Here is a regular safety property: "the first time that a becomes true, it must remain true until b is true". Stated formally, for a path w, this is as follows: for the smallest index i such that w(i) |= a, either there exists an index k > i such that w(k) |= b and w (j) |= a for all i i. 1. Give a deterministic finite automaton that represents the finite violations ("bad pre- fixes") of this property. 2. Construct the product of this DFA and the DTMC below, which has the following labelling function: L (S_0) = {b}, L (s_1) = {a, b, c} and L(s_3) = theta. 3. Compute the probability that the safety property of part (a) is satisfied from the initial state of the DTMC in par (b). Here is a regular safety property: "the first time that a becomes true, it must remain true until b is true". Stated formally, for a path w, this is as follows: for the smallest index i such that w(i) |= a, either there exists an index k > i such that w(k) |= b and w (j) |= a for all i i. 1. Give a deterministic finite automaton that represents the finite violations ("bad pre- fixes") of this property. 2. Construct the product of this DFA and the DTMC below, which has the following labelling function: L (S_0) = {b}, L (s_1) = {a, b, c} and L(s_3) = theta. 3. Compute the probability that the safety property of part (a) is satisfied from the initial state of the DTMC in par (b)