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

Scott Aaronson (MIT)

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**P vs. NP: I Assume You’ve Heard of It**

Frank Wilczek (Physics Nobel 2004) was recently asked: “If you could ask a superintelligent alien one yes-or-no question, what would it be?” His response: “P vs. NP. That basically contains all the other questions, doesn’t it?”

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**A Depressing Possibility…**

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 doesn’t hold! We’ve 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 we’re not at the beginning either.

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**Achievement 1: Increased Confidence That P vs**

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

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**PCP NP-completeness Goal: Lift this box**

What NP-completeness accomplished… SAT

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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 “But can’t soap bubbles solve the Minimum Steiner Tree problem in an instant, rendering P vs. NP irrelevant?” Would-be PNP provers: don’t get discouraged! Likewise for spin glasses, folding proteins, DNA computers, analog computers…

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**More serious challenges to the Polynomial-Time Church-Turing Thesis have also been addressed…**

Randomness: P = BPP under plausible assumptions (indeed, assumptions that will probably have to be proved before PNP) [NW94], [IW97], … Nonuniform Algorithms: P/poly is “almost the same as P,” for P vs. NP purposes [Karp-Lipton 82] Quantum Computing: BQP probably is larger than P. But even NPBQP doesn’t look like a “radically” different conjecture from PNP Quantum Gravity? What little we know is consistent with BQP being the “end of the line” E.g., topological quantum field theories can be simulated in BQP [FKLW’02]

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Achievement 2: Half a Century of Experience with Efficient Computation, Increasing One’s Confidence That PNP

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**P Trivial Problems #P Problems with Miraculous Cancellation**

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

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**Experimental Complexity Theory**

We now have a fairly impressive “statistical physics understanding” of the hardness of NP-complete problems [Achlioptas, Ricci-Tersenghi 2006] Known heuristic CSP 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 would’ve long ago declared PNP a law of nature**

Feynman apparently had trouble accepting that P vs. NP was an open problem at all! When people say: “What if P=NP? What if there’s an n10000 algorithm for SAT? Or an nlogloglog(n) algorithm?” Response: What if the aliens killed JFK to keep him from discovering that algorithm?

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“But couldn’t you have said the same about Linear Programming before Khachiyan, or primality before AKS?” No. In those cases we had plenty of hints about what was coming, from both theory and practice. “But haven’t there been lots of surprises in complexity?”

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**Achievement 3: Knowing What A Nontrivial Lower Bound Looks Like**

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**Can P vs. NP Be Solved By A “Fool’s Mate?”**

(A 5-line observation that everyone somehow missed?) Suppose P=NP. Then clearly PA=NPA for all oracles A. But this is known to be false; hence PNP. Fact [A.-Wigderson ’08]: Given a 3SAT formula , suppose a randomized verifier needs (polylog n) queries to to decide if 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 PNP. Proof: If P=NP, then NEXP=EXP=RG (where RG = Refereed Games), and indeed NEXPA=EXPA=RGA for all oracles A.

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**So What Does A Real Chess Match Look Like?**

Lower bounds on proof complexity Monotone lower bound for Clique [Razborov] Lower bounds for constant-depth circuits [FSS, Ajtai, RS] Circuit lower bounds based on algebraic degree [Strassen, Mulmuley…] Lower bounds for specific algorithms (DPLL, GSAT…) nlog(n) lower bound on multilinear formula size [Raz] Bounds on spectral gaps for NP-complete problems [DMV, FGG] Time-space tradeoffs for SAT Circuit lower bounds for PP, MAEXP, etc. [BFT, Vinodchandran, Santhanam]

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Metaquestion: Given how short these results fall of proving PNP, can we infer anything from them about what a proof of PNP would look like? Yes! Any proof of PNP (or at least of NPP/poly, NPcoNP, etc.) will have to contain most of the known lower bounds as special cases Analogy: We don’t 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 PNP is so hard, as well as a criterion for evaluating proposed approaches

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**Achievement 4: Formal Barriers**

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**The known barriers, in one sentence each**

Relativization [BGS’75]: Any proof of PNP (or even NEXPP/poly, etc.) will need to use something specific about NP- complete problems—something that wouldn’t be true in a fantasy universe where P and NP machines could both solve PSPACE-complete problems for free Algebrization [AW’08]: Any proof of PNP (or even NEXPP/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 [RR’97]: Any proof of NPP/poly (or even NPTC0, etc.) will need to use something specific about NP- complete problems—some property that can’t 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 don’t serious mathematicians ignore all these barriers, and just plunge ahead and tackle hard problems—their minds unpolluted by pessimism? If you like to be unpolluted by pessimism, why are you thinking about P vs. NP?

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**P EXP [Hartmanis-Stearns]**

P NP RELATIVIZATION ALGEBRIZATION MAEXP P/poly [BFT] Parity AC0 [FSS, Ajtai] NATURAL PROOFS NP AC0 [Furst-Saxe-Sipser, Ajtai] NP ACC0 NP TC0 NP NC NP P/poly MAEXP P/poly [Buhrman-Fortnow-Thierauf] NEXP P/poly PSPACE P/poly EXP P/poly NP P/poly

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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=F2), 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=F2, 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: Permanent vs. Determinant**

[Valiant 70’s]: Given an nn matrix A, suppose you can’t write per(A) as det(B), where B is a poly(n)poly(n) matrix of linear combinations of the entries of A. Then AlgNCAlg#P. This is important! It reduces a barrier problem in circuit lower bounds to algebraic geometry—a subject about which there are yellow books.

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**Mulmuley’s GCT Program: The String Theory of Computer Science**

To each (real) complexity class C, one can associate a (real) algebraic variety XC Dream: Show that X#P(n) has “too little symmetry” to be embedded into XNC(m). This would imply AlgNCAlg#P. X#P(n) = “Orbit closure” of the nn Permanent function, under invertible linear transformations of the entries XNC(m) = “Orbit closure” of the mm Determinant function, for some m=poly(n)

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**Mulmuley’s 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 functions—the fact that these functions can be uniquely characterized by their symmetries—to 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 we’d 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 theory—see [A.’03] If P NP is a “universal mathematical statement”, why shouldn’t the proof require an infinite number of mathematical ideas? More concretely: if the proof needs to “know” that Matching is in P, Linear Programming is in P, etc., what doesn’t 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 PNP might have to be the greatest synthesis of mathematical ideas ever But don’t let that discourage you “Obvious” 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

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Open Problems P vs. NP Use GCT (or pieces of it) to prove something new about computation One natural place to look: Polynomial Identity Testing Evade the algebrization and natural proofs barriers, by exploiting additional “magic properties” of NP- and #P- complete problems Beyond the locality of 3SAT and the low degree / self- correctibility of the permanent “Experimental complexity theory”: What else can we do?

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