CMPS 260 Spring 2024
HW #2: DFA's and Regular Languages
Due: 2pm, Wednesday, Feb. 14

The names that you give to the states in the minimized DFA should make clear which set of states each one came from. For example, if you find that {0,3,4} is an equivalence class of the indistinguishability relation of the given DFA, the corresponding state in the minimized DFA should be named {0,3,4}.

δ state a b Initial? Final? 0 5 4 ✓   1 5 2     2 2 0   ✓ 3 2 6   ✓ 4 4 1   ✓ 5 3 4     6 2 3

2. Apply the Cartesian Product Construction upon the two DFAs shown below, M₁ and M₂, for the purpose of obtaining a DFA that accepts the intersection of L(M₁) and L(M₂). Present your answer in tabular form rather than (or in addition to, if you want) a transition graph. Each state in your DFA should be named so as to make it clear from which pair of states in M₁ and M₂ it arises. Feel free to use a name such as [0,2] rather than [q₀, r₂]. Don't include states that are unreachable from the start state.

M₁

State δ Start? Accepting? State ID a b ✓ q0 q0 q1 q1 q2 q0 ✓ q2 q2 q1

M₂

State δ Start? Accepting? State ID a b ✓ ✓ r0 r1 r1 r1 r2 r1 ✓ r2 r1 r0

3. Expressed in terms of the 5-tuple description of DFAs, the Cartesian Product Construction that aids in proving that the regular languages are closed under intersection goes like this:

For i = 1,2, let M_i = (Q_i, Σ, δ_i, q_i,0, F_i) be DFAs (with the same alphabet Σ). Then a DFA that accepts L(M₁) ∩ L(M₂) is M = (Q, Σ, δ, q₀, F), where

• Q = Q₁×Q₂,
• q₀ = [q_1,0,q_2,0],
• For all p ∈ Q₁, q ∈ Q₂, and c ∈ Σ:   δ([p,q],c) = [δ₁(p,c), δ₂(q,c)], and
• F = F₁ × F₂ (equivalently, F = { [p,q]  |  p ∈ F₁ and q ∈ F₂ } )

(a) By changing a small detail of this construction, the language accepted by DFA M would be the difference L(M₁) − L(M₂) of L(M₁) and L(M₂) rather than their intersection. (For sets S and T, S−T = { a  |  a ∈S and a ∉ T }.) Describe that small change. Be precise.

(b) Similarly, a different small change in the construction yields a DFA that accepts the symmetric difference of L(M₁) and L(M₂). The symmetric difference of two sets S and T, sometimes denoted by S Δ T, contains those elements are are members of exactly one among S and T. In other words, S Δ T = (S − T) ∪ (T − S). Describe that small change. Be precise.

4. Let M = (Q, Σ δ, q₀, F) be a DFA such that L = L(M) (i.e., M accepts language L).

Consider the language

Note: The notation Σ⁺ denotes the set of all non-empty strings over Σ. End of note.

In words, x is a member of L' if and only if x is a member of L but x is not a proper prefix of any member of L.

Describe how to modify M so as to obtain a DFA M' that accepts L'.

Bonus: Assume that M is a minimal DFA (meaning that every state is distinguishable from every other state). Make whatever meaningful observations you can about the accepting states in M'.

5. Let #: {0,1}⁺ ⟶ ℕ be the function that maps nonempty bit strings to natural numbers in accord with the binary numeral system. For example, #(11001) = 25 and #(100110) = 38. Where it is understood that x ∈ {0,1}⁺ (i.e., x is a non-empty bit string), a recursive characterization of # is as follows:

#(0)	=	0	(base case)
#(1)	=	1	(base case)
#(x0)	=	2·#(x)	(recursive case)
#(x1)	=	2·#(x) + 1	(recursive case)

Present a DFA that accepts the language consisting of all those bit strings x such that #(x) is one more than a multiple of four. A more formal way to describe this language (using % as the remainder (or modulo) operator) is

Your DFA should have four states, one corresponding to each of the possible remainders of a division by four.