Everybody Knows

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Everybody Knows
Foundations of knowledge representation and
reasoning

• Leon van der Torre

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Content

• How to represent agent A knows that?
• The semantics of knowing and believing
• Properties of knowing and believing
• Everybody knows and common knowledge
• Other modal operators

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Knowing what others know

• Knowing what others know is important in
• communication,
• negotiations,
• teaching,
• strategic behavior (no incentive compatibility)
• .......

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How to describe knowledge

• A special type of logical operator
• KI? agent I knows ?
• BI? agent I believes ?
• Nesting of operators
• KIKYKI? I know that You know that I know ?

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The meaning of knowledge

• Semantics possible abstract description of the
  real world
• Example
• r
• s
• t

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Propositional semantics
s
Proposition
?t
s ? ?t
s ? ?t
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Semantics of knowledge belief
RB
RA
s ? ?t holds
In s1 agent A believes s ? ?t BA (s ? ?t)
In s3 agent B believes that A believes s ? ?t
BB BA (s ? ?t)
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Semantics of knowledge belief

• The semantics is defined with respect to a
  structure M (S, ?, RA,...)
• S is a set of states (possible worlds)
• ? is a truth assignment per state
• RA the relation describing an agents knowledge
• A agent knows KA ? in a state s
• (M,s) KA ? iff (M,t) ? for every state t
  such that (s,t) ? RA
• Is this a good semantics of knowledge?

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Doxastic Logic

• Representing what an agent believes
• Axioms
• B(? ?y)?(B? ? By)
• B?
• B? ? BBj
• B? ? BBj

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Epistemic Logic

• Representing what an agent knows
• Axioms
• K(? ?y)?(K? ? Ky)
• K? ? j
• K? ? KKj
• K? ? KKj

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Idealizations

• No distinction between implicit and explicit
  beliefs or knowledge.
• We know what we dont know.
• As a well known politician said

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Rumsfeld

• ... for 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 dont 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."

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Everyone Knows

• Several agent A1,...,n
• Every agent knows j
• Ej K1j ? ... ? Knj
• If everyone knows, does everyone knows that
  everyone knows?
• If everyone knows that everyone knows, is there
  common knowledge?

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JPR know!

• Chandler and Monica have a secret j
• Chandler and Monica know the secret
• KCM j
• Joey, Phoebe and Rachel know the secret
• KJPR j
• Chandler and Monica do not known that, Phoebe and
  Rachel know the secret
• ?KCM KPR j
• Everyone knows the secret
• E j KC j ? KM j ? KJ j ? KP j ? KR j
• Not everyone knows that everyone knows
• ?E E j

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CM know they know!

• Chandler and Monica known that Joey, Phoebe and
  Rachel know their secret
• KCM KJPR j
• Joey knows that Chandler and Monica know that
  Phoebe and Rachel know their secret
• KJ KCM KPR j
• Phoebe and Rachel do not know that Chandler and
  Monica know that they know their secret
• KPR KCM KPR j
• Chandler and Monica know that everyone knows
  their secret
• KCM E j

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JPR know CM know JPR know!

• Joey, Phoebe and Rachel know that Chandler and
  Monica known they know their secret
• KJPR KCM KJPR j
• Chandler and Monica do not know that Joey, Phoebe
  and Rachel know that they known that Joey, Phoebe
  and Rachel know their secret
• KCM KJPR KCM KJPR j
• Everyone knows that everyone knows the secret
• E E j
• Not everyone knows that everyone knows that
  everyone knows the secret
• ?E E E j

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Common Knowledge

• After everybody finds out
• C j E j ? E E j ? E E E j ? ...

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Muddy children problem

• There are N children of which K children have mud
  on their heads.
• The father tells his children that at least one
  of them has mud on the head.
• The father asks the children that currently know
  that they have mud on their head to step forward.
• If no child steps forward the father repeats the
  question.
• After K times asking the question all muddy
  children step forward.
• Why?

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C
B
D
A
E
F

• B considers it possible that there are 3 muddy
  children.
• B considers it possible that C considers it
  possible that there are 2 muddy children.
• B considers it possible that C considers it
  possible that E considers it possible that there
  is 1 muddy child.

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Modal logic

• philosophical logic
• a formal treatment of intensional notions
• various flavours
• epistemic / doxastic
• temporal (time, future past)
• dynamic (action logic)
• deontic (obligations permissions)
• combinations (BDI, KARO)

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Modal logics

• modal logic logic of ? and ?
• neutral reading necessity / possibility
• ?? ? ????, ?? ? ????
• basic property (K)
• ?????????? ????????or equivalently
• (?? ?? ??????)?? ???

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Modal logics, semantics

• Kripke structure M (S, ?, R)
• S set of states / possible worlds
• ? truth assignment function
• R accessibility relation
• ?t R(s,t)??the set of states accessible from s

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Modal logics, semantics

• (M,s) ?? iff (M,t) ? for every t such
  that (s,t) ? R
• (M,s) ?? iff (M,t) ? for some t such that
  (s,t) ? R

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Semantic relations

• for any R ?(???) ? (?????) K
• R reflexive ???j T
• R serial ????j D
• R transitive ?????j 4
• R symmetric ????j
• R euclidean ?????j 5

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Varieties of modal logics

• Dynamic logic no restrictions system K
• describing actions
• Deontic logic R serial system KD
• describing obligations
• Temporal logic R refl, trans system S4 KT4
• describing time
• Doxastic logic R ser, trans, eucl system KD45
• describing belief
• Epistemic logic R refl, trans, symm system S5
  KT45
• describing knowledge

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BDI architecture

• Philosophical foundation practical reasoning
  consists of 2 activities.
• Deliberation
• Deciding what to do.
• Means-ends analysis
• Deciding how to do it.
• The results of the activities are intensions.

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BDI architecture
The general picture
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Naïve BDI logic

• B(? ?y)?(B? ? By)
• B?
• B? ? BBj
• B? ? BBj
• D(? ?y)?(D? ? Dy)
• I(? ?y)?(I? ? Iy)
• I?

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Other modal operators

• Agent A tells that ....
• A modal operator is an operator that has a
  proposition as a parameter and the truth value of
  this proposition in the world does not determine
  the truth value of the operator the proposition.

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Other modal operators

• lies(left) lies(right)(lies(left) ?
  ?lies(right)) ? (?lies(left) ? lies(right))
• leads-to(left,castle) leads-to(right,castle
  )
• TellsA? ? lies(A) ? ??
• TellsA? ? ?lies(A) ? ?

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Other modal operators

• lies(left) lies(right)(lies(left) ?
  ?lies(right)) ? (?lies(left) ? lies(right))
• leads-to(left,castle) leads-to(right,castle
  )
• TellsA? ? lies(A) ? ??
• TellsA? ? ?lies(A) ? ?
• Tellsleft(Tellsrightleads-to(left,castle))
• 1,2,3,4,5 ?leads-to(left,castle) ?
  leads-to(right,castle)

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Questions
?
You think I know the answer to the question
but is the answer common knowledge after
answering?
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Acknowledgement

• Slides are based on presentations of Wiebe van
  der Hoek.and Nico Roos