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Checking Computations in Polylogarithmic Time
, 1991
"... . 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 no ..."
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Cited by 262 (11 self)
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. 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 errorcorrecting 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)...
The surveyability of long proofs
"... Abstract. The specific characteristics of mathematical argumentation all depend on the centrality that writing has in the practice of mathematics, but blindness to this fact is near universal. What follows concerns just one of those characteristics, justification by proof. There is a prevalent view ..."
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Cited by 2 (0 self)
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Abstract. The specific characteristics of mathematical argumentation all depend on the centrality that writing has in the practice of mathematics, but blindness to this fact is near universal. What follows concerns just one of those characteristics, justification by proof. There is a prevalent view that long proofs pose a problem for the thesis that mathematical knowledge is justified by proof. I argue that there is no such problem: in fact, virtually all the justifications of mathematical knowledge are ‘long proofs’, but because these real justifications are distributed in the written archive of mathematics, proofs remain surveyable, hence good.