Probabilistic checking of proofs

Arora, Sanjeev; Safra, Shmuel

doi:10.1145/273865.273901

Cited by 854 publications

(417 citation statements)

References 14 publications

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“…For a long time only a few hardness results for approximation were known. A breakthrough in the study of this fundamental problem was made with the proof of the PCP Theorem in [AS98,ALM98]. The connection between such theorems and hardness of approximation was established in [FGL91].…”

Section: Convergence In Entropymentioning

confidence: 99%

Expander graphs and their applications

Hoory

Linial

2006

Bull. Amer. Math. Soc.

1,501

1,355

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A major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists, since expander graphs, the protagonists of our story, come up in numerous and often surprising contexts in both fields.But, perhaps, we should start with a few words about graphs in general. They are, of course, one of the prime objects of study in Discrete Mathematics. However, graphs are among the most ubiquitous models of both natural and human-made structures. In the natural and social sciences they model relations among species, societies, companies, etc. In computer science, they represent networks of communication, data organization, computational devices as well as the flow of computation, and more. In mathematics, Cayley graphs are useful in Group Theory. Graphs carry a natural metric and are therefore useful in Geometry, and though they are "just" one-dimensional complexes, they are useful in certain parts of Topology, e.g. Knot Theory. In statistical physics, graphs can represent local connections between interacting parts of a system, as well as the dynamics of a physical process on such systems.The study of these models calls, then, for the comprehension of the significant structural properties of the relevant graphs. But are there nontrivial structural properties which are universally important? Expansion of a graph requires that it is simultaneously sparse and highly connected. Expander graphs were first defined by Bassalygo and Pinsker, and their existence first proved by Pinsker in the early '70s. The property of being an expander seems significant in many of these mathematical, computational and physical contexts. It is not surprising that expanders are useful in the design and analysis of communication networks. What is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness. In mathematics, we will encounter e.g. their role in the study of metric embeddings, and in particular in work around the Baum-Connes Conjecture. Expansion is closely related to the convergence rates of Markov Chains, and so they play a key role in the study of Monte-Carlo algorithms in statistical mechanics and in a host of practical computational applications. The list of such interesting and fruitful connections goes on and on with so many applications we will not even be able to mention. This universality of expanders is becoming more evident as more connections are discovered. It transpires that expansion is a fundamental mathematical concept, well deserving to be thoroughly investigated on its own.In hindsight, one reason that expanders are so ubiquitous is that their very definition can be given in at least three languages: combinatorial/geometric, probabilistic and algebraic. Combinatorially, expanders are graphs which are highly connected; to disconnect a large part of the graph, one has to sever many edges. Equivalently, using the geometric notion of isoperimetry, every s...

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Section: Convergence In Entropymentioning

confidence: 99%

Expander graphs and their applications

Hoory

Linial

2006

Bull. Amer. Math. Soc.

1,501

1,355

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“…The PCP theorem provides insights into the ability to verify proofs probabilistically and has spawned many important results [3,2,1]. The relevant consequence of the theorem in this context, is the implied hardness of approximation for some NP-complete problems.…”

Section: Accuracymentioning

confidence: 99%

The Computational Complexity of Density Functional Theory

Whitfield

Schuch

Verstraete

2014

Many-Electron Approaches in Physics, Chemistry and Mathematics

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Density functional theory is a successful branch of numerical simulations of quantum systems. While the foundations are rigorously defined, the universal functional must be approximated resulting in a 'semi'-ab initio approach. The search for improved functionals has resulted in hundreds of functionals and remains an active research area. This chapter is concerned with understanding fundamental limitations of any algorithmic approach to approximating the universal functional. The results based on Hamiltonian complexity presented here are largely based on [20]. In this chapter, we explain the computational complexity of DFT and any other approach to solving electronic structure Hamiltonians. The proof relies on perturbative gadgets widely used in Hamiltonian complexity and we provide an introduction to these techniques using the Schrieffer-Wolff method. Since the difficulty of this problem has been well appreciated before this formalization, practitioners have turned to a host approximate Hamiltonians. By extending the results of [20], we show in DFT, although the introduction of an approximate potential leads to a noninteracting Hamiltonian, it remains, in the worst case, an NP-complete problem.

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“…Probabilistically checkable proofs [1] allow to verify a possibly long proof by querying a small number of its bits. Micali [30] has presented computationally sound proofs, where the verification is not perfect, and the proof can be forged, but it is computationally hard to do.…”

Section: Related Workmentioning

confidence: 99%

Verifiable Computation in Multiparty Protocols with Honest Majority

Laud

Pankova

2014

Provable Security

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Abstract. We present a generic method for turning passively secure protocols into protocols secure against covert attacks. The method adds a post-execution verification phase to the protocol that allows a misbehaving party to escape detection only with negligible probability. The execution phase, after which the computed protocol result is already available for parties, has only negligible overhead added by our method. The checks, based on linear probabilistically checkable proofs, are done in zero-knowledge, thereby preserving the privacy guarantees of the original protocol. Our method is inspired by recent results in verifiable computation, adapting them to multiparty setting and significantly lowering their computational costs for the provers.

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Probabilistic checking of proofs

Cited by 854 publications

References 14 publications

Expander graphs and their applications

Expander graphs and their applications

The Computational Complexity of Density Functional Theory

Verifiable Computation in Multiparty Protocols with Honest Majority

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