On the Impossibility of Probabilistic Proofs in Relativized Worlds

We initiate the systematic study of probabilistic proofs in relativized worlds, where the goal is to understand, for a given oracle, the possibility of "non-trivial" proof systems for deterministic or nondeterministic computations that make queries to the oracle. This question is intimately related to a recent line of work that seeks to improve the efficiency of probabilistic proofs for computations that use functionalities such as cryptographic hash functions and digital signatures, by instantiating them via constructions that are "friendly" to known constructions of probabilistic proofs. Informally, negative results about probabilistic proofs in relativized worlds provide evidence that this line of work is inherent and, conversely, positive results provide a way to bypass it. We prove several impossibility results for probabilistic proofs relative to natural oracles. Our results provide strong evidence that tailoring certain natural functionalities to known probabilistic proofs is inherent

Distinguishing between inhomogeneous model and ΛCDM\Lambda\textrm{CDM} model with the cosmic age method

Cosmological observables could be used to construct cosmological models, however, a fixed number of observables limited on the light cone is not enough to uniquely determine a certain model. A reconstructed spherically symmetric, inhomogeneous model that share the same angular-diameter-distance-redshift relationship $d_A(z)$ and Hubble parameter $H(z)$ besides $\Lambda\textrm{CDM}$ model (which we call LTB-$\Lambda\textrm{CDM}$ model in this paper), may provide another solution. Cosmic age, which is off the light cone, could be employed to distinguish these two models. We derive the formulae for age calculation with origin conditions. From the data given by 9-year WMAP measurement, we compute the likelihood of the parameters in these two models respectively by using the Distance Prior method and do likelihood analysis by generating Monte Carlo Markov Chain for the purpose of breaking the degeneracy of $\Omega_m$ and $H_0$ (the parameters that we use for calculation). The results yield to be: $t_{\Lambda\textrm{CDM}} =13.76 \pm 0.09 ~\rm Gyr$, $t_{\rm {LTB}-\Lambda\textrm{CDM}} =11.38 \pm 0.15 ~\rm Gyr$, both in $1\sigma$ agreement with the constraint of cosmic age given by metal-deficient stars. The cosmic age method that is set in this paper enables us to distinguish between the inhomogeneous model and $\Lambda\textrm{CDM}$ model.Comment: 10 pages, 2 figures, accepted by Physics Letters B. arXiv admin note: text overlap with arXiv:0911.3852 by other author

High-Dimensional Expanders from Expanders

We present an elementary way to transform an expander graph into a simplicial complex where all high order random walks have a constant spectral gap, i.e., they converge rapidly to the stationary distribution. As an upshot, we obtain new constructions, as well as a natural probabilistic model to sample constant degree high-dimensional expanders. In particular, we show that given an expander graph G, adding self loops to G and taking the tensor product of the modified graph with a high-dimensional expander produces a new high-dimensional expander. Our proof of rapid mixing of high order random walks is based on the decomposable Markov chains framework introduced by [Jerrum et al., 2004]