P is not equal to NP

SAT is not in P, is true and provable in a simply consistent extension B' of a first order theory B of computing, with a single finite axiom characterizing a universal Turing machine. Therefore, P is not equal to NP, is true and provable in a simply consistent extension B" of B.Comment: In the 2nd printing the proof, in the 1st printing, of theorem 1 is divided into three parts a new lemma 4, a new corollary 8, and the remaining part of the original proof. The 2nd printing contains some simplifications, more explanations, but no error has been correcte

Query-driven learning for predictive analytics of data subspace cardinality

Fundamental to many predictive analytics tasks is the ability to estimate the cardinality (number of data items) of multi-dimensional data subspaces, defined by query selections over datasets. This is crucial for data analysts dealing with, e.g., interactive data subspace explorations, data subspace visualizations, and in query processing optimization. However, in many modern data systems, predictive analytics may be (i) too costly money-wise, e.g., in clouds, (ii) unreliable, e.g., in modern Big Data query engines, where accurate statistics are difficult to obtain/maintain, or (iii) infeasible, e.g., for privacy issues. We contribute a novel, query-driven, function estimation model of analyst-defined data subspace cardinality. The proposed estimation model is highly accurate in terms of prediction and accommodating the well-known selection queries: multi-dimensional range and distance-nearest neighbors (radius) queries. Our function estimation model: (i) quantizes the vectorial query space, by learning the analysts’ access patterns over a data space, (ii) associates query vectors with their corresponding cardinalities of the analyst-defined data subspaces, (iii) abstracts and employs query vectorial similarity to predict the cardinality of an unseen/unexplored data subspace, and (iv) identifies and adapts to possible changes of the query subspaces based on the theory of optimal stopping. The proposed model is decentralized, facilitating the scaling-out of such predictive analytics queries. The research significance of the model lies in that (i) it is an attractive solution when data-driven statistical techniques are undesirable or infeasible, (ii) it offers a scale-out, decentralized training solution, (iii) it is applicable to different selection query types, and (iv) it offers a performance that is superior to that of data-driven approaches

Dynamical Optimal Transport on Discrete Surfaces

We propose a technique for interpolating between probability distributions on discrete surfaces, based on the theory of optimal transport. Unlike previous attempts that use linear programming, our method is based on a dynamical formulation of quadratic optimal transport proposed for flat domains by Benamou and Brenier [2000], adapted to discrete surfaces. Our structure-preserving construction yields a Riemannian metric on the (finite-dimensional) space of probability distributions on a discrete surface, which translates the so-called Otto calculus to discrete language. From a practical perspective, our technique provides a smooth interpolation between distributions on discrete surfaces with less diffusion than state-of-the-art algorithms involving entropic regularization. Beyond interpolation, we show how our discrete notion of optimal transport extends to other tasks, such as distribution-valued Dirichlet problems and time integration of gradient flows