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Has There Been Progress on the P vs. NP Question? Scott Aaronson

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P vs. NP: I Assume Youve Heard of It Class of decision problems solvable in polynomial time by a deterministic Turing machine P NP Class of decision problems for which a YES answer can be verified in polynomial time (given an appropriate witness) NP-complete

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Importance of the P vs. NP Question NP problems everywhere: machine learning, AI, biology… Gödel (in 1956 letter to von Neumann): If P=NP, then apart from the postulation of axioms, the mental effort of the mathematician could be completely replaced by machine One of the seven Clay Millennium Problems (and far and away the most important)

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Wait … hasnt PNP already been proved? My controversial blog-response (from vacation): If Vinay Deolalikar is awarded the $1,000,000 Clay Millennium Prize for his proof of PNP, then I, Scott Aaronson, will personally supplement his prize by the amount of $200,000. Long Story Short - Heroic rapid response by computer scientists and mathematicians (including Richard Lipton, Neil Immerman, Ryan Williams, Terry Tao, Timothy Gowers…) - Communication mostly via blog comments - After a few days, multiple fatal problems with the proof had emerged - Deolalikar himself hasnt retracted anything, but my condo seems safe…

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How can you tell if a claimed PNP proof is wrong (without spending too much time on it) ? 0. Proof invokes Nazis, Jesus Christ, the authors parents… 1.Lower bound argument for 3SAT would work equally well for problems known to be in P (2SAT, XORSAT) 2.No easier lower bounds proved along the way 3.Known results not encompassed as special cases 4.No coherent plan of attack 5.Proof is nonlinear spaghetti: no hierarchical structure, lemmas arent well-isolated, important concepts used before being defined… 6.Argument runs afoul of one or more formal barriers (about which more later)

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A Depressing Possibility… From the standpoint of P vs. NP, the last 50 years of theoretical computer science have taken us around in circles and been a complete waste of time. Rest of the talk: We might be nowhere close to a proof, but at least the depressing possibility doesnt hold! Weve found (and continue to find) nontrivial insights that will play a role in the solution, assuming there is one.

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Achievement 1: Increased Confidence That P vs. NP Was The Right Question To Ask

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The Unreasonable Robustness of P A half-century of speculation about alternative computational models has taken us only slightly beyond P Would-be P NP provers: dont get discouraged! But cant soap bubbles solve the Minimum Steiner Tree problem in an instant, rendering P vs. NP irrelevant? Spin glasses, folding proteins, DNA computers, analog computers, even probabilistic and quantum computers still seem unable to solve all of NP in polynomial time

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Achievement 2: Half a Century of Experience with Efficient Computation, Increasing Ones Confidence That P NP

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P Dynamic Programming Linear Programming Semidefinite/Convex Programming #P Problems with Miraculous Cancellation Determinant, counting planar perfect matchings, 3- regular-planar-mod-7-SAT… #P Problems with Miraculous Positivity Test Matching, Littlewood- Richardson coefficients… Polynomial Identity Testing (assuming P=BPP) Matrix Group Membership (modulo discrete log) Polynomial Factoring Trivial Problems

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Experimental Complexity Theory We now have a pretty detailed statistical physics understanding of when and where NP-complete problems become hard [Achlioptas, Ricci-Tersenghi 2006] Known heuristic algorithms fail when a large connected cluster of solutions melts into exponentially many disconnected pieces

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Claim: Had we been physicists, we wouldve long ago declared P NP a law of nature When people say: What if P=NP? What if theres an n algorithm for SAT? Or an n logloglog(n) algorithm? Feynman apparently had trouble accepting that P vs. NP was an open problem at all! Response: What if the aliens killed JFK to keep him from discovering that algorithm?

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Achievement 3: Knowing What A Nontrivial Lower Bound Looks Like

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So What Does One Look Like? Time-space tradeoffs for SAT Monotone lower bound for C LIQUE [Razborov] Lower bounds for constant- depth circuits [FSS, Ajtai, RS] Lower bounds on proof complexity n log(n) lower bound on multilinear formula size [Raz] Lower bounds for specific algorithms (DPLL, GSAT…) Bounds on spectral gaps for NP- complete problems [DMV, FGG] Circuit lower bounds for PP, MA EXP, etc. [BFT, Vinodchandran, Santhanam] Circuit lower bounds based on algebraic degree [Strassen, Mulmuley…]

