Consider a process in which information is transmitted from a given root node on a noisy tree network T. We start with an unbiased random bit R at the root of the tree and send it down the edges of T. On every edge the bit can be reversed with probability ε, and these errors occur independently. The goal is to reconstruct R from the values which arrive at the nth level of the tree. This model has been studied in information theory, genetics and statistical mechanics. We bound the reconstruction probability from above, using the maximum flow on T viewed as a capacitated network, and from below using the electrical conductance of T. For general infinite trees, we establish a sharp threshold: the probability of correct reconstruction tends to 1/2 as n → ∞ if 1 − 2ε 2 < p c T , but the reconstruction probability stays bounded away from 1/2 if the opposite inequality holds. Here p c T is the critical probability for percolation on T; in particular p c T = 1/b for the b + 1-regular tree. The asymptotic reconstruction problem is equivalent to purity of the "free boundary" Gibbs state for the Ising model on a tree. The special case of regular trees was solved in 1995 by Bleher, Ruiz and Zagrebnov; our extension to general trees depends on a coupling argument and on a reconstruction algorithm that weights the input bits by the electrical current flow from the root to the leaves.

In recent years there has been an increasing trend toward the incorpor ation of computers into a variety of devices where the amount of memory available is limited. This makes it desirable to try to reduce the size of applications where possible. This article explores the use of compiler techniques to accomplish code compaction to yield smaller executables. The main contribution of this article is to show that careful, aggressive, interprocedural optimization, together with procedural abstraction of repeated code fragments, can yield significantly better reductions in code size than previous approaches, which have generally focused on abstraction of repeated instruction sequences. We also show how “equivalent” code fragments can be detected and factored out using conventional compiler techniques, and without having to resort to purely linear treatments of code sequences as in suffix-tree-based approaches, thereby setting up a framework for code compaction that can be more flexible in its treatment of what code fragments are considered equivalent. Our ideas have been implemented in the form of a binary-rewriting tool that reduces the size of executables by about 30% on the average.

The information carried by a signal decays when the signal is corrupted by random noise. This occurs when a message is transmitted over a noisy channel, as well as when a noisy component performs computation. We first study this signal decay in the context of communication and obtain a tight bound on the rate at which information decreases as a signal crosses a noisy channel. We then use this information theoretic result to obtain depth lower bounds in the noisy circuit model of computation defined by von Neumann. In this model, each component fails (produces 1 instead of 0 or vice-versa) independently with a fixed probability, and yet the output of the circuit is required to be correct with high probability. Von Neumann showed how to construct circuits in this model that reliably compute a function and are no more than a constant factor deeper than noiseless circuits for the function. We provide a lower bound on the multiplicative increase in circuit depth necessary for reliable computation, and an upper bound on the maximum level of noise at which reliable computation is possible. A preliminary version of this work appeared in the first author's thesis [1].

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Abstract-We determine the precise threshold of component noise below which formulas composed of odd degree components can reliably compute all Boolean functions.

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It is shown that if a formula is constructed from noisy 2-input NAND gates, with each gate failing independently with probability ", then reliable computation can or cannot take place according as " is less than or greater than " 0 = (3? p 7)=4 = 0:08856 : : : .