure models, is usually infinite. We discuss how it can be quotiented to an equivalent finite automaton, where emptiness can be checked effectively. In order to show the applicability of the approach, we discuss how it instantiates to finite safe Petri nets. As a proof of concept we provide a model checking tool implementing the technique

A genome-wide scan for common alleles affecting risk for autism

Although autism spectrum disorders (ASDs) have a substantial genetic basis, most of the known genetic risk has been traced to rare variants, principally copy number variants (CNVs). To identify common risk variation, the Autism Genome Project (AGP) Consortium genotyped 1558 rigorously defined ASD families for 1 million single-nucleotide polymorphisms (SNPs) and analyzed these SNP genotypes for association with ASD. In one of four primary association analyses, the association signal for marker rs4141463, located within MACROD2, crossed the genome-wide association significance threshold of P < 5 × 10−8. When a smaller replication sample was analyzed, the risk allele at rs4141463 was again over-transmitted; yet, consistent with the winner's curse, its effect size in the replication sample was much smaller; and, for the combined samples, the association signal barely fell below the P < 5 × 10−8 threshold. Exploratory analyses of phenotypic subtypes yielded no significant associations after correction for multiple testing. They did, however, yield strong signals within several genes, KIAA0564, PLD5, POU6F2, ST8SIA2 and TAF1C