http://learningcomputation.com/blog/2007/04/why-i-blog.html There exist problems, intractable to decide, yet easy to check. Fri, 07 May 2010 22:25:51 -0400 http://wordpress.org/?v=2.9.2 hourly 1 http://learningcomputation.com/blog/2007/04/why-i-blog.html/comment-page-1#comment-56 Kurt Tue, 28 Aug 2007 03:42:59 +0000 http://learningcomputation.wordpress.com/2007/04/25/why-i-blog#comment-56 Andy, Thanks so much for that! That was much easier to understand than anything else I've seen trying to explain what the PCP theorem is.Not to put any pressure on you or anything, but you know that if you wrote that up and expanded on it a bit that would make a heck of a blog post. Andy, Thanks so much for that! That was much easier to understand than anything else I’ve seen trying to explain what the PCP theorem is.Not to put any pressure on you or anything, but you know that if you wrote that up and expanded on it a bit that would make a heck of a blog post.

]]> http://learningcomputation.com/blog/2007/04/why-i-blog.html/comment-page-1#comment-57 Andy D Tue, 28 Aug 2007 00:29:54 +0000 http://learningcomputation.wordpress.com/2007/04/25/why-i-blog#comment-57 Hi Kurt! Just happened upon your blog/post, thanks for the mention... I'd be happy to discuss PCPs (or theory of comp more generally) with you, to the limited extent of my expertise. Today, just like 15 years ago, the PCP theorem is a hard slog. But it's gotten much less hard with the recent work of Irit Dinur. My exposition was aimed at the part of her work that I found hardest to understand.The *statement* of the PCP theorem, however, is not too complex. It has multiple forms, but here's perhaps the most intuitive one:There exists a constant eps > 0 such that the following is true:Given any language L in NP, there exists a polynomial-time reduction R mapping bitstrings to 3-SAT formulae, such that-if x is in L, then R(x) is a satisfiable formula;-if x isn't in L, then no assignment to the variables of R(x) can satisfy more than a (1 - eps) fraction of the clauses of R(x).What does this result have to do with 'probabilistically checkable proofs'?Suppose we are an all-knowing prover and want to prove to a poly-time verifier that x is in L. We send the verifier a description of a satisfying assignment b to R(x).The verifier picks a random subset of the clauses of R(x), of size (1/eps), and checks that b satisfies these clauses; this takes 3/eps queries, a constant not depending on n = |x| (!!!). The verifier accepts iff each of these 1/eps clauses are satisfied by b.For the analysis:-If x is in L, then R(x) is satisfiable by some b, and sending b causes the verifier to accept with probability 1.-on the other hand, if x isn't in L, then (by the assumed property of the reduction R) any assignment b' to R(x) that we send will fail to satisfy an eps fraction of clauses, so there is a constant nonzero probability that at least one of the 1/eps clauses the verifier chooses will be unsatisfied; hence the verifier rejects with fair probability. This probabilistic, constant-query-complexity protocol for L is called a PCP, and we've sketched how any L in NP has one as a consequence of the NP-hardness of approximating 3-SAT. Hi Kurt! Just happened upon your blog/post, thanks for the mention… I’d be happy to discuss PCPs (or theory of comp more generally) with you, to the limited extent of my expertise. Today, just like 15 years ago, the PCP theorem is a hard slog. But it’s gotten much less hard with the recent work of Irit Dinur. My exposition was aimed at the part of her work that I found hardest to understand.The *statement* of the PCP theorem, however, is not too complex. It has multiple forms, but here’s perhaps the most intuitive one:There exists a constant eps > 0 such that the following is true:Given any language L in NP, there exists a polynomial-time reduction R mapping bitstrings to 3-SAT formulae, such that-if x is in L, then R(x) is a satisfiable formula;-if x isn’t in L, then no assignment to the variables of R(x) can satisfy more than a (1 – eps) fraction of the clauses of R(x).What does this result have to do with ‘probabilistically checkable proofs’?Suppose we are an all-knowing prover and want to prove to a poly-time verifier that x is in L. We send the verifier a description of a satisfying assignment b to R(x).The verifier picks a random subset of the clauses of R(x), of size (1/eps), and checks that b satisfies these clauses; this takes 3/eps queries, a constant not depending on n = |x| (!!!). The verifier accepts iff each of these 1/eps clauses are satisfied by b.For the analysis:-If x is in L, then R(x) is satisfiable by some b, and sending b causes the verifier to accept with probability 1.-on the other hand, if x isn’t in L, then (by the assumed property of the reduction R) any assignment b’ to R(x) that we send will fail to satisfy an eps fraction of clauses, so there is a constant nonzero probability that at least one of the 1/eps clauses the verifier chooses will be unsatisfied; hence the verifier rejects with fair probability. This probabilistic, constant-query-complexity protocol for L is called a PCP, and we’ve sketched how any L in NP has one as a consequence of the NP-hardness of approximating 3-SAT.

]]> http://learningcomputation.com/blog/2007/04/why-i-blog.html/comment-page-1#comment-58 Foxy Thu, 26 Apr 2007 03:43:40 +0000 http://learningcomputation.wordpress.com/2007/04/25/why-i-blog#comment-58 Oh dear ... one of the hardest questions I ever have to answer is why I do math. Let me think about this. Oh dear … one of the hardest questions I ever have to answer is why I do math. Let me think about this.

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