1. of 25 January 7, 2016 CBS: Math, Proofs, Computing 1 Mathematics, Proofs and Computation Madhu Sudan Harvard

2. of 25 Logic, Mathematics, Proofs Reasoning: Reasoning: Start with body of knowledge. Start with body of knowledge. Add to body of knowledge by new observations, and new deductions Add to body of knowledge by new observations, and new deductions Process susceptible to errors: Process susceptible to errors: One erroneous observation may propagate. One erroneous observation may propagate. Constant process of “consistency checking”. Constant process of “consistency checking”. Mathematics = Language of Precision Mathematics = Language of Precision Captures (subset of) knowledge precisely. Captures (subset of) knowledge precisely. Proofs: Enable checking of consistency of precisely stated facts. Proofs: Enable checking of consistency of precisely stated facts. January 7, 2016CBS: Math, Proofs, Computing2

3. of 25 In this talk: Proofs and Computation “Computer Assisted Proofs ?” “Computer Assisted Proofs ?” [Appel-Haken] – 4-color theorem [Appel-Haken] – 4-color theorem [Hales] – Kepler Conjecture [Hales] – Kepler Conjecture [Petkovsky,Wilf,Zeilberger] – “A=B” [Petkovsky,Wilf,Zeilberger] – “A=B” January 7, 2016CBS: Math, Proofs, Computing3 No! Mathematics Computing Proofs

4. of 25 Formal Logic Attempts to convert reasoning to symbolic manipulation. Attempts to convert reasoning to symbolic manipulation. Remarkably powerful. Remarkably powerful. Originated independently, and with different levels of impact, in different civilizations … Originated independently, and with different levels of impact, in different civilizations … January 7, 2016CBS: Math, Proofs, Computing4 "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 Mathematics Proofs

5. of 25 George Boole (1815-1864) January 7, 2016CBS: Math, Proofs, Computing5 (page 34) Mathematics Proofs

6. of 25 Boole’s Mathematics: Boole’s Mathematics: Focus on tiny part of mathematical universe. Focus on tiny part of mathematical universe. January 7, 2016CBS: Math, Proofs, Computing6 {0,1} Progress In Math

7. 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] “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] January 7, 2016CBS: Math, Proofs, Computing7 {0,1}

8. of 25 Whither Computing? How well does the logic capture mathematics? How well does the logic capture mathematics? January 7, 2016CBS: Math, Proofs, Computing8 Cantor‘1890: Logic may face some problems? Hilbert ‘1900: Should capture everything! Godel ‘1920s: Incompleteness Church-Turing 1930s: Incompleteness holds for any effective reasoning procedure. This statement is not true provable

9. of 25 January 7, 2016CBS: Math, Proofs, Computing9 Turing’s Machine Model of computer Model of computer FiniteState Control Control R/W UniversalMachine Encodings of other machines One machine to rule them all! → von Neumann architecture CPU RAM - Universal! Mathematics Computing Proofs

10. of 25 Proofs: Story so far Proof: Has to be mechanically verifiable. Proof: Has to be mechanically verifiable. Theorem: Statement with a proof. Theorem: Statement with a proof. Incompleteness: There exist statements consistent with the system of logic that do not admit a proof. Incompleteness: There exist statements consistent with the system of logic that do not admit a proof. Unaddressed: What difference does proof make? Unaddressed: What difference does proof make? January 7, 2016CBS: Math, Proofs, Computing10 Both mechanically verifiable!

11. of 25 Origins of Modern Complexity January 7, 2016CBS: Math, Proofs, Computing11

12. of 25 Proofs, Complexity & Optimization! January 7, 2016CBS: Math, Proofs, Computing12 [Cook ’70] Complexity of Theorem Proving [Levin ’71] Universal Search problems

13. of 25 Proofs, Complexity & Optimization - 2 January 7, 2016CBS: Math, Proofs, Computing13 [Karp ‘72] Reducibility among combinatorial optimization problems

14. of 25 Some NP-complete Problems Map Coloring: Can you color a given map with 3- colors, s.t. bordering states have diff. colors? Map Coloring: Can you color a given map with 3- colors, s.t. bordering states have diff. colors? January 7, 2016CBS: Math, Proofs, Computing14

15. of 25 Some NP-Complete Problems Travelling Salesman Problem: (TSP) – Find tour of minimum length visiting given set of cities. Travelling Salesman Problem: (TSP) – Find tour of minimum length visiting given set of cities. January 7, 2016CBS: Math, Proofs, Computing15 Image due to [Applegate, Bixby, Chvatal, Cook]. Optimal TSP visiting ~13000 most populated cities in US.

