Mathematics, Proofs and Computation

Transcription

1 Mathematics, Proofs and Computation Madhu Sudan Harvard December 16, 2016 TIMC: Math, Proofs, Computing 1 of 25

2 Logic, Mathematics, Proofs Reasoning: Start with body of knowledge. Add to body of knowledge by new observations, and new deductions Process susceptible to errors: One erroneous observation may propagate. Constant process of consistency checking. Mathematics = Language of Precision Captures (subset of) knowledge precisely. Proofs: Enable checking of consistency of precisely stated facts. December 16, 2016 TIMC: Math, Proofs, Computing 2 of 25

3 In this talk: Proofs and Computation Computer Assisted Proofs? [Appel-Haken] 4-color theorem [Hales] Kepler Conjecture [Petkovsky,Wilf,Zeilberger] A=B No! Mathematics Proofs Computing December 16, 2016 TIMC: Math, Proofs, Computing 3 of 25

4 Formal Logic Attempts to convert reasoning to symbolic manipulation. Remarkably powerful. Originated independently, and with different levels of impact, in different civilizations Mathematics Proofs "Aristotle Altemps Inv8575" by Copy of Lysippus - Jastrow (2006). Licensed under Public Domain via Commons - December 16, 2016 TIMC: Math, Proofs, Computing 4 of 25

5 George Boole ( ) Mathematics Proofs The strange math of ( 0,1 ;,, ) Typical Derivation: Axiom: Repetition does not add knowledge Formally: xxxx = xx Example: Object is Good and Good Object is Good Consequence: Principle of Contradiction it is impossible for any being to possess a quality and at the same time to not possess it. Proof: xx 2 = xx xx 2 xx = 0 xx xx 1 = 0 xx = 0 or xx 1 xx = 0 xx or xx does not hold (page 34) December 16, 2016 TIMC: Math, Proofs, Computing 5 of 25

6 Whither Computing? How well does the logic capture mathematics? Cantor 1890: Logic may face some problems? Hilbert 1900: Should capture everything! Godel 1920s: Incompleteness This statement Church-Turing 1930s: Incompleteness holds for any effective reasoning procedure. is not provable true December 16, 2016 TIMC: Math, Proofs, Computing 6 of 25

7 Turing s Machine Mathematics Computing Proofs Encodings of other machines Model of computer - Universal! von Neumann architecture Universal Machine Finite State CPU Control R/W RAM One machine to rule them all! December 16, 2016 TIMC: Math, Proofs, Computing 7 of 25

8 Proofs: Story so far Proof: Has to be mechanically verifiable. Theorem: Statement with a proof. Incompleteness: There exist statements consistent with the system of logic that do not admit a proof. Unaddressed: What difference does proof make? Theorem: TT Theorem: TT #steps ~l Proof: Π 1 Π 2 Π 3 Π 4 Π l = TT Both mechanically verifiable! Proof: Has l lines ~2 l December 16, 2016 TIMC: Math, Proofs, Computing 8 of 25

9 Origins of Modern Complexity [Gödel 1956] in letter to von Neumann: Is there a more effective procedure to find proof of length l if one exists? (in l 2 steps? l l 2?) [Cobham, Edmonds, Hartmanis, Stearns 60s]: Time Complexity is a (coarse) measure. 10l 2 = 5l 2! But l 2 > l 1.9. PP problems solvable in time l cc for constant cc Edmonds Conjecture: Travelling Salesman Problem is not solvable in PP December 16, 2016 TIMC: Math, Proofs, Computing 9 of 25

10 Proofs, Complexity & Optimization! [Cook 70] Complexity of Theorem Proving [Levin 71] Universal Search problems Formalized Edmond s Conjecture: NNNN = Problems w. efficiently verifiable solutions NNNN-complete = Hardest problem in NP Theorem-Proving NP-Complete SAT (simple format of proofs) NP-complete Domino tiling NP-Complete Godel s question Is NNNN = PP? December 16, 2016 TIMC: Math, Proofs, Computing 10 of 25

11 Proofs, Complexity & Optimization - 2 [Karp 72] Reducibility among combinatorial optimization problems Showed central importance of NNNN. Nineteen problems NNNN-Complete! Cover optimization, logic, combinatorics, graph theory, chip design. December 16, 2016 TIMC: Math, Proofs, Computing 11 of 25

12 Some NP-complete Problems Map Coloring: Can you color a given map with 3- colors, s.t. bordering states have diff. colors? December 16, 2016 TIMC: Math, Proofs, Computing 12 of 25

13 Some NP-Complete Problems Travelling Salesman Problem: (TSP) Find tour of minimum length visiting given set of cities. Image due to [Applegate, Bixby, Chvatal, Cook]. Optimal TSP visiting ~13000 most populated cities in US. December 16, 2016 TIMC: Math, Proofs, Computing 13 of 25

