Ex. 4.1 Semantics of epistemic logic. Recall the muddy forehead puzzle. We have three agents: A (for Alice), B (for Bob), and C (for Claire). A state (a situation) can be described by a sequence of length three, each item being either of m (for muddy) or c (for clean). For example, the state cmm describes a situation where Alice has a clean forehead (c at first position), and Bob and Claire each have a muddy forehead (m at second and third position, respectively). Analogously, in mcm Alice and Claire have a muddy forehead, but Bob does not. Let muddyA , muddyB and muddyC be three propositional symbols that encode whether Alice, Bob and Claire have a muddy forehead at a specific situation, e.g. in mcm the atoms muddyA and muddyC would be true, but muddyB would be false. Consider the situation where one of the parents already asserted that at least one of the three has a muddy forehead. An epistemic model M for the possible combinations of all remaining situations is as follows: mmm A cmm C C B A mmc mcm B B cmc A ccm C mcc where the accesibility relations are marked with the respective agents they are associated with. The accesibility relations are S5 relations (i.e. they are reflexive, symmetric and transtive), the reflexive self-loop is omitted in the graph above..

1. Ex. 4.1 Semantics of epistemic logic. Recall the muddy forehead puzzle. We have three agents: A
(for Alice), B (for Bob), and C (for Claire). A state (a situation) can be described by a sequence of
length three, each item being either of m (for muddy) or c (for clean). For example, the state cmm
describes a situation where Alice has a clean forehead (c at first position), and Bob and Claire
each have a muddy forehead (m at second and third position, respectively). Analogously, in mcm
Alice and Claire have a muddy forehead, but Bob does not. Let muddyA , muddyB and muddyC
be three propositional symbols that encode whether Alice, Bob and Claire have a muddy forehead
at a specific situation, e.g. in mcm the atoms muddyA and muddyC would be true, but muddyB
would be false. Consider the situation where one of the parents already asserted that at least one
of the three has a muddy forehead. An epistemic model M for the possible combinations of all
remaining situations is as follows: mmm A cmm C C B A mmc mcm B B cmc A ccm C mcc where
the accesibility relations are marked with the respective agents they are associated with. The
accesibility relations are S5 relations (i.e. they are reflexive, symmetric and transtive), the reflexive
self-loop is omitted in the graph above.