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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

The Mathematics of Knowledge, Rationality, & Action

Courtesy of Prof. Wiebe van der Hoek

Computer Science

University of Liverpool

United Kingdom

agents paradigm
``..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
perceive reason act cycle each agent

sense

sense

act

act

goal

info

sense

act

Perceive-Reason-Act Cycle (Each Agent)

reason

perceive reason act cycle multiple agents

sense

sense

act

act

goal

info

sense

act

Perceive-Reason-Act Cycle (Multiple Agents)

reason

rationality
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

rationality1
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

rationality2
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

rationality3
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

rationality4
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

rationality5
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

rationality6
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

e pistemic l ogic relies on 4 axioms f and y are facts
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)
slide13

Multiagent Knowledge

KAm: agent A knows m

A sends m to B

KAm ?

KBm ? Em ?

KAKB m?

KBKAKB m? KAKBKAKB m, ..., Cm?

knowledge
Knowledge

: Monica and Chandler are ...

¬: they are just friends

nested knowledge
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)

rachel finds out
Rachel Finds Out …

Pre:

  • Kj, ¬Kr, Br
  • ¬KjKr, ¬Kj ¬Kr
rachel found out
Rachel Found Out 

Pre:

  • Kj, ¬Kr, Br
  • ¬KjKr, ¬Kj ¬Kr

Post:

  • Kj, Kr
  • KjKr, KrKj
  • E{j,r}, E{j,r} E{j,r} 
phoebe found out
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

phoebe found out1
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)

m and c learn e f
m and c Learn EF

Pre:

  • EFEF
  • ¬KmEF, ¬KcEF
  • ¬ELEF

(Chandler tells Monica about the attempted seduction)

m and c learn e f1
m and c Learn EF

Pre:

  • EFEF
  • ¬KmEF, ¬KcEF
  • ¬ELEF

Post:

  • KmEF, KcEF
  • EL EF
  • ¬EF EL EF
p and r learn e l e f
p and r Learn EL EF

Pre:

  • KmEF, KcEF
  • EL EF
  • ¬EF EL EF, BF EL EF
p and r learn e l e f1
p and r learn EL EF

Pre:

  • KmEF, KcEF
  • EL EF
  • ¬EF EL EF, BF EL EF

Post:

  • EF EL EF
  • ¬E LEFELEF
common knowledge
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
enemies
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

secretary of defense
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.”
knowledge rationality action
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
ad