ICT role in 21st century education and it's challenges.
Ā
Has There Been Progress on the P vs. NP Question?,
1. Has There Been Progress on the P vs. NP Question? Scott Aaronson (MIT)
2. P vs. NP : I Assume Youāve Heard of It Frank Wilczek (Physics Nobel 2004) was recently asked: āIf you could ask a superintelligent alien one yes-or-no question, what would it be?ā His response: ā P vs. NP . That basically contains all the other questions, doesnāt it?ā
3. A Depressing Possibilityā¦ From the standpoint of P vs. NP , the last 50 years of complexity theory have taken us around in circles and been a complete waste of time. This talk: We might be nowhere close to a proof, but at least the depressing possibility doesnāt hold! Weāve found (and continue to find) nontrivial insights that will play a role in the solution, assuming there is one. The end is not in sight, but weāre not at the beginning either.
6. The Unreasonable Robustness of P A half-century of speculation about alternative computational models has taken us only slightly beyond P Would-be P ļ¹ NP provers: donāt get discouraged! ā But canāt soap bubbles solve the Minimum Steiner Tree problem in an instant, rendering P vs. NP irrelevant?ā Likewise for spin glasses, folding proteins, DNA computers, analog computersā¦
7. More serious challenges to the Polynomial-Time Church-Turing Thesis have also been addressedā¦ Randomness: P = BPP under plausible assumptions (indeed, assumptions that will probably have to be proved before P ļ¹ NP ) [NW94], [IW97], ā¦ Nonuniform Algorithms: P/poly is āalmost the same as P ,ā for P vs. NP purposes [Karp-Lipton 82] Quantum Computing: BQP probably is larger than P . But even NP ļ BQP doesnāt look like a āradicallyā different conjecture from P ļ¹ NP Quantum Gravity? What little we know is consistent with BQP being the āend of the lineā E.g., topological quantum field theories can be simulated in BQP [FKLWā02]
8. Achievement 2: Half a Century of Experience with Efficient Computation, Increasing Oneās Confidence That P ļ¹ NP
9. P Dynamic Programming Linear Programming Semidefinite/Convex Programming #P Problems with Miraculous Cancellation Determinant, counting planar perfect matchings, 3-regular-planar-mod-7-SATā¦ #P Problems with Miraculous Positivity Test Matching, Littlewood-Richardson coefficientsā¦ Polynomial Identity Testing (assuming P = BPP ) Matrix Group Membership (modulo discrete log) Polynomial Factoring Trivial Problems
10. Experimental Complexity Theory We now have a fairly impressive āstatistical physics understandingā of the hardness of NP -complete problems [Achlioptas, Ricci-Tersenghi 2006] Known heuristic CSP algorithms fail when a large connected cluster of solutions āmeltsā into exponentially many disconnected pieces
11. Claim: Had we been physicists, we wouldāve long ago declared P ļ¹ NP a law of nature When people say: āWhat if P = NP ? What if thereās an n 10000 algorithm for SAT? Or an n logloglog(n) algorithm?ā 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?
12. ā But couldnāt you have said the same about Linear Programming before Khachiyan, or primality before AKS?ā No. In those cases we had plenty of hints about what was coming, from both theory and practice. ā But havenāt there been lots of surprises in complexity?ā
14. Can P vs. NP Be Solved By A āFoolās Mate?ā Fact [A.-Wigderson ā08] : Given a 3SAT formula ļŖ , suppose a randomized verifier needs ļ· (polylog n) queries to ļŖ to decide if ļŖ is satisfiable, even given polylog(n) communication with a competing yes-prover and no-prover (both of whom can exchange private messages not seen by the other prover) . Then P ļ¹ NP . (A 5-line observation that everyone somehow missed?) Proof: If P = NP , then NEXP = EXP = RG (where RG = Refereed Games), and indeed NEXP A = EXP A = RG A for all oracles A. Suppose P = NP . Then clearly P A = NP A for all oracles A. But this is known to be false; hence P ļ¹ NP .
15. So What Does A Real Chess Match Look Like? Time-space tradeoffs for SAT Monotone lower bound for CLIQUE [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, GSATā¦) 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, Mulmuleyā¦]
18. The known barriers, in one sentence each Relativization [BGSā75] : Any proof of P ļ¹ NP (or even NEXP ļ P/poly , etc.) will need to use something specific about NP -complete problemsāsomething that wouldnāt be true in a fantasy universe where P and NP machines could both solve PSPACE -complete problems for free Algebrization [AWā08] : Any proof of P ļ¹ NP (or even NEXP ļ P/poly , etc.) 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 [RRā97] : Any proof of NP ļ P/poly (or even NP ļ TC 0 , etc.) will need to use something specific about NP -complete problemsāsome property that canāt be exploited to efficiently certify a random Boolean function as hard (thereby breaking pseudorandom generators, and doing many of very things we were trying to prove intractable) But donāt serious mathematicians ignore all these barriers, and just plunge ahead and tackle hard problemsātheir minds unpolluted by pessimism? If you like to be unpolluted by pessimism, why are you thinking about P vs. NP ?
21. 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 āmathierā 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 āeasierā than the discrete version
22. Even Simpler: PERMANENT vs. DETERMINANT [Valiant 70ās]: Given an n ļ“ n matrix A, suppose you canāt write per(A) as det(B), where B is a poly(n) ļ“ poly(n) matrix of linear combinations of the entries of A. Then AlgNC ļ¹ Alg#P . This is important! It reduces a barrier problem in circuit lower bounds to algebraic geometryāa subject about which there are yellow books.
23. Mulmuleyās 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) = āOrbit closureā of the n ļ“ n Permanent function, under invertible linear transformations of the entries X NC (m) = āOrbit closureā of the m ļ“ m Determinant function, for some m=poly(n) Dream: Show that X #P (n) has ātoo little symmetryā to be embedded into X NC (m) . This would imply AlgNC ļ¹ Alg#P .
24.
25.
26. Conclusions A proof of P ļ¹ NP might have to be the greatest synthesis of mathematical ideas ever But donāt let that discourage you ā Obviousā starting point is PERMANENT vs. DETERMINANT My falsifiable 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