Gödel’s Incompleteness Theorem and the Birth of the Computer

1. Gödel’s Incompleteness Theoremand the Birth of the Computer Christos H. Papadimitriou UC Berkeley

2. 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

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

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

5. 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

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

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

8. 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 (1902-1911) • Result is highly unsatisfactory (but inspires what comes next) CS294, Lecture 1

9. 20 years later: the disaster Gödel 1931 The Incompleteness Theorem “sometimes, we cannot know” Theorems that have no proof CS294, Lecture 1

10. 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

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

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

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

14. “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

15. In a world without Turing… CS294, Lecture 1

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

17. 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 CS294, Lecture 1

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

19. Theory of Computation since Turing:Efficient algorithms • Some problems can be solved in polynomial time (n, n log n, n2, n3, 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

20. Polynomial algorithms deliver Moore’s Law to the world • A 2n algorithm for SAT, run for 1 hour: CS294, Lecture 1

21. NP-completenessCook, 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

22. 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

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25. Recall: Hilbert’s Quest axioms + conjecture always answers “yes/no” Turing’s halting problem CS294, Lecture 1

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

27. 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

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

29. 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

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

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