Approximations by OBDDs and the Variable Ordering Problem

Ordered binary decision diagrams (OBDDs) and their variants are motivated by the need to represent Boolean functions in applications. Research concerning these applications leads also to problems and results interesting from theoretical point of view. In this paper, methods from communication complexity and information theory are combined to prove that the direct storage access function and the inner product function have the following property. They have linear pi-OBDD size for some variable ordering pi and, for most variable orderings pi0 , all functions which approximate them on considerably more than half of the inputs, need exponential pi0-OBDD size. These results have implications for the use of OBDDs in genetic programming

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strong nonrandom pattern in Matlab default rando

Tensor rank-one decomposition of probability tables

We propose a new additive decomposition of probability tables- tensor rank-one decomposition. The basic idea is to decompose a probability table into a series of tables, such that the table that is the sum of the series is equal to the original table. Each table in the series has the same domain as the original table but can be expressed as a product of one-dimensional tables. Entries in tables are allowed to be any real number, i.e. they can be also negative numbers. The possibility of having negative numbers, in contrary to a multiplicative decomposition, opens new possibilities for a compact representation of probability tables. We show that tensor rank-one decomposition can be used to reduce the space and time requirements in probabilistic inference. We provide a closed form solution for minimal tensor rank-one decomposition for some special tables and propose a numerical algorithm that can be used in cases when the closed form solution is not known.

On the Bent Boolean Functions Which Are Symmetric

Bent functions are the boolean functions having the maximal possible Hamming distance from the linear boolean functions. Bent functions were introduced and studied rst by Rothaus (1976). We prove that there are exactly four symmetric bent functions on every even number of variables. These functions are exactly the four symmetric quadratic polynomials of the given number of variables. 1 Introduction It is well-known (see [6]) that for every boolean function of n variables there exists a linear boolean function such that the Hamming distance of these two functions is at most 2 n 1 2 n=2 1 . If n is even then there exist boolean functions such that the Hamming distance of any of them from every linear boolean function is at least 2 n 1 2 n=2 1 . These functions are called bent in [6]. They are naturally dened in [6] through their discrete Fourier transform. If n is odd, the n 1 variable bent function has the distance 2 n 1 2 (n 1)=2 from all the linear boolean functions of ..

randtoolbox: Toolbox for Pseudo and Quasi Random Number Generation and Random Generator Tests

The randtoolbox package provides (1) pseudo random generators - general linear congruential generators, multiple recursive generators and generalized feedback shift register (SF-Mersenne Twister) algorithm and WELL; (2) quasi random generators - the Torus algorithm, the Sobol sequence, the Halton sequence (including the Van der Corput sequence) and (3) some generator tests - the gap test, the serial test, the poker test, see, e.g., Gentle (2003)