Mathematics Proofs and Computation Madhu Sudan Harvard January

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

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

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

Formal Logic n n n Attempts to convert reasoning to symbolic manipulation. Remarkably powerful. Originated independently, and with different levels of impact, in different civilizations … Mathematics Proofs "Aristotle Altemps Inv 8575" by Copy of Lysippus - Jastrow (2006). Licensed under Public Domain via Commons - https: //commons. wikimedia. org/wiki/File: Aristotle_Altemps_Inv 8575. jpg#/media/File: Aristotle_Altemps_Inv 8575. jpg January 7, 2016 CBS: Math, Proofs, Computing 4 of 25

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

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

Boole’s “modest” ambition n “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] n n {0, 1} January 7, 2016 CBS: Math, Proofs, Computing 7 of 25

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

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

Proofs: Story so far n n 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? n n January 7, 2016 n Both mechanically verifiable! CBS: Math, Proofs, Computing n 10 of 25

Origins of Modern Complexity n January 7, 2016 CBS: Math, Proofs, Computing 11 of 25

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

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

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

Some NP-Complete Problems n 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 7, 2016 CBS: Math, Proofs, Computing 15 of 25

Some NP-Complete Problems n n 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 7, 2016 CBS: Math, Proofs, Computing 16 of 25

Consequences to Proof Checking n n January 7, 2016 CBS: Math, Proofs, Computing 17 of 25

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

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

Interactive Proofs n January 7, 2016 CBS: Math, Proofs, Computing 20 of 25

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

Probabilistically Checkable Proofs n n January 7, 2016 n CBS: Math, Proofs, Computing 22 of 25

PCPs and Optimization n January 7, 2016 CBS: Math, Proofs, Computing 23 of 25

Summary and Conclusions n Computing is a science: n Goes to the very heart of scientific inquiry. n n n What big implications follow from local steps? Search for proofs captures essence of all search and optimization. “Is P=NP? ” Central mathematical question. n Still open. n But lots of progress … n “Khot’s UGC” (Unique Games Conjecture): Cutting edge of optimization. January 7, 2016 CBS: Math, Proofs, Computing 24 of 25

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