16. of 25 Some NP-Complete Problems Biology: Fold DNA sequence so as to minimize energy. Biology: Fold DNA sequence so as to minimize energy. Economics: Finding optimal portfolio of stocks subject to budget constraint. Economics: Finding optimal portfolio of stocks subject to budget constraint. Industrial Engineering: Schedule tasks subject to precedence constraints to minimize completion time. Industrial Engineering: Schedule tasks subject to precedence constraints to minimize completion time. … January 7, 2016CBS: Math, Proofs, Computing16

17. of 25 Consequences to Proof Checking January 7, 2016CBS: Math, Proofs, Computing17

18. of 25 Don’t know … Don’t know … If P=NP … If P=NP … Cryptography might well be impossible (current systems all broken simultaneously) Cryptography might well be impossible (current systems all broken simultaneously) All optimization problems become “easy” All optimization problems become “easy” … You get whatever you wish … if you can verify satisfaction. … You get whatever you wish … if you can verify satisfaction. Mathematicians replaced by computers. Mathematicians replaced by computers. If P≠NP … If P≠NP … … Consistent with current thinking, so no radical changes. … Consistent with current thinking, so no radical changes. Proof would be very educational. Proof would be very educational. Might provide sound cryptosystems. Might provide sound cryptosystems. Independent of Peano’s axioms, Choice …? Independent of Peano’s axioms, Choice …? January 7, 2016CBS: Math, Proofs, Computing18 Is P=NP? “Of all the Clay Problems, this might be the one to find the shortest solution, by an amateur mathematician.” - Devlin, The Millenium Problems (Possibly thinking P=NP) “If someone shows P=NP, then they prove any theorem they wish. So they would walk away not just with $1M, but $6M by solving all the Clay Problems!” - Lance Fortnow, Complexity Blog “P = NP?” is Mathematics-Complete !!

19. of 25 Post-Modern Complexity Emphasis on Randomness. Emphasis on Randomness. Randomness can potentially speed up algorithms. Randomness can potentially speed up algorithms. Essential for Essential for Equilibrium behavior Equilibrium behavior Coordination among multiple players Coordination among multiple players Cryptography Cryptography But it probably can’t help with Logic – right? But it probably can’t help with Logic – right? Actually – it does!! Actually – it does!! January 7, 2016CBS: Math, Proofs, Computing19

20. of 25 Interactive Proofs January 7, 2016CBS: Math, Proofs, Computing20

21. of 25 Probabilistically Checkable Proofs Do proofs have to be read in entirety to verify? Do proofs have to be read in entirety to verify? January 7, 2016CBS: Math, Proofs, Computing21

22. of 25 Probabilistically Checkable Proofs January 7, 2016CBS: Math, Proofs, Computing22

23. of 25 PCPs and Optimization January 7, 2016CBS: Math, Proofs, Computing23

24. of 25 Summary and Conclusions Computing is a science: Computing is a science: Goes to the very heart of scientific inquiry. Goes to the very heart of scientific inquiry. What big implications follow from local steps? What big implications follow from local steps? Search for proofs captures essence of all search and optimization. Search for proofs captures essence of all search and optimization. “Is P=NP?” Central mathematical question. “Is P=NP?” Central mathematical question. Still open. Still open. But lots of progress … But lots of progress … “Khot’s UGC” (Unique Games Conjecture): Cutting edge of optimization. “Khot’s UGC” (Unique Games Conjecture): Cutting edge of optimization. Khot’s Unique Games Conjecture: Cutting edge of optimization. Sharp thresholds almost everywhere? Khot’s Unique Games Conjecture: Cutting edge of optimization. Sharp thresholds almost everywhere? Communication of Knowledge: Communication of Knowledge: Will we use PCPs to communicate math proofs? Will we use PCPs to communicate math proofs? Will we use PCPs anywhere? Will we use PCPs anywhere? Computing is a science!! Computing is a science!! What global changes can/can not be effected by sequence of local changes? What global changes can/can not be effected by sequence of local changes? January 7, 2016CBS: Math, Proofs, Computing24

25. of 25 Thank You! January 7, 2016CBS: Math, Proofs, Computing25