Gödel’s Incompleteness Theorem and the Birth of the Computer Christos H. Papadimitriou UC Berkeley

CS294, Lecture 1 Outline The Foundational Crisis in Math (1900 – 31) How it Led to the Computer (1931 – 46) And to P vs NP (1946 – 72)

CS294, Lecture 1 The prehistory of computation Pascal’s Calculator 1650 Babbage & Ada, 1850 the analytical engine Jacquard’s looms 1805

CS294, Lecture 1 Trouble in Math Non-euclidean geometries Cantor, 1880: sets and infinity ∞

CS294, Lecture 1 Logic! Boole’s logic is inadequate (how do you say “for all integers x”?) Frege introduces First-Order Logic And writes a two-volume opus on the foundations of Arithmetic (~1890 – 1900)

CS294, Lecture 1 The quest for foundations Hilbert, 1900: “We must know, we can know we shall know!”

CS294, Lecture 1 The two quests An axiomatic system that comprises all of Mathematics A machine that finds a proof for every theorem

CS294, Lecture 1 The Paradox and the Book Russell discovers in 1901 the paradox about “the set of all sets that don’t contain themselves” And writes with Whitehead the Principia, trying to restore set theory and logic ( ) Result is highly unsatisfactory (but inspires what comes next)

CS294, Lecture 1 20 years later: the disaster Gödel 1931 The Incompleteness Theorem “sometimes, we cannot know” Theorems that have no proof

CS294, Lecture 1 Recall the two quests Find an axiomatic system that comprises all of Mathematics ? Find a machine that finds a proof for every theorem

CS294, Lecture 1 Also impossible? but what is a machine?

CS294, Lecture 1 The mathematical machines (1934 – 37) Post Kleene Church Turing

CS294, Lecture 1 Universal Turing machine Powerful and crucial idea which anticipates software …and radical too: dedicated machines were favored at the time

CS294, Lecture 1 “If it should turn out that the basic logics of a machine designed for the numerical solution of differential equations coincide with the logics of a machine intended to make bills for a department store, I would regard this as the most amazing coincidence that I have ever encountered” Howard Aiken, 1939

CS294, Lecture 1 In a world without Turing… WELCOME TO THE COMPUTER STORE! First Floor: Web browsers, ers Second Floor: Database engines, Word processors Third Floor: Accounting computers, Business machines Basement: Game engines, Video and Music computers SPECIAL TODAY: All number crunchers 40% off!

CS294, Lecture 1 And finally… von Neumann 1946 EDVAC and report

CS294, Lecture 1 Johnny come lately von Neumann and the Incompleteness Theorem “Turing has done good work on the theories of almost periodic functions and of continuous groups” (1939) Zuse (1936 – 44), Turing (1941 – 52), Atanasoff/Berry (1937 – 42), Aiken (1939 – 45), etc. The meeting at the Aberdeen, MD train station The “logicians” vs the “engineers” at UPenn Eckert, Mauchly, Goldstine, and the First Draft

Madness in their method? the painful human story G. Cantor D. Hilbert K. Gödel E. Post A. M. Turing J. Von Neumann G. Frege

CS294, Lecture 1 Theory of Computation since Turing: Efficient algorithms Some problems can be solved in polynomial time (n, n log n, n 2, n 3, etc.) Others, like the traveling salesman problem and Boolean satisfiability, apparently cannot (because they involve exponential search) Important dichotomy (von Neumann 1952, Edmonds 1965, Cobham 1965, others)

CS294, Lecture 1 Polynomial algorithms deliver Moore’s Law to the world A 2 n algorithm for SAT, run for 1 hour: n = 15n = 23n = 31n = 38n = 45n = 53 An n or n log n algorithmn3n3 n7n7 × 100 every decade× 5× 2

CS294, Lecture 1 NP-completeness Cook, Karp, Levin (1971 – 73) Efficiently solvable problems: P Exponential search: NP Many common problems capture the full power of exponential search: NP-complete Arguably the most influential concept to come out of Computer Science Is P = NP? Fundamental open question

CS294, Lecture 1 Intellectual debt to Gödel/Turing? Negative results are an important intellectual tradition in Computer Science (and Logic too) The Incompleteness Theorem and Turing’s halting problem are the archetypical negative results The Gödel letter (discovered 1992)

CS294, Lecture 1

Recall: Hilbert’s Quest axioms + conjecture always answers “yes/no” Turing’s halting problem

CS294, Lecture 1 Gödel’s revision axioms + conjecture if there is a proof of length n it finds it in time k n (this is trivial, just try all proofs)

CS294, Lecture 1 Hilbert’s last stand Gödel asked von Neumann in the 1956 letter: “Can this be done in time n ? n 2 ? n c ?” This would still mechanize Mathematics…

CS294, Lecture 1 Surprise! Gödel’s question is equivalent to “P = NP” He seems to be optimistic about it…

CS294, Lecture 1 So… Hilbert’s foundations quest and the Incompleteness Theorem have started an intellectual Rube Goldberg that eventually led to the computer Some of the most important concepts in today’s Computer Science, including P vs NP, owe a debt to that tradition

CS294, Lecture 1 And this is the story we tell in…

CS294, Lecture 1 LOGICOMIX: A graphic novel of reason, madness and the birth of the computer By Apostolos Doxiadis and Christos Papadimitriou Art: Alecos Papadatos and Annie Di Donna Bloomsbury, 2008