Comments on: Distinguished Lecture Series I: Avi Wigderson, “The power and weakness of randomness in computation” http://terrytao.wordpress.com/2008/01/10/distinguished-lecture-series-i-avi-wigderson-the-power-and-weakness-of-randomness-in-computation/ Updates on my research and expository papers, discussion of open problems, and other maths-related topics. By Terence Tao Wed, 14 May 2008 08:39:42 +0000 http://wordpress.org/?v=MU By: Terence Tao http://terrytao.wordpress.com/2008/01/10/distinguished-lecture-series-i-avi-wigderson-the-power-and-weakness-of-randomness-in-computation/#comment-24418 Terence Tao Tue, 15 Jan 2008 17:37:00 +0000 http://terrytao.wordpress.com/2008/01/10/distinguished-lecture-series-i-avi-wigderson-the-power-and-weakness-of-randomness-in-computation/#comment-24418 Dear Anonymous, It is currently not known whether quantum computers are powerful enough to solve NP-complete problems in polynomial time. (I believe the name for this assertion is $latex NP \subset BQP$.) Quantum computing has some aspects of parallelism but it is not as strong as a massively parallel classical computer, due to the sharply limited number of operations one is permitted to perform on a superposition of states. This article by Scott Aaronson (who, incidentally, states the above point in the very title of his blog) has more discussion on this topic: http://scottaaronson.com/blog/?p=266 Dear Anonymous,

It is currently not known whether quantum computers are powerful enough to solve NP-complete problems in polynomial time. (I believe the name for this assertion is NP \subset BQP.) Quantum computing has some aspects of parallelism but it is not as strong as a massively parallel classical computer, due to the sharply limited number of operations one is permitted to perform on a superposition of states.

This article by Scott Aaronson (who, incidentally, states the above point in the very title of his blog) has more discussion on this topic:

http://scottaaronson.com/blog/?p=266

]]> By: Anonymous http://terrytao.wordpress.com/2008/01/10/distinguished-lecture-series-i-avi-wigderson-the-power-and-weakness-of-randomness-in-computation/#comment-24401 Anonymous Tue, 15 Jan 2008 13:46:35 +0000 http://terrytao.wordpress.com/2008/01/10/distinguished-lecture-series-i-avi-wigderson-the-power-and-weakness-of-randomness-in-computation/#comment-24401 i was wondering about the suggestive statement 'creativity cannot be automated' in the context of quantum computation, for example theorem proving. i have heard a characterization of modern computers as built largely on newtonian principles and hence capable of solving problems of newtonian complexity: iterative ode/pde etc. but quantum computers built on quantum principles can process information at the quantum level, hence solve schroedinger eqn at the atomic level across an entire material or simultaneously calculate on all numbers etc. it doesnt seem implausible that a quantum computer might be able to do theorem proving and exhibit 'creativity' in the sense of this blog entry, does this have any philosophical or scientific implications? for instance, if a quantum computer can do theorem proving, then should we not consider theorem proving essentially 'creative'? i was wondering about the suggestive statement ‘creativity cannot be automated’ in the context of quantum computation, for example theorem proving. i have heard a characterization of modern computers as built largely on newtonian principles and hence capable of solving problems of newtonian complexity: iterative ode/pde etc. but quantum computers built on quantum principles can process information at the quantum level, hence solve schroedinger eqn at the atomic level across an entire material or simultaneously calculate on all numbers etc.

it doesnt seem implausible that a quantum computer might be able to do theorem proving and exhibit ‘creativity’ in the sense of this blog entry, does this have any philosophical or scientific implications? for instance, if a quantum computer can do theorem proving, then should we not consider theorem proving essentially ‘creative’?

]]> By: Johan Richter http://terrytao.wordpress.com/2008/01/10/distinguished-lecture-series-i-avi-wigderson-the-power-and-weakness-of-randomness-in-computation/#comment-24095 Johan Richter Sun, 13 Jan 2008 12:04:55 +0000 http://terrytao.wordpress.com/2008/01/10/distinguished-lecture-series-i-avi-wigderson-the-power-and-weakness-of-randomness-in-computation/#comment-24095 There is an explanation of the PCP theorem and its proof in volume 44, number 1 of the Bulletin of the AMS. http://www.ams.org/bull/2007-44-01/S0273-0979-06-01143-8/S0273-0979-06-01143-8.pdf There is an explanation of the PCP theorem and its proof in volume 44, number 1 of the Bulletin of the AMS. http://www.ams.org/bull/2007-44-01/S0273-0979-06-01143-8/S0273-0979-06-01143-8.pdf

