1 / 32

Topic 16

Topic 16. Limits of Computation (optional). Survey!. History of (Theoretical) Computing. Cantor and infinities Bertrand Russell and self-reference Godel and Incompleteness Turing and the Halting Problem. Book Plugs. Logicomix Godel , Escher, and Bach. Godel , Escher, and Bach.

ayala
Download Presentation

Topic 16

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Topic 16 Limits of Computation (optional)

  2. Survey!

  3. History of (Theoretical) Computing • Cantor and infinities • Bertrand Russell and self-reference • Godel and Incompleteness • Turing and the Halting Problem

  4. Book Plugs • Logicomix • Godel, Escher, and Bach

  5. Godel, Escher, and Bach

  6. Cantor and Infinity • Sets invented by George Cantor • Size is defined by cardinality

  7. How many people can you fit in an Infinite Hotel?

  8. Countable • Bijection between the (sub)set of natural numbers and some other set. • {1,2,3}  {-333,135,3.14159…} • {1,2,3,4, …}  {1,-1,2,-2,…}

  9. Are rational numbers countable?

  10. Real Numbers are Uncountable (Diagonalization)

  11. Survey!!!

  12. Russel’s Paradox (Participation)

  13. The Liar Paradox

  14. Aside: Why Proofs? • Knowing for sure. • Understanding the problem • Opening up new ways of thinking • Examples • Euclid’s fifth postulate and projective geometry. • Fermat’s Last Theorem

  15. Russell & Whitehead (Principia Mathematica)

  16. Aside: Why Set Theory? • A foundation for mathematics. • Which allows (in principle) every mathematical statement to be proven

  17. Aside: why do we care about provability? • To get a better understanding of the concepts • To understand its implications • Example: Newton’s calculus • Troll math

  18. Godel’s Incompleteness Theorem • Hilbert’s program • All mathematical statements should be written in a precise language with well-defined rules • Completeness • Consistency • Etc.

  19. Are all mathematical statements* complete and consistent? *from a formal theory whose set of axioms are a recursively enumerable set.

  20. Proof ideas • Construct a bijection between logical statements and natural numbers • Every statement has a unique ID (Godel numbering) • Can construct statements with numbers like “statement S is provable in system” • Make a self-referential statement that says “this statement is unprovable in this system” • Diagonalization

  21. Turing Machines

  22. Theoretical Computation • Automata, Grammars, and Computability (473) • DFA  PDA  Turing Machines • P, NP, NP-hard, NP-Complete

  23. Halting Problem • Can we decide (say ‘yes’ or ‘no’ in finite time) whether any mathematical statement is even provable? • Can a program determine whether any program will halt eventually?

  24. Diagonalization in the Halting Problem

More Related