Mathematics, Proofs and Computation Madhu Sudan Harvard January 4, 2016 IIT-Bombay: Math, Proofs, Computing 1 of 25
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. January 4, 2016 IIT-Bombay: Math, Proofs, Computing 2 of 25
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 January 4, 2016 IIT-Bombay: Math, Proofs, Computing 3 of 25
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 - https://commons.wikimedia.org/wiki/file:aristotle_altemps_inv8575.jpg#/media/file:aristotle_altemps_inv8575.jpg January 4, 2016 IIT-Bombay: Math, Proofs, Computing 4 of 25
George Boole (1815-1864) Mathematics Proofs The strange math of ( 0,1 ;,, ) Typical Derivation: Axiom: Repitition 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) January 4, 2016 IIT-Bombay: Math, Proofs, Computing 5 of 25
R, Q, Z vs. {00, 11} Boole s Mathematics: Focus on tiny part of mathematical universe. R: Algebra/Calculus Progress In Math Q: optimization Z: number theory {0,1} January 4, 2016 IIT-Bombay: Math, Proofs, Computing 6 of 25
Boole s modest ambition The design of the following treatise is to investigate the fundamental laws of those operations of the mind by which reasoning is performed; to give expression to them in the symbolical language of a Calculus, and upon this foundation to establish the science of Logic and construct its method; to make that method itself the basis of a general method for the application of the mathematical doctrine of Probabilities; and, finally, to collect from the various elements of truth brought to view in the course of these inquiries some probable intimations concerning the nature and constitution of the human mind. [G.Boole, On the laws of thought p.1] R: Algebra/Calculus Q: optimization Z: number theory Mathematics All of reasoning {0,1} {0,1} January 4, 2016 IIT-Bombay: Math, Proofs, Computing 7 of 25
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 January 4, 2016 IIT-Bombay: Math, Proofs, Computing 8 of 25
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! January 4, 2016 IIT-Bombay: Math, Proofs, Computing 9 of 25
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 January 4, 2016 IIT-Bombay: Math, Proofs, Computing 10 of 25
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 3 + 10l 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 January 4, 2016 IIT-Bombay: Math, Proofs, Computing 11 of 25
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? January 4, 2016 IIT-Bombay: Math, Proofs, Computing 12 of 25
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. January 4, 2016 IIT-Bombay: Math, Proofs, Computing 13 of 25
Some NP-complete Problems Map Coloring: Can you color a given map with 3- colors, s.t. bordering states have diff. colors? January 4, 2016 IIT-Bombay: Math, Proofs, Computing 14 of 25
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. January 4, 2016 IIT-Bombay: Math, Proofs, Computing 15 of 25
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. January 4, 2016 IIT-Bombay: Math, Proofs, Computing 16 of 25
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) January 4, 2016 IIT-Bombay: Math, Proofs, Computing 17 of 25
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!!? January 4, 2016 IIT-Bombay: Math, Proofs, Computing 18 of 25
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!! January 4, 2016 IIT-Bombay: Math, Proofs, Computing 19 of 25
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 January 4, 2016 IIT-Bombay: Math, Proofs, Computing 20 of 25
Probabilistically Checkable Proofs Do proofs have to be read in entirety to verify? January 4, 2016 IIT-Bombay: Math, Proofs, Computing 21 of 25
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! January 4, 2016 IIT-Bombay: Math, Proofs, Computing 22 of 25
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!! January 4, 2016 IIT-Bombay: Math, Proofs, Computing 23 of 25
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. January 4, 2016 IIT-Bombay: Math, Proofs, Computing 24 of 25
Thank You! January 4, 2016 IIT-Bombay: Math, Proofs, Computing 25 of 25