]]> By: Anonymous http://terrytao.wordpress.com/2008/01/10/distinguished-lecture-series-i-avi-wigderson-the-power-and-weakness-of-randomness-in-computation/#comment-23934 Anonymous Fri, 11 Jan 2008 22:50:46 +0000 http://terrytao.wordpress.com/2008/01/10/distinguished-lecture-series-i-avi-wigderson-the-power-and-weakness-of-randomness-in-computation/#comment-23934 Thanks. This theorem sounds amazing and I'm going to have to try to learn something about it. (Note: anonymouses 1 and 2 and 3 (me) are all the same person). Thanks. This theorem sounds amazing and I’m going to have to try to learn something about it. (Note: anonymouses 1 and 2 and 3 (me) are all the same person).

]]> By: Terence Tao http://terrytao.wordpress.com/2008/01/10/distinguished-lecture-series-i-avi-wigderson-the-power-and-weakness-of-randomness-in-computation/#comment-23924 Terence Tao Fri, 11 Jan 2008 19:58:41 +0000 http://terrytao.wordpress.com/2008/01/10/distinguished-lecture-series-i-avi-wigderson-the-power-and-weakness-of-randomness-in-computation/#comment-23924 Dear Anonymous 1, I think Avi's point was that based purely on empirical evidence of one can actually do in polynomial time, it would seem that both $latex P \neq NP$ and $latex P \neq BPP$, but once one sees the connection between hardness and pseudorandom number generation one would be led instead to the presumably more correct position that $latex P \neq NP$ and $latex P = BPP$. Dear Anonymous 2, I am not an expert in PCP (apart from a conversation this morning with Avi), but I believe the answer is yes, that after computing N digits of pi and then doing poly(N) more work to encode that computation in a suitable format, that the resulting encoded computation can be verified probabilistically using only O(1) inspections of that code and O(log N) work. An analogy would be with locally testable (and locally decodable) codes. There are error-correcting codes in which one can take an N-bit string and verify quickly (just by looking at O(1) of the bits) whether this string is indeed close to a codeword. In particular, those O(1) queries can be used as a "proof" that the prover possesses a codeword. If I understand correctly, the original proofs of the PCP theorem proceeded by showing that the assertion that one possesses a proof of a statement S can be converted (roughly speaking) after a polynomial amount of work into an assertion that one possesses a codeword of a certain locally testable error-correcting code. (There is a later and simpler proof of Dinur which proceeds in a slightly different manner, combining error correcting codes with expander graphs in a clever way.) Dear Anonymous 1,

I think Avi’s point was that based purely on empirical evidence of one can actually do in polynomial time, it would seem that both P \neq NP and P \neq BPP, but once one sees the connection between hardness and pseudorandom number generation one would be led instead to the presumably more correct position that P \neq NP and P = BPP.

Dear Anonymous 2,

I am not an expert in PCP (apart from a conversation this morning with Avi), but I believe the answer is yes, that after computing N digits of pi and then doing poly(N) more work to encode that computation in a suitable format, that the resulting encoded computation can be verified probabilistically using only O(1) inspections of that code and O(log N) work.

An analogy would be with locally testable (and locally decodable) codes. There are error-correcting codes in which one can take an N-bit string and verify quickly (just by looking at O(1) of the bits) whether this string is indeed close to a codeword. In particular, those O(1) queries can be used as a “proof” that the prover possesses a codeword. If I understand correctly, the original proofs of the PCP theorem proceeded by showing that the assertion that one possesses a proof of a statement S can be converted (roughly speaking) after a polynomial amount of work into an assertion that one possesses a codeword of a certain locally testable error-correcting code. (There is a later and simpler proof of Dinur which proceeds in a slightly different manner, combining error correcting codes with expander graphs in a clever way.)

]]> By: Distinguished Lecture Series II: Avi Wigderson, “Expander graphs - constructions and applications” « What’s new http://terrytao.wordpress.com/2008/01/10/distinguished-lecture-series-i-avi-wigderson-the-power-and-weakness-of-randomness-in-computation/#comment-23915 Distinguished Lecture Series II: Avi Wigderson, “Expander graphs - constructions and applications” « What’s new Fri, 11 Jan 2008 18:31:41 +0000 http://terrytao.wordpress.com/2008/01/10/distinguished-lecture-series-i-avi-wigderson-the-power-and-weakness-of-randomness-in-computation/#comment-23915 [...] with his second lecture “Expander Graphs - Constructions and Applications“. As in the previous lecture, he spent some additional time after the talk on an “encore”, which in this case was [...] [...] with his second lecture “Expander Graphs - Constructions and Applications“. As in the previous lecture, he spent some additional time after the talk on an “encore”, which in this case was [...]

