Course 02286 - 2013, Week 12 Exercises on epistemic logics

<strong>Exercises</strong> on the three cards scenarioFor the three cards scenario introduced in the lecture:1. Construct an epistemic model M describing the scenario.Take as states all possible distributions of the cards.2. Formalize in epistemic logic the statements:2.1 Player 1 knows that everybody holds exactly one of the cards.2.2 Player 1 knows the card (s)he is holding, but does not knowthe cards of the other two players.3. Determine whether the formulaP 1,A → K 1 (¬K 2 P 2,A ∧ ¬K 2 P 3,A )is true at every state of the epistemic model M.4. Update the model M after player 2 shows his card to 3 only.5. Using the updated model, argue that, after the action ofplayer 2, player 1 still does not know the card of player 3.

<strong>Exercises</strong> on the Three Muddy Children problemThree boys, A, B, and C, have played in the mud and some ofthem may have soiled their foreheads. Each of them can see theothers’ foreheads but not his own. The boys have perfect logicalreasoning skills and are truthful in their announcements.1. Construct the epistemic model describing the scenario.2. Using the propositions m X to mean “X has a muddyforehead”, for X ∈ {A, B, C}, describe with formulae ofepistemic logic the following statements:2.1 “A knows that he has a muddy forehead or a clean foreheadbut does not know which one.”2.2 “A knows whether the forehead of B is muddy or clean, andboth A and C know that B does not know whether B’sforehead is clean or muddy.”2.3 “If the forehead of C is clean, then that fact, and the fact thatC does not know it, are common knowledge among A and B.”

<strong>Exercises</strong> on the Three Muddy Children problem, continued3. Determine the truth in the 3 muddy children epistemic modelof the epistemic formulae on the lecture slides (26, 27).4. Suppose hereafter that only A has a muddy forehead.The father comes and says“At least one of you has a muddy forehead.”Update the model accordingly and use it to determinewhether each of the following is true.4.1 K A m A .4.2 K A ¬m B .4.3 K B ¬m B .4.4 K B K A m A .4.5 K A ¬K B ¬m B .4.6 C {A,B} ¬m C .4.7 D {A,B} (m A ∧ ¬m B ).