Question

Question 1: Design a DFA with at most 5 states for the language

L₁ = {w ∈ {0, 1}^∗ | w contains at most one 1 and |w| is odd}.

Provide a state diagram for your DFA.

Approaching the Solution --since we haven’t really practiced this type of assignment (i.e. had to define our machine based on only having the language given; not the formal 5 tuples), I am providing the steps for how to work through this; you are required to draw the DFA for your answer to this question and provide three test cases (additional instructions for this question are at the end of the explanation below)

To get an idea for the design of the DFA M, it is probably easiest to think about what the strings in L₁ look like. A string w ∈ L₁ (i.e. a string (w) that belong to L₁ ) takes on one of the following forms:

• w consists of an odd number of 0s and contains no 1s;

• w starts with an odd number of 0s, followed by one 1, followed by an odd number of zeros; • w starts with an even number of 0s, followed by one 1, followed by an even number of zeros;

Thus, our DFA M needs to keep track of the parity of the number of leading 0s (i.e. whether that number is even or odd). If no 1 is encountered, accept if the number of leading 0s is odd; otherwise go to a dead state. If a 1 is encountered, keep track of the number of trailing 0s and accept if that number has the same parity as the number of leading 0s (if applicable as a single “1” can be complete as input)

Our 5 states keep track of the following situations:

• state “even