The average amount of information lost in multiplication

We show that if X and Y are integers independently and uniformly distributed in the set {1, ..., N}, then the information lost in forming their product (which is given by the equivocation H(X,Y | XY)), is of order log log N. We also prove two extremal results regarding cases in which X and Y are not necessarily independently or uniformly distributed. First, we note that the information lost in multiplication can of course be 0. We show that the condition H(X,Y | XY) = 0 implies that 2log_2 N - H(X, Y) is of order at least log log N. Furthermore, if X and Y are independent and uniformly distributed on disjoint sets of primes, it is possible to have H(X,Y | XY) = 0 with log_2 N - H(X) and log_2 N - H(Y) each of order at most log log N. Second, we show that however X and Y are distributed, H(X,Y | XY) is of order at most log N/log log N. Furthermore, there are distributions (in which X and Y are independent and uniformly distributed over sets of numbers having only small and distinct prime factors) for which H(X,Y | XY) is of order log log N.Comment: i+11 p

The Linking Probability of Deep Spider-Web Networks

We consider crossbar switching networks with base $b$ (that is, constructed from $b\times b$ crossbar switches), scale $k$ (that is, with $b^k$ inputs, $b^k$ outputs and $b^k$ links between each consecutive pair of stages) and depth $l$ (that is, with $l$ stages). We assume that the crossbars are interconnected according to the spider-web pattern, whereby two diverging paths reconverge only after at least $k$ stages. We assume that each vertex is independently idle with probability $q$, the vacancy probability. We assume that $b\ge 2$ and the vacancy probability $q$ are fixed, and that $k$ and $l = ck$ tend to infinity with ratio a fixed constant $c>1$. We consider the linking probability $Q$ (the probability that there exists at least one idle path between a given idle input and a given idle output). In a previous paper it was shown that if $c\le 2$, then the linking probability $Q$ tends to 0 if $0<q<q_c$ (where $q_c = 1/b^{(c-1)/c}$ is the critical vacancy probability), and tends to $(1-\xi)^2$ (where $\xi$ is the unique solution of the equation $(1-q (1-x))^b=x$ in the range $0<x<1$) if $q_c<q<1$. In this paper we extend this result to all rational $c>1$. This is done by using generating functions and complex-variable techniques to estimate the second moments of various random variables involved in the analysis of the networks.Comment: i+21 p

Fault Tolerance in Cellular Automata at High Fault Rates

A commonly used model for fault-tolerant computation is that of cellular automata. The essential difficulty of fault-tolerant computation is present in the special case of simply remembering a bit in the presence of faults, and that is the case we treat in this paper. We are concerned with the degree (the number of neighboring cells on which the state transition function depends) needed to achieve fault tolerance when the fault rate is high (nearly 1/2). We consider both the traditional transient fault model (where faults occur independently in time and space) and a recently introduced combined fault model which also includes manufacturing faults (which occur independently in space, but which affect cells for all time). We also consider both a purely probabilistic fault model (in which the states of cells are perturbed at exactly the fault rate) and an adversarial model (in which the occurrence of a fault gives control of the state to an omniscient adversary). We show that there are cellular automata that can tolerate a fault rate $1/2 - \xi$ (with $\xi>0$) with degree $O((1/\xi^2)\log(1/\xi))$, even with adversarial combined faults. The simplest such automata are based on infinite regular trees, but our results also apply to other structures (such as hyperbolic tessellations) that contain infinite regular trees. We also obtain a lower bound of $\Omega(1/\xi^2)$, even with purely probabilistic transient faults only