We present a new class of asymptotically good, linear error-correcting codes based upon expander graphs. These codes have linear time sequential decoding algorithms, logarithmic time parallel decoding algorithms with a linear number of processors, and are simple to understand. We present both randomized and explicit constructions for some of these codes. Experimental results demonstrate the extremely good performance of the randomly chosen codes. 1. Introduction We present a new class of error correcting codes derived from expander graphs. These codes have the advantage that they can be decoded very efficiently. That makes them particularly suitable for devices which must decode cheaply, such as compact disk players and remote satellite receivers. We hope that the connection we draw between expander graphs and error correcting codes will stimulate research in both fields. 1.1. Error correcting codes An error correcting code is a mapping from messages to codewords such that the mappi...

. Motivated by Manuel Blum's concept of instance checking, we consider new, very fast and generic mechanisms of checking computations. Our results exploit recent advances in interactive proof protocols [LFKN92], [Sha92], and especially the MIP = NEXP protocol from [BFL91]. We show that every nondeterministic computational task S(x; y), defined as a polynomial time relation between the instance x, representing the input and output combined, and the witness y can be modified to a task S 0 such that: (i) the same instances remain accepted; (ii) each instance/witness pair becomes checkable in polylogarithmic Monte Carlo time; and (iii) a witness satisfying S 0 can be computed in polynomial time from a witness satisfying S. Here the instance and the description of S have to be provided in error-correcting code (since the checker will not notice slight changes). A modification of the MIP proof was required to achieve polynomial time in (iii); the earlier technique yields N O(log log N)...

We present a new class of asymptotically good, linear error-correcting codes. These codes can be both encoded and decoded in linear time. They can also be encoded by logarithmic-depth circuits of linear size and decoded by logarithmic depth circuits of size 0 (n log n). We present both randomized and explicit constructions of these codes.

Machine (a virtual machine model which underlies most of the current Prolog implementations and incorporates crucial optimization techniques) starting from a more abstract EA for Prolog developed by Borger in [Bo1--Bo3]. Q: How do you tailor an EA machine to the abstraction level of an algorithm whose individual steps are complicated algorithms all by themselves? For example, the algorithm may be written in a high level language that allows, say, multiplying integer matrices in one step. A: You model the given algorithm modulo those algorithms needed to perform single steps. In your case, matrix multiplication will be built in as an operation. Q: Coming back to Turing, there could be a good reason for him to speak about computable functions rather than algorithms. We don't really know what algorithms are. A: I agree. Notice, however, that there are different notions of algorithm. On the one hand, an algorithm is an intuitive idea which you have in your head before writing code. Th...

Query answers from on-line databases can easily be corrupted by hackers or malicious database publishers. Thus it is important to provide mechanisms which allow clients to trust the results from on-line queries. Authentic publication is a novel approach which allows untrusted publishers to securely answer queries from clients on behalf of trusted off-line data owners. Publishers validate answers using compact, hard-to-forge verification objects (VOs), which clients can check efficiently. This approach provides greater scalability (by adding more publishers) and better security (on-line publishers don't need to be trusted).

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We survey a major collective accomplishment of the theoretical computer science community on efficiently verifiable proofs. Informally, a formal proof is transparent (or holographic) if it can be verified with large confidence by a small number of spot-checks. Recent work by a large group of researchers has shown that this seemingly paradoxical concept can be formalized and is feasible in a remarkably strong sense; every formal proof in ZF, say, can be rewritten in transparent format (proving the same theorem in a different proof system) without increasing the length of the proof by too much. This result in turn has surprising implications for the intractability of approximate solutions of a wide range of discrete optimization problems, extending the pessimistic predictions of the P-NP theory to approximate solvability. We discuss the main results on transparent proofs and their implications to discrete optimization. We give an account of several links between the two subjects as well ...

foremost recognition of technical contributions to the computing community. The citation of Cook's achievements noted that "Dr. Cook has advanced our understanding of the complexity of computation in a significant and profound way. His seminal paper, The Complexity of Theorem Proving Procedures, presented at the 1971 ACM SIGACT Symposium on the Theory of Computing, laid the foundations for the theory of NP-completeness. The ensuing exploration of the boundaries and nature of the NP-complete class of problems has been one of the most active and important research activities in computer science for the last decade. Cook is well known for his influential results in fundamental areas of computer science. He has made significant contributions to complexity theory, to time-space tradeoffs in computation, and to logics for programming languages. His work is characterized by elegance and insights and has illuminated the very nature of computation." During 1970-1979, Cook did extensive work under grants from the

This paper is a survey of concepts and results related to simple Kolmogorov complexity, prefix complexity and resource-bounded complexity. We also consider a new type of complexity--- statistical complexity closely related to mathematical statistics. Unlike other discoverers of algorithmic complexity, A. N. Kolmogorov's leading motive was developing on its basis a mathematical theory more adequately substantiating applications of probability theory, mathematical statistics and information theory. Kolmogorov wanted to deduce properties of a random object from its complexity characteristics without use of the notion of probability. In the first part of this paper we present several results in this direction. Though the subsequent development of algorithmic complexity and randomness was different, algorithmic complexity has successful applications in a traditional probabilistic framework. In the second part of the paper we consider applications to the estimation of parameters and the definition of Bernoulli sequences. All considerations have finite combinatorial character. 1.

A "Pointer Machine" is many things. Authors who consider referring to this term are invited to read the following note first. 1 Introduction In a 1992 paper by Galil and the author we referred to a "pointer machine " model of computation. A subsequent survey of related literature has produced over twenty references to papers having to do with "pointer machines", naturally containing a large number of cross-references. These papers address a range of subjects that range from the model considered in the above paper to some other ones which are barely comparable. The fact that such different notions have been discussed under the heading of "pointer machines" has produced the regrettable effect that cross references are sometimes found to be misleading. Clearly, it is easy for a reader who does not follow a paper carefully to misinterpret its claims when a term that is so ill-defined is used. This note is an attempt to rectify the situation. We start with a survey of the different notions...