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The Mathematics of Knowledge, Rationality, & Action

The Mathematics of Knowledge, Rationality, & Action. Courtesy of Prof. Wiebe van der Hoek Computer Science University of Liverpool United Kingdom.

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The Mathematics of Knowledge, Rationality, & Action

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  1. The Mathematics of Knowledge, Rationality, & Action Courtesy of Prof. Wiebe van der Hoek Computer Science University of Liverpool United Kingdom

  2. ``..eventually, when you put enough feelingless calculations together in a huge coordinated organization, you’ll get something that has properties on another level. You can see it .... as a system of tendencies and desires and beliefs and so on’’ Hofstadter, 1981, Scientific American Agents’ Paradigm

  3. sense sense act act goal info sense act Perceive-Reason-Act Cycle (Each Agent) reason

  4. sense sense act act goal info sense act Perceive-Reason-Act Cycle (Multiple Agents) reason

  5. Rationality A B B A A A $1 $4 $4 $3 $1 $5 $2 $4 $7 $3 $6 $5 $1 for A $6 for B

  6. Rationality A B B A A A 1 4 4 3 1 5 2 4 7 3 6 5 $1 for A $6 for B

  7. Rationality A B B A A A 1 4 4 3 1 5 2 4 7 3 6 5 $1 for A $6 for B

  8. Rationality A B B A A A 1 4 4 3 1 5 2 4 7 3 6 5 $1 for A $6 for B

  9. Rationality A B B A A A 1 4 4 3 1 5 2 4 7 3 6 5 $1 for A $6 for B

  10. Rationality A B B A A A 1 4 4 3 1 5 2 4 7 3 6 5 $1 for A $6 for B

  11. Rationality A B B A A A 1 4 4 3 1 5 2 4 7 3 6 5 $1 for A $6 for B

  12. K(f y)  (Kf  Ky) An agent reasons logicallyf is “Bob is standing in the rain”y is “Bob is getting wet” Kf  f An agent only knows true things Kf  KKf Positive introspection ¬Kf  K¬Kf Negative introspection Epistemic Logic Relies on 4 Axioms(f and y are facts)

  13. Multiagent Knowledge KAm: agent A knows m A sends m to B KAm ? KBm ? Em ? KAKB m? KBKAKB m? KAKBKAKB m, ..., Cm?

  14. Knowledge : Monica and Chandler are ... ¬: they are just friends

  15. Nested Knowledge Post: Kc, Km, Kj E{cmj} , E{cmj}E{cmj}  E{cmj} (¬Kr & ¬Kp &¬Kro) Goal{m,c}(¬Kr & ¬Kp &¬Kro) Goalj(Kr & Kp &Kro)

  16. Rachel Finds Out … Pre: • Kj, ¬Kr, Br • ¬KjKr, ¬Kj ¬Kr

  17. Rachel Found Out  Pre: • Kj, ¬Kr, Br • ¬KjKr, ¬Kj ¬Kr Post: • Kj, Kr • KjKr, KrKj • E{j,r}, E{j,r} E{j,r} 

  18. Phoebe Found Out  Pre: • Kj, Km, Kp, Kr • E{j,r,p} • ¬E{j,r,p}E{j,r,p} Let us call this team the Friends team (r, p, and j), EF and let {c, m, and j } be the Lovers team, EL : ¬EFEF and ¬ELEF

  19. Phoebe Found Out  Pre: • Kj, Km, Kp, Kr, Kc • EF • ¬EFEF Post: • EFEF • ¬KmEF, ¬KcEF • ¬ELEF, Goalp ¬ELEF, Goalj ELEF (Phoebe tries to seduce Chandler)

  20. m and c Learn EF Pre: • EFEF • ¬KmEF, ¬KcEF • ¬ELEF (Chandler tells Monica about the attempted seduction)

  21. m and c Learn EF Pre: • EFEF • ¬KmEF, ¬KcEF • ¬ELEF Post: • KmEF, KcEF • EL EF • ¬EF EL EF

  22. p and r Learn EL EF Pre: • KmEF, KcEF • EL EF • ¬EF EL EF, BF EL EF

  23. p and r learn EL EF Pre: • KmEF, KcEF • EL EF • ¬EF EL EF, BF EL EF Post: • EF EL EF • ¬E LEFELEF

  24. E (Friends) Happy?

  25. Common Knowledge • Ej: Everybody knows j • Ej  EEj • Cj: it is Common Knowledge that j • idea: Cj = Ej & EEj & EEEj & EEEEj & ….. • so: ¬Cj if somebody • considers it possible that • somebody considers it possible that somebody considers it possible that …………. j does not hold

  26. Enemies “The message is, that There are known knowns, there are things we know that we know. There are known unknowns, that is to say, there are things that we now know, we don’t know. But there are also unknown unknowns—there are things we do not know we don’t know. And each year, we discover a few more of those unknown unknowns.’’ Play

  27. Secretary of Defense • “Despite Mr. Rumsfeld’s rather outlandish mode of speaking, fans of the tough-talking U.S. defense secretary argued that he was misunderstood.”

  28. Knowledge, Rationality, Action • Knowledge • about world, other’s knowledge, & rationality • uncertainty, beliefs, suspicion • dynamic aspects: updating, learning • Rationality • An agent should do what is best for itself • Action • Emphasis on choice • Abilities: physical (robots), computational • Epistemic actions

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