14 An almost NP-Complete Problem Divisor: Given integers 1 AA, BB, CC < 2 nn does AA have a divisor between BB and CC. [Kilian 90] Simplest NP-complete problem? (for this audience)? Almost =? Under Number-Theory Conjecture Or under randomized reduction. December 16, 2016 TIMC: Math, Proofs, Computing 14 of 25

15 Some NP-Complete Problems Biology: Fold DNA sequence so as to minimize energy. Economics: Finding optimal portfolio of stocks subject to budget constraint. Industrial Engineering: Schedule tasks subject to precedence constraints to minimize completion time. December 16, 2016 TIMC: Math, Proofs, Computing 15 of 25

16 Consequences to Proof Checking NP-Complete problem Format for proofs. 3-coloring is NP-complete exists function ff ff TT, l = Map with l cc regions s.t. TT has proof of length l Map is 3-colorable no proofs of length l Map not 3-colorable Format? Rather than convention proof, can simply give coloring of map! Verifier computes ff(tt, l) and verifies coloring is good Advantage: Error is local (two improperly colored regions) December 16, 2016 TIMC: Math, Proofs, Computing 16 of 25

17 Is P=NP? Don t know If P=NP Of all the Clay Problems, this might be the one to find the shortest solution, by an amateur mathematician. Cryptography might well be impossible (current systems all broken simultaneously) All optimization problems become easy You get whatever you wish if you can verify satisfaction. Mathematicians replaced by computers. - Devlin, The Millenium Problems (Possibly thinking P=NP) If someone shows P=NP, then they prove any theorem they If P NP wish. So they would walk away not just with $1M, but $6M by solving all the Clay Problems! Consistent with current thinking, so no radical changes. Proof would be very educational. Might provide sound cryptosystems. - Lance Fortnow, Complexity Blog Independent P = NP? of Peano s is Mathematics-Complete axioms, Choice!!? December 16, 2016 TIMC: Math, Proofs, Computing 17 of 25

18 Post-Modern Complexity Emphasis on Randomness. Randomness can potentially speed up algorithms. Essential for Equilibrium behavior Coordination among multiple players Cryptography But it probably can t help with Logic right? Actually it does!! December 16, 2016 TIMC: Math, Proofs, Computing 18 of 25

19 Interactive Proofs [Goldwasser, Micali, Rackoff], [Babai] ~1985 Verifier asks questions and Prover responds: Space of questions exponentially large in the length! Prover has to be ready for all! Many striking examples: Pepsi Coke! ( Graphs not isomorphic ) Can prove theorem has no short proof. IP = PSPACE [LFKN, Shamir] Zero Knowledge Protocols Foundations of Secure communication December 16, 2016 TIMC: Math, Proofs, Computing 19 of 25

20 Probabilistically Checkable Proofs Do proofs have to be read in entirety to verify? December 16, 2016 TIMC: Math, Proofs, Computing 20 of 25

21 Probabilistically Checkable Proofs Do proofs have to be read in entirety to verify? Conventional formats for proofs YES! But we can change the format! Format Verification Algorithm Any verifier is ok, provided: If TT has proof of length l in standard system, then VV should accept some proof of length poly(l) If TT has no proofs, then VV should not accept any proof with probability 1 2X.001 PCP Theorem [Arora, Lund, Motwani, Safra, Sudan, Szegedy 92]: A format exists where V reads only constant number of bits of proof! December 16, 2016 TIMC: Math, Proofs, Computing 21 of 25

22 Whither PCPs? Classical NP-completeness: TT, l Graph 3-coloring. PCP step: Graph GG Operator Φ on Polynomials (non-linear) GG 3-colorable poly s pp, qq, rr s.t. Φ pp, qq, rr 0 Key to local checking: Non-zero polynomials are almost always non-zero. not-so-fundamental theorem of algebra December 16, 2016 TIMC: Math, Proofs, Computing 22 of 25

23 PCPs and Optimization Classical connection: [Cook Karp]: Solving optimization problems finding proofs New Connection: [Feige et al., Arora et al.] Solving optimization problems approximately finding nearly valid proofs. Existence of nearly valid proofs Existence of perfectly valid proofs (due to PCPs)! Conclude: Solving (some/many) optimizations approximately is as hard as solving them exactly! 1992-today: PCP-induced revolution in understanding approximability!! December 16, 2016 TIMC: Math, Proofs, Computing 23 of 25

24 Summary and Conclusions Computing is a science: Goes to the very heart of scientific inquiry. What big implications follow from local steps? Search for proofs captures essence of all search and optimization. Is P=NP? Central mathematical question. Still open. But lots of progress Khot s UGC (Unique Games Conjecture): Cutting edge of optimization. December 16, 2016 TIMC: Math, Proofs, Computing 24 of 25

25 Thank You! December 16, 2016 TIMC: Math, Proofs, Computing 25 of 25