1. Has There Been Progress on the P vs. NP Question?
Scott Aaronson (MIT)

2. 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?”

3. 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.

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

5. PCP NP-completeness Goal: Lift this box
What NP-completeness accomplished…
SAT

6. 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 PNP provers: don’t get discouraged!
Likewise for spin glasses, folding proteins, DNA computers, analog computers…

7. 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 PNP) [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 NPBQP doesn’t look like a “radically” different conjecture from PNP
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]

8. Achievement 2: Half a Century of Experience with Efficient Computation, Increasing One’s Confidence That PNP

9. 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

10. 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

11. Claim: Had we been physicists, we would’ve long ago declared PNP 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?

12. “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?”

13. Achievement 3: Knowing What A Nontrivial Lower Bound Looks Like

14. 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 PNP.
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 PNP.
Proof: If P=NP, then NEXP=EXP=RG (where RG = Refereed Games), and indeed NEXPA=EXPA=RGA for all oracles A.

15. 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]

16. 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 (or at least of NPP/poly, NPcoNP, 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 PNP is so hard, as well as a criterion for evaluating proposed approaches

17. Achievement 4: Formal Barriers

18. The known barriers, in one sentence each
Relativization [BGS’75]: Any proof of PNP (or even NEXPP/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 PNP (or even NEXP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 [RR’97]: Any proof of NPP/poly (or even NPTC0, 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?

19. 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

20. Achievement 5: Connections to “Real” Math

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=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

22. Even Simpler: Permanent vs. Determinant
[Valiant 70’s]: Given an nn 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 AlgNCAlg#P.
This is important! It reduces a barrier problem in circuit lower bounds to algebraic geometry—a subject about which there are yellow books.

23. 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 AlgNCAlg#P.
X#P(n) = “Orbit closure” of the nn Permanent function, under invertible linear transformations of the entries
XNC(m) = “Orbit closure” of the mm Determinant function, for some m=poly(n)

24. 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

25. 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)

26. Conclusions
A proof of PNP 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

27. 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?