Spielman, Daniel
Overview
Works:  3 works in 3 publications in 1 language and 6 library holdings 

Roles:  Author 
Publication Timeline
.
Most widely held works by
Daniel Spielman
The perceptron strikes back by
R Beigel(
Book
)
1 edition published in 1990 in English and held by 2 WorldCat member libraries worldwide
Abstract: "We show that circuits composed of a symmetric gate at the root with ANDOR subcircuits of constant depth can be simulated by probabilistic depth2 circuits with essentially the same symmetric gate at the root and AND gates of small fanin at the bottom. In particular, every language recognized by a depthd AC⁰ circuit is decidable by a probabilistic perceptron of size 2[superscript O(log 5d n)] and order O(log[superscript 5d] n) that uses O(log³n) probabilistic bits. As a corollary, we present a new proof that depthd ANDOR circuits computing the parity of n binary inputs require size 2[superscript n omega](1/d)]."
1 edition published in 1990 in English and held by 2 WorldCat member libraries worldwide
Abstract: "We show that circuits composed of a symmetric gate at the root with ANDOR subcircuits of constant depth can be simulated by probabilistic depth2 circuits with essentially the same symmetric gate at the root and AND gates of small fanin at the bottom. In particular, every language recognized by a depthd AC⁰ circuit is decidable by a probabilistic perceptron of size 2[superscript O(log 5d n)] and order O(log[superscript 5d] n) that uses O(log³n) probabilistic bits. As a corollary, we present a new proof that depthd ANDOR circuits computing the parity of n binary inputs require size 2[superscript n omega](1/d)]."
PP is closed under intersection by
R Beigel(
Book
)
1 edition published in 1990 in English and held by 2 WorldCat member libraries worldwide
Abstract: "In his seminal paper on probabilistic Turing machines, Gill [Gil77] asked whether the class PP is closed under intersection and union. We give a positive answer to this question. In fact, PP is closed under polynomialtime multilinear reductions. In circuits, this allows us to combine several threshold gates into a single threshold gate, while increasing depth by only a constant. Consequences in complexity theory include definite collapse and plausible separation of certain query hierarchies over PP
1 edition published in 1990 in English and held by 2 WorldCat member libraries worldwide
Abstract: "In his seminal paper on probabilistic Turing machines, Gill [Gil77] asked whether the class PP is closed under intersection and union. We give a positive answer to this question. In fact, PP is closed under polynomialtime multilinear reductions. In circuits, this allows us to combine several threshold gates into a single threshold gate, while increasing depth by only a constant. Consequences in complexity theory include definite collapse and plausible separation of certain query hierarchies over PP
Computationally efficient errorcorrecting codes and holographic proofs by D. A Spielman(
Book
)
1 edition published in 1996 in English and held by 2 WorldCat member libraries worldwide
Abstract: "We present computationally efficient errorcorrecting codes and holographic proofs. Our errorcorrecting codes are asymptotically good and can be encoded and decoded in linear time. Our construction of holographic proofs provide, for every proof of any theorem, a slightly larger 'holographic' proof whose accuracy can be probabilistically checked by an algorithm that only reads a constant number of the bits of the holographic proof and runs in polylogarithmic time (such proofs have also been called 'transparent proofs' and 'probabilistically checkable proofs'). We explain how these constructions are related and how improvements of these constructions should result in a strengthening of this relationship. For every constant r such that 0 <r <1, we construct an infinite family of systematic linear block error correcting codes that have an encoding circuit with a linear number of wires. There is a constant [epsilon]> 0 and a lineartime decoding algorithm for these codes that maps every word of relative distance at most [epsilon] from a codeword to that codeword. The encoding circuits have logarithmic depth. The decoding algorithm can be implemented as a circuit with O(n log n) wires and logarithmic depth. These constructions make use of explicit constructions of expander graphs and superconcentrators. Our constructions of holographic proofs improve on the theorem PCP(log n, 1) = NP, proved by Arora, Lund, Motwani, Sudan, and Szegedy, by providing, for every [epsilon]> 0, constantquery checkable proofs of size O(n[superscript 1 + [epsilon]]). That is, we design a probabilistic poly logarithmic time proof checking algorithm that takes two inputs: a theorem candidate and a proof candidate. After reading a constant number of bits from each input, the proof checker decides whether to accept or reject its inputs. For every rigorous proof of length n of any theorem, there is an easily computable holographic proof of that theorem of size O(n[superscript 1 + [epsilon]]) such that, with probability one, the proof checker will accept the holographic proof and an encoding of the theorem. Conversely, if the proof checker accepts a theorem candidate and a proof candidate with probability greater than onehalf, then the theorem candidate is close to a unique encoding of a true theorem and the proof candidate constitutes a proof of that theorem."
1 edition published in 1996 in English and held by 2 WorldCat member libraries worldwide
Abstract: "We present computationally efficient errorcorrecting codes and holographic proofs. Our errorcorrecting codes are asymptotically good and can be encoded and decoded in linear time. Our construction of holographic proofs provide, for every proof of any theorem, a slightly larger 'holographic' proof whose accuracy can be probabilistically checked by an algorithm that only reads a constant number of the bits of the holographic proof and runs in polylogarithmic time (such proofs have also been called 'transparent proofs' and 'probabilistically checkable proofs'). We explain how these constructions are related and how improvements of these constructions should result in a strengthening of this relationship. For every constant r such that 0 <r <1, we construct an infinite family of systematic linear block error correcting codes that have an encoding circuit with a linear number of wires. There is a constant [epsilon]> 0 and a lineartime decoding algorithm for these codes that maps every word of relative distance at most [epsilon] from a codeword to that codeword. The encoding circuits have logarithmic depth. The decoding algorithm can be implemented as a circuit with O(n log n) wires and logarithmic depth. These constructions make use of explicit constructions of expander graphs and superconcentrators. Our constructions of holographic proofs improve on the theorem PCP(log n, 1) = NP, proved by Arora, Lund, Motwani, Sudan, and Szegedy, by providing, for every [epsilon]> 0, constantquery checkable proofs of size O(n[superscript 1 + [epsilon]]). That is, we design a probabilistic poly logarithmic time proof checking algorithm that takes two inputs: a theorem candidate and a proof candidate. After reading a constant number of bits from each input, the proof checker decides whether to accept or reject its inputs. For every rigorous proof of length n of any theorem, there is an easily computable holographic proof of that theorem of size O(n[superscript 1 + [epsilon]]) such that, with probability one, the proof checker will accept the holographic proof and an encoding of the theorem. Conversely, if the proof checker accepts a theorem candidate and a proof candidate with probability greater than onehalf, then the theorem candidate is close to a unique encoding of a true theorem and the proof candidate constitutes a proof of that theorem."
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 Beigel, Richard Author
 Reingold, Nick
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