Proof, Computation, & Randomness Kurt Gödel John von Neumann and Theoretical Computer Science Avi Wigderson School of Mathematics Institute for Advanced Study

Fundamental concepts: Proof vs. Truth Gödel’s Incompleteness Theorem Computation and its limitations, computation in nature Efficient Computation Gödel’s letter, and the P vs. NP problem Limits of Knowledge Randomness: Probabilistic algorithms, pseudorandomness, weak random sources Cryptography: Secrets Probabilistic Proofs Game Strategy, Rationality, Through computational complexity glasses

Proof Axioms: self evident statements (eg = 1 ) Deduction rules: simple, local (eg A, B  A  B ) A proof of statement T is a sequence A 1, A 2, …,A k, A k+1 …., A i, …, A n =T, such that A 1, A 2, …., A k are axioms Each subsequent A i is derived from two previous statements by a deduction rule. Truth of axioms implies truth of T Verification: fast&simple mechanical procedure.

Computation Inputs: available data (eg bits, numbers,…) Operations: simple, local (eg x,y  x  y ) A computation of a function f is a sequence x 1, x 2, …,x k, x k+1 …., x i, …, x n =f(x 1,…,x k ), s.t. x 1, x 2, …,x k are inputs Each x i is computed from two previously computed values by a basic operation A finite program specifies which op to apply next

Computation is everywhere Church-Turing Thesis: “Turing machine captures every realistic physical model of computation” Computation: evolution of an environment via repeated application of simple, local rules [von Neumann’s cellular automata] (Almost) all Physics and Biology theories satisfy! -Weather - Proteins in a cell -Magnetization -Ant hills - Fission - Epidemics -Brain - Populations - Burning fire Nature provides ideas/technologies for computing Understanding algorithms is essential to science.

Computation vs. Proof Common feature: Simple, local steps Can every true statement be proven? Gödel’s Incompleteness Theorem [1931]: Some true statements are unprovable! Can every function be computed? Turing’s Undecidability Theorem [1936]: Some functions are not computable! Proof: f(T) = 1 if statement T is true. f(T) = 0 if T is false. A program for f is a complete proof system #

Limits of Knowledge (I) Undecidable Problems Example of input Diophantine equations (  integer solutions?) Bug-free programs (Does P 0 always halt?) Infection (will it die out?) Input: x (integer) Program P 0 (1) if x=1 halt (2) if x is even, set x  x/2 and go to (1) (3) if x is odd, set x  3x+1 and go to (1) X n +Y n =Z n, n>2

Limits of Knowledge (II) Decidable: Solvable in finite time. Which problems are efficiently solvable? [’60s Cobham, Edmonds, Rabin] Efficient:Time grows moderately with data size. P – problems which can be solved efficiently [’70s Cook, Levin, Karp] NP – problems which can be verified efficiently

Efficient Computation & P vs. NP Gödel’s letter to von Neumann [1954]: Theorem f(T,n)=1 if T has a proof of length n Proving f(T,n)=0 otherwise. This problem is decidable! Is it tractable?? Algorithm for f: Try all possible sequences of length n; for each, check (easy) if it proves T. Time complexity: >2 n steps (terribly inefficient) Is there an efficient algorithm (n or n 2 steps)? Can one avoid “brute force” search, when efficient verification exists? This is the P vs. NP question – central to Math & Science!

Limits of Knowledge (II) P – problems for which solutions can be efficiently found. Everything we can know. NP – problems for which solutions can be easily verified. Everything we want to know. Mathematician: Given a statement, find a proof Scientist: Given data on some phenomena, find a theory explaining it. Engineer: Given constraints (size,weight,energy), find a design (bridge, medicine, phone) All expect to know when they succeeded!

Does P=NP ? NP-complete – Hardest problems in NP 1000’s of them across the sciences. -Math Is a given knot is unknotted? -CS Can we satisfy a set of constraints? -Biology Find the shape of protein folding -Physics Determining min energy All equally difficult! If any is easy: P=NP. - Strong evidence that P  NP - Sources of new technologies: Quantum, DNA..

Probabilistic Algorithms - Use independent, unbiased coin tosses … - Perform task with very high probability Are probabilistic algorithms much faster than deterministic ones? YES: Examples all over CS, Statistical Physics, Math, Biology, Operations Research, Economics… No: Theorem [Blum-Micali-Yao, Nisan-Impagliazzo-Wigderson] Assuming (something like) “P≠NP” ! Hard problems  Pseudo-random generators

Randomness Unbiased, independent Probabilisti c algorithms Cryptograph y Game Theory Applications: Analyzed on perfect randomness biased, dependent Reality: Sources of imperfect randomness Stock market fluctuations Sun spots Radioactive decay Extractor Theory

Unreasonable usefulness of hard problems - Cryptography Assumption: Factoring Integers has no efficient algorithm (assumption implies P  NP) Secret communication Digital signatures Electronic cash Zero-knowledge proofs Poker on the telephone E-commerce Everything! Efficient players! Impossible in information- theoretic setting! Can we base cryptography on P  NP? p,qpqpq Easy Hard

The Millionaires’ Problem - Both want to know who is richer - Neither gets any other information Theorem[Yao] Possible if: - Factoring is hard - Players are efficient $B $A

Probabilistic Proof Systems Is a mathematical statement claim true? E.g. claim: “No integers x, y, z, n>2 satisfy x n +y n = z n “ claim: “The map of Africa is 3-colorable” An efficient Verifier V(claim, argument) satisfies: *) If claim is true then V(claim, argument) = TRUE for some argument (in which case claim=theorem, argument=proof) **) If claim is false then V(claim, argument) = FALSE for every argument probabilistic with probability > 99% always Prover

Probabilistically Checkable Proofs (PCPs) claim: The Riemann Hypothesis Prover: (argument) Verifier: (editor/referee/amateur) Verifier’s concern: Is the argument correct? PCPs: Verifier reads 100 (random) bits of the argument. Thm [Arora-Safra, Arora-Lund-Motwani-Sudan-Szegedy]: Every proof can be efficiently transformed to a PCP Refereeing (even by amateurs) in a jiffy! Major application – approximation algorithms

Zero-Knowledge (ZK) proofs [Goldwasser-Micali-Rackoff] claim: The Riemann Hypothesis Prover: (argument) Verifier: (editor/referee/amateur) Prover’s concern: Will Verifier publish first? ZK proofs: argument reveals only correctness! Theorem [Goldreich-Micali-Wigderson]: Every proof can be efficiently transformed to ZK proof assuming Factoring is HARD Major application - cryptography

Game Theory von Neumann 1928: The minimax Theorem von Neumann & Morgenstern 1944: Game Theory Make conflicts in Economics, Politics, War & life, amenable to mathematical analysis Is there always a rational solution to conflicts? zero-sum games: YES [von Neumann 1928] Reasonable theory? YES: computing it is in P strategic games: YES [Nash 1950] Reasonable theory? NO: PPAD-complete Algorithmic Game Theory Agents: efficient algs

Conclusions Efficient computation: A lens through which basic notions get new meaning: - Proof - Game / Strategy - Randomness - Secret / Privacy - Learning - Knowledge - Fault-tolerance - Small-world phenomena Every natural theory must be efficient! P vs NP captures how much we can know!

There is much more I haven’t told you about..