5. Knowledge and Information Flow

Question 1

How to write ‘agent i knows that p’ without the square?

Answer 1

KiP

Question 2

What is Uncertainty?

Answer 2

The set of current options for the actual world

Question 3

How to describe that an agent cannot distinguish two situations?

Answer 3

The line with 2 arrowheads

Question 4

What does the self loop say?

Answer 4

Indicate that we cannot distinguish an option from itself.

Question 5

Describe the knowledge information model

Answer 5

(W, {→i| i ∈ I}, V )
W: set of worlds
→i: binary accessibility relations between worlds
V: valuation assigned truth values to proposition letters at words

Question 6

What are the equivalence relations?

Answer 6

(1) reflexivity For all w, Rww
(2) symmetry For all w, v: if Rwv, then Rvw
(3) transitivity For all w, v, u, if Rwv and Rvu, then Rwu

Question 7

When is P called invalid?

Answer 7

if there is at least one model M with a world s where P is false

Question 8

Veridicality

Answer 8

■ϕ → ϕ

Question 9

Positive Introspection

Answer 9

■ϕ → ■■ϕ

Question 10

Negative Introspection

Answer 10

¬■ϕ → ■¬■ϕ

Question 11

Predicate logic for veridicality

Answer 11

Reflexivity, ∀xRxx

Question 12

Predicate logic for Positive Introspection

Answer 12

Transitivity ∀x∀y∀z((Rxy ∧ Ryz) → Rxz)

Question 13

Predicate logic for Negative Introspection

Answer 13

Euclidicity ∀x∀y∀z((Rxy ∧ Rxz) → Ryz)

Question 14

Proof system for the minimal modal logic K:

Answer 14

(1) All propositional tautologies are theorems
(2) K axiom schema, All formulas of the form ■(ϕ → ψ) → (■ϕ → ■ψ)
(3) modus ponens rule, If ϕ and ϕ → ψ are theorems, then ψ is a theorem.
(4) necessitation rule, If ϕ is a theorem, then ■ϕ is also a theorem

Question 15

How to announce p

Answer 15

!p

Question 16

What does [!ϕ]ψ say?

Answer 16

That we know ψ after announcing ϕ

Question 17

What does <!ϕ>ψ say?

Answer 17

That after announcing ϕ we know that ψ is true in some world