]]> By: Anonymous http://terrytao.wordpress.com/2008/01/10/distinguished-lecture-series-i-avi-wigderson-the-power-and-weakness-of-randomness-in-computation/#comment-23883 Anonymous Fri, 11 Jan 2008 12:08:18 +0000 http://terrytao.wordpress.com/2008/01/10/distinguished-lecture-series-i-avi-wigderson-the-power-and-weakness-of-randomness-in-computation/#comment-23883 Question about PCP theorem: suppose it turns out that the first occurrence of the ten-digit sequence 0123456789 in the decimal expansion of pi starts at the 37 trillionth digit. We could write the statement S: statement S: FIRST_PI('0123456789') = N (where N = 37 trillion) Obviously we can confirm S in poly(N) time by computing all those digits. Does the PCP theorem really say that if we do enough work, we should be able to convince another person of S in O(log N) steps by answering O(1) queries that they send us? Question about PCP theorem: suppose it turns out that the first occurrence of the ten-digit sequence 0123456789 in the decimal expansion of pi starts at the 37 trillionth digit. We could write the statement S:

statement S: FIRST_PI(’0123456789′) = N (where N = 37 trillion)

Obviously we can confirm S in poly(N) time by computing all those digits.

Does the PCP theorem really say that if we do enough work, we should be able to convince another person of S in O(log N) steps by answering O(1) queries that they send us?

]]> By: Anonymous http://terrytao.wordpress.com/2008/01/10/distinguished-lecture-series-i-avi-wigderson-the-power-and-weakness-of-randomness-in-computation/#comment-23877 Anonymous Fri, 11 Jan 2008 11:09:39 +0000 http://terrytao.wordpress.com/2008/01/10/distinguished-lecture-series-i-avi-wigderson-the-power-and-weakness-of-randomness-in-computation/#comment-23877 I didn't realize it was considered surprising that beliefs 1 and 2 can't be simultaneously true. I thought most theorists (certainly anyone who does cryptography) believes P=BPP because otherwise there would be no pseudo-random generators in NP. Russell Impagliazzo has an interesting essay about different levels at which P=NP could be true or false. A summary and link is here: http://weblog.fortnow.com/2004/06/impagliazzos-five-worlds.html I didn’t realize it was considered surprising that beliefs 1 and 2 can’t be simultaneously true. I thought most theorists (certainly anyone who does cryptography) believes P=BPP because otherwise there would be no pseudo-random generators in NP.

Russell Impagliazzo has an interesting essay about different levels at which P=NP could be true or false. A summary and link is here:

http://weblog.fortnow.com/2004/06/impagliazzos-five-worlds.html

]]> By: Terence Tao http://terrytao.wordpress.com/2008/01/10/distinguished-lecture-series-i-avi-wigderson-the-power-and-weakness-of-randomness-in-computation/#comment-23779 Terence Tao Thu, 10 Jan 2008 16:47:16 +0000 http://terrytao.wordpress.com/2008/01/10/distinguished-lecture-series-i-avi-wigderson-the-power-and-weakness-of-randomness-in-computation/#comment-23779 Thanks for the corrections and suggestions! Thanks for the corrections and suggestions!

]]> By: Allen Knutson http://terrytao.wordpress.com/2008/01/10/distinguished-lecture-series-i-avi-wigderson-the-power-and-weakness-of-randomness-in-computation/#comment-23772 Allen Knutson Thu, 10 Jan 2008 15:19:37 +0000 http://terrytao.wordpress.com/2008/01/10/distinguished-lecture-series-i-avi-wigderson-the-power-and-weakness-of-randomness-in-computation/#comment-23772 Maybe you pointed out this out somewhere and I missed it: it seems worth reminding the reader that the idea of using a computationally subtle function to produce bits to then hand off to a randomized algorithm is, indeed, exactly how randomized algorithms are run on computers in practice. Maybe you pointed out this out somewhere and I missed it: it seems worth reminding the reader that the idea of using a computationally subtle function to produce bits to then hand off to a randomized algorithm is, indeed, exactly how randomized algorithms are run on computers in practice.

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