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 muddy AA,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 mem the atoms muddy A and muddy yC would be true, but muddy yB 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: 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. (a) Which of the following statements hold? Formally justify your answer with respect to the model M, and give an intuitive explanation with reference to the situation in the puzzle scenario. (i) M,mccEABCmuddyA (ii) M,mccCABCmuddyA (iii) M(KCmuddyC)(muddyAmuddyB) (b) Give a common knowledge formula that is true in M (but not a trivial one which is a tautology anyway)..

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 muddy
AA,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 mem the atoms muddy A and muddy
yC would be true, but muddy yB 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: 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. (a) Which of the following statements hold? Formally justify your
answer with respect to the model M, and give an intuitive explanation with reference to the
situation in the puzzle scenario. (i) M,mccEABCmuddyA (ii) M,mccCABCmuddyA (iii)
M(KCmuddyC)(muddyAmuddyB) (b) Give a common knowledge formula that is true in M (but
not a trivial one which is a tautology anyway).