Auto-Epistemic Logic

1
Auto-Epistemic Logic

• Proposed by Moore (1985)
• Contemplates reflection on self knowledge
  (auto-epistemic)
• Permits to talk not just about the external
  world, but also about the knowledge I have of it

2
Syntax of AEL

• 1st Order Logic, plus the operator L (applied to
  formulas)
• L j signifies I know j
• Examples
• place ? L place (or ? L place ? ? place)
• young (X) ? ?L ?studies (X) ? studies (X)

3
Meaning of AEL

• What do I know?
• What I can derive (in all models)
• And what do I know not?
• What I cannot derive
• But what can be derived depends on what I know
• Add knowledge, then test

4
Semantics of AEL

• T is an expansion of theory T iff
• T Th(T?Lj T j ? ?Lj T ? j)
• Assuming the inference rule j/Lj
• T CnAEL(T ? ?Lj T ? j)
• An AEL theory is always two-valued in L, that is,
  for every expansion
• ? j Lj ? T ? ?Lj ? T

5
Knowledge vs. Belief

• Belief is a weaker concept
• For every formula, I know it or know it not
• There may be formulas I do not believe in,
  neither their contrary
• The Auto-Epistemic Logic of knowledge and belief
  (AELB), introduces also operator B j I believe
  in j

6
AELB Example

• I rent a film if I believe Im neither going to
  baseball nor football games
• B?baseball ? B?football ? rent_filme
• I dont buy tickets if I dont know Im going to
  baseball nor know Im going to football
• ? L baseball ? ? L football ? ? buy_tickets
• Im going to football or baseball
• baseball ? football
• I should not conclude that I rent a film, but do
  conclude I should not buy tickets

7
Axioms about beliefs

• Consistency Axiom
• ?B?
• Normality Axiom
• B(F ? G) ? (B F ? B G)
• Necessitation rule
• F
• B F

8
Minimal models

• In what do I believe?
• In that which belongs to all preferred models
• Which are the preferred models?
• Those that, for one same set of beliefs, have a
  minimal number of true things
• A model M is minimal iff there does not exist a
  smaller model N, coincident with M on Bj e Lj
  atoms
• When j is true in all minimal models of T, we
  write T min j

9
AELB expansions

• T is a static expansion of T iff
• T CnAELB(T ? ?Lj T ? j
• ? Bj T min j)
• where CnAELB denotes closure using the axioms of
  AELB plus necessitation for L

10
The special case of AEB

• Because of its properties, the case of theories
  without the knowledge operator is especially
  interesting
• Then, the definition of expansion becomes
• T YT(T)
• where YT(T) CnAEB(T ? Bj T min j)
• and CnAEB denotes closure using the axioms of AEB

11
Least expansion

• Theorem Operator Y is monotonic, i.e.
• T ? T1 ? T2 ? YT(T1) ? YT(T2)
• Hence, there always exists a minimal expansion of
  T, obtainable by transfinite induction
• T0 CnAEB(T)
• Ti1 YT(Ti)
• Tb Ua lt b Ta (for limit ordinals b)

12
Consequences

• Every AEB theory has at least one expansion
• If a theory is affirmative (i.e. all clauses have
  at least a positive literal) then it has at least
  a consistent expansion
• There is a procedure to compute the semantics