The computational power and complexity of discrete feedforward neural networks

The number of binary functions that can be defined on a set of L vectors in R^N equals 2^L . For L>N the total number of threshold functions in N-dimensional space grows polynomially (2^N(N-1))while the total number of Boolean functions, definable on N binary inputs, growsexponentially ( 2^2^2), and as N increases a percentage of threshold functions in relation to the total number of Boolean functions - goes to zero. This means that for the realization of a majority of tasks a neural network must possess at least two or three layers. The examples of small computational problems are arithmetic functions, like multiplication, division, addition, exponentiation or comparison and sorting. This article analyses some aspects of two- and more than two layers of threshold and Boolean circuits (feedforward neural nets), connected with their computational power and node, edge and weight complexity

The Connectivity of Boolean Satisfiability: Dichotomies for Formulas and Circuits

For Boolean satisfiability problems, the structure of the solution space is characterized by the solution graph, where the vertices are the solutions, and two solutions are connected iff they differ in exactly one variable. In 2006, Gopalan et al. studied connectivity properties of the solution graph and related complexity issues for CSPs, motivated mainly by research on satisfiability algorithms and the satisfiability threshold. They proved dichotomies for the diameter of connected components and for the complexity of the st-connectivity question, and conjectured a trichotomy for the connectivity question. Recently, we were able to establish the trichotomy [arXiv:1312.4524]. Here, we consider connectivity issues of satisfiability problems defined by Boolean circuits and propositional formulas that use gates, resp. connectives, from a fixed set of Boolean functions. We obtain dichotomies for the diameter and the two connectivity problems: on one side, the diameter is linear in the number of variables, and both problems are in P, while on the other side, the diameter can be exponential, and the problems are PSPACE-complete. For partially quantified formulas, we show an analogous dichotomy.Comment: 20 pages, several improvement

The systematic study of B→πB\to \pi form factors in pQCD approach and its reliability

The study of exclusive B decays in perturbative QCD are complicated by the endpoint problem. In order to perform the perturbative calculation, the Sudakov effects are introduced to regulate the endpoint singularity. We provide a systematic analysis with leading and next-to-leading twist corrections for $B\to\pi$ form factors in pQCD approach. The intrinsic transverse momentum dependence of hadronic wave function and threshold resummation effects are included in pQCD approach. There are two leading twist B meson distribution amplitudes (or generally wave functions) in general. The QCD equations of motion provide important constraints on B meson wave functions. The reliability of pQCD approach in \bpi form factors is discussed. 70% of the result comes from the region $\alpha_s(t)/\pi<0.2$ and 38% comes from the region where the momentum transfer t\geq 1\GeV. The conceptual problems of pQCD approach are discussed in brief. Our conclusion is that pQCD approach in the present form cannot provide a precise prediction for \bpi transition form factors.Comment: 30 pages, latex, some typos corrected, to appear in Nucl. Phys.

Many Hard Examples in Exact Phase Transitions with Application to Generating Hard Satisfiable Instances

This paper first analyzes the resolution complexity of two random CSP models (i.e. Model RB/RD) for which we can establish the existence of phase transitions and identify the threshold points exactly. By encoding CSPs into CNF formulas, it is proved that almost all instances of Model RB/RD have no tree-like resolution proofs of less than exponential size. Thus, we not only introduce new families of CNF formulas hard for resolution, which is a central task of Proof-Complexity theory, but also propose models with both many hard instances and exact phase transitions. Then, the implications of such models are addressed. It is shown both theoretically and experimentally that an application of Model RB/RD might be in the generation of hard satisfiable instances, which is not only of practical importance but also related to some open problems in cryptography such as generating one-way functions. Subsequently, a further theoretical support for the generation method is shown by establishing exponential lower bounds on the complexity of solving random satisfiable and forced satisfiable instances of RB/RD near the threshold. Finally, conclusions are presented, as well as a detailed comparison of Model RB/RD with the Hamiltonian cycle problem and random 3-SAT, which, respectively, exhibit three different kinds of phase transition behavior in NP-complete problems.Comment: 19 pages, corrected mistakes in Theorems 5 and