1 _____ 2 _____ 3 _____ 4 _____ 5 _____ Total _____

There are five problems. You will be graded based upon your best four answers.

1. Let L₁ be the language over the alphabet {a,b} containing precisely those strings that end with b and have an even number of occurrences of a.
More formally, L₁ = { xb | x ∈ {a,b}^* and #_a(x) is even }

(a) Present a (preferably, minimal) DFA that accepts L₁.
(b) Present a regular expression that denotes L₁.

2.

M1	M2

Consider the DFA's M₁ and M₂ pictured to the right.

(a) In the space below, fill in the missing entries in the table that describes the semi-DFA M obtained by applying the cartesian product construction. (A semi-DFA is a DFA except that the final/accepting states are left unspecified.) It is understood that [p0,q0] is the start/initial state, due to p0 (respectively, q0) being the initial state of M₁ (resp., M₂). (You can draw the transition diagram describing M in the space to the right, but it is not required.)

	δ
state	a	b
[p0,q0]
[p0,q1]
[p1,q0]
[p1,q1]

(b) Which state(s) of M should be designated as final/accepting for the resulting DFA to accept the intersection of L(M₁) and L(M₂)?

(c) Which state(s) of M should be designated as final/accepting for the resulting DFA to accept the union of L(M₁) and L(M₂)?

(d) Suppose that [p0,q1] were the lone state in M to be designated as final/accepting. What language is accepted by that DFA? (Try to express it in terms of L(M₁), L(M₂), and set operators, as opposed to referring to a's and b's.)

3. Let L₃ be the language over the alphabet {a,b} containing precisely those strings that either have aba as a suffix or do not have baa as a substring. Present an NFA that accepts L₃.

Hint: Make use of the relaxed requirements of NFA's (compared to DFA's) to make the task easier. Among these are λ-transitions, not having to explicitly show the dead state (or transitions going to it), and multiple transitions from the same state with the same label.

Hint: As building blocks, devise regular expressions describing

• the set of strings by which to first arrive at q₁ from q₀ without passing through q₂.
• the set of strings by which to first arrive at q₁ from q₀ via q₂.
• the set of strings by which to first arrive at q₀ from q₁.

(a) Suppose that L(M) is finite. Describe L(M), exactly. (There is only one possibility.)
(b) Suppose that L(M) is infinite and that aa is the shortest member of L(M). Then L(M) is one of three possible languages. Describe them, exactly.