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Metaquestion: Given how short these results fall of proving P NP, can we infer anything from them about what a proof of P NP would look like? Yes! Any proof of P NP will have to contain most of the known lower bounds as special cases Analogy: We dont have a quantum theory of gravity, but the fact that it has to contain the existing theories (QM and GR) as limiting cases constrains it pretty severely This provides another explanation for why P NP is so hard, as well as a criterion for evaluating proposed approaches

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Achievement 4: Formal Barriers

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Relativization [BGS75]: Any proof of P NP (or even much weaker results) will need to use something specific about NP-complete problemssomething that wouldnt be true in a fantasy universe where P and NP machines could both solve PSPACE-complete problems for free Algebrization [AW08]: Any proof of P NP (or even much weaker results) will need to use something specific about NP-complete problems, besides the extendibility to low-degree polynomials used in IP=PSPACE and other famous non-relativizing results Natural Proofs [RR97]: Any proof of P NP will need to use some specific property of NP-complete problems, which couldnt be exploited to certify a random Boolean function as hard (For otherwise, we could turn the proof around and use it to break pseudorandom generatorsthereby solving many of the very same problems we were trying to prove intractable!) The known barriers, in one sentence each

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NP AC 0 [Furst-Saxe-Sipser, Ajtai] NP ACC 0 NP TC 0 NP NC NP P/poly MA EXP P/poly [Buhrman-Fortnow-Thierauf] NEXP P/poly PSPACE P/poly EXP P/poly NP P/poly P EXP [Hartmanis-Stearns] P ARITY AC 0 [FSS, Ajtai] MA EXP P/poly [BFT] RELATIVIZATION NATURAL PROOFS ALGEBRIZATION P NP

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Achievement 5: Connections to Real Math

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The Blum-Cucker-Shub-Smale Model One can define analogues of P and NP over an arbitrary field F When F is finite (e.g., F=F 2 ), we recover the usual P vs. NP question When F=R or F=C, we get an interesting new question with a mathier feel All three cases (F=F 2, F=R, and F=C) are open, and no implications are known among them But the continuous versions (while ridiculously hard themselves) seem likely to be easier than the discrete version

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Even Simpler: P ERMANENT vs. D ETERMINANT [Valiant 70s]: Given an n n matrix A, suppose you cant write per(A) as det(B), where B is a poly(n) poly(n) matrix of linear combinations of the entries of A. Then AlgNC Alg#P. This is important! It reduces a P vs. NP-like complexity question to algebraic geometrya subject about which there are yellow books.

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Mulmuleys Geometric Complexity Theory (GCT) Program: The String Theory of Computer Science To each (real) complexity class C, one can associate a (real) algebraic variety X C X #P (n) = Orbit closure of the n n Permanent function, under invertible linear transformations of the entries X NC (m) = Orbit closure of the m m Determinant function, for some m=poly(n) Dream: Show that X #P (n) has too little symmetry to be embedded into X NC (m). This would imply AlgNC Alg#P.

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Mulmuleys GCT Program: The String Theory of Computer Science But where do we get any new leverage? Proposal: Exploit the exceptional nature of the Permanent and Determinant functionsthe fact that these functions can be uniquely characterized by their symmetriesto reduce the embeddability problem to a problem in representation theory (Which merely requires a generalization of a generalization of a generalization of the Riemann Hypothesis over finite fields) Indeed, we already knew from Relativization / Algebrization / Natural Proofs that wed have to exploit some special properties of the Permanent and Determinant, besides their being low-degree polynomials

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Metaquestion: Why should P NP be provable at all? Indeed, people have speculated since the 70s about its possible independence from set theorysee [A.03] If P NP is a universal statement about mathematics, then why shouldnt the proof require an infinite number of mathematical ideas? More concretely: if the proof needs to know that M ATCHING is in P, L INEAR P ROGRAMMING is in P, etc., what doesnt it need to know is in P? GCT suggests one possible answer: the proof would only need to know about exceptional problems in P (e.g., problems characterized by their symmetries)

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Conclusions A proof of P NP might have to be the greatest synthesis of mathematical ideas ever (but dont let that discourage you) Obvious starting point is P ERMANENT vs. D ETERMINANT My prediction: Progress will come not by ignoring the last half-century of complexity theory and starting afresh, but by subsuming the many disparate facts we already know into something terrifyingly bigger If nothing else, this provides a criterion for evaluating proposed P vs. NP attempts, like the one we saw this summer

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