Susskind Black Hole War Pdf
Shtetl Optimized Blog Archive The Scientific Case for PNPOut there in the wider worldOK, OK, among Lubo Motl, and a few others who comment on this blogthere appears to be a widespread opinion that PNP is just a fashionable dogma of the so called experts, something thats no more likely to be true than false. The doubters can even point to at least one accomplished complexity theorist, Dick Lipton, who publicly advocates agnosticism about whether PNP. Of course, not all the doubters reach their doubts the same way. For Lipton, the thinking is probably something like as scientists, we should be rigorously open minded, and constantly question even the most fundamental hypotheses of our field. For the outsiders, the thinking is more like computer scientists are just not very smartcertainly not as smart as real scientistsso the fact that they consider something a fundamental hypothesis provides no information of value. Consider, for example, this comment of Ignacio Mosqueira If there is no proof that means that there is no reason a priori to prefer your arguments over those of Lubos. Expertise is not enough. And the fact that Lubos is difficult to deal with doesnt change that. In my response, I wondered how broadly Ignacio would apply the principle if theres no proof, then theres no reason to prefer any argument over any other one. For example, would he agree with the guy interviewed on Jon Stewart who earnestly explained that, since theres no proof that turning on the LHC will destroy the world, but also no proof that it wont destroy the world, the only rational inference is that theres a 5. John Olivers deadpan response was classic Im not sure thats how probability worksIn a lengthy reply, Lubo bites this bullet with relish and mustard. In physics, he agrees, or even in continuous mathematics that is more physics wise, its possible to have justified beliefs even without proof. For example, he admits to a 9. Susskind Black Hole War Pdf To Excel
Riemann hypothesis is true. But, he goes on, partial evidence in discrete mathematics just cannot exist. Discrete math and computer science, you see, are so arbitrary, manmade, and haphazard that every question is independent of every other no amount of experience can give anyone any idea which way the next question will go. No, Im not kidding. The Theoretical Minimum What You Need To Know To Start Thinking Like a Physicist. This page is designed to support the book, The Theoretical Minimum by Leonard. Stephen William Hawking, CH, CBE, FRS 8. Januar 1942 in Oxford, Oxfordshire, England ist ein britischer theoretischer Physiker und Astrophysiker. Responses to The Scientific Case for PNP Philip White Says Comment 1 March 7th, 2014 at 816 am. Great postI really liked the greenyellow frog. I was very surprised by the amazing twist in the Syria plot, when her access was submitted to the UN treaty about chemical weapons. I considered myself very well. List of All Philadelphia Orchestra Musicians since 1900. Susskind Black Hole War Pdf FreeThats his argument. I couldnt help wondering what about number theory Arent the positive integers a discrete structure
And isnt the Riemann Hypothesis fundamentally about the distribution of primes Or does the Riemann Hypothesis get counted as an honorary physics wise continuous problem because it can also be stated analytically But then what about Goldbachs ConjectureIs Lubo 5. Dell Optiplex Gx280 Bios more. Better yet, what about continuous, analytic problems that are closely related to P vs. NP For example, Valiants Conjecture says you cant linearly embed the permanent of an nn matrix as the determinant of an mm matrix, unless mexpn. Mulmuley and others have connected this continuous cousin of PNP to issues in algebraic geometry, representation theory, and even quantum groups and Langlands duality. So, does that make it kosher The more I thought about the proposed distinction, the less sense it made to me. But enough of this. In the rest of this post, I want to explain why the odds that you should assign to PNP are more like 9. This post supersedes my 2. I hereby retire. While that post was mostly OK as far as it went, I now feel like I can do a much better job articulating the central point. And also, I made the serious mistake in 2. That works great for readers who already know the issues inside and out, and just want to be amused. Alas, it doesnt work so well for readers who dont know the issues, are extremely literal minded, and just want ammunition to prove their starting assumption that Im a doofus who doesnt understand the basics of his own field. So, OK, why should you believe PNPHeres why Because, like any other successful scientific hypothesis, the PNP hypothesis has passed severe tests that it had no good reason to pass were it false. What kind of tests am I talking about By now, tens of thousands of problems have been proved to be NP complete. They range in character from theorem proving to graph coloring to airline scheduling to bin packing to protein folding to auction pricing to VLSI design to minimizing soap films to winning at Super Mario Bros. Meanwhile, another cluster of tens of thousands of problems has been proved to lie in P or BPP. Those range from primality to matching to linear and semidefinite programming to edit distance to polynomial factoring to hundreds of approximation tasks. Like the NP complete problems, many of the P and BPP problems are also related to each other by a rich network of reductions. For example, countless other problems are in P because linear and semidefinite programming are. So, if we were to draw a map of the complexity class NP according to current knowledge, what would it look like Thered be a huge, growing component of NP complete problems, all connected to each other by an intricate network of reductions. Thered be a second huge component of P problems, many of them again connected by reductions. Then, much like with the map of the continental US, thered be a sparser population in the middle stuff like factoring, graph isomorphism, and Unique Games that for various reasons has thus far resisted assimilation onto either of the coasts. Of course, to prove PNP, it would suffice to find a single linkthat is, a single polynomial time equivalencebetween any of the tens of thousands of problems on the P coast, and any of the tens of thousands on the NP complete one. In half a century, this hasnt happened even as theyve both ballooned exponentially, the two giant regions have remained defiantly separate from each other. But thats not even the main point. The main point is that, as people explore these two regions, again and again there are close calls places where, if a single parameter had worked out differently, the two regions would have come together in a cataclysmic collision. Yet every single time, its just a fake out. Again and again the two regions touch, and their border even traces out weird and jagged shapes. But even in those border zones, not a single problem ever crosses from one region to the other. Its as if theyre kept on their respective sides by an invisible electric fence. As an example, consider the Set Cover problem i. S1,Sm1,n, of finding as few subsets as possible whose union equals the whole set. Chvatal showed in 1. This raises an obvious question can you do better What about 0. Alas, building on a long sequence of prior works in PCP theory, it was recentlyshown that, if you could find a covering set at most 1 lnn times larger than the optimum one, then youd be solving an NP complete problem, and P would equal NP. Notice that, conversely, if the hardness result worked for lnn or anything above, then wed also get PNP. So, why do the algorithm and the hardness result happen to meet at exactly lnn, with neither one venturing the tiniest bit beyond Well, we might say, lnn is where the invisible electric fence is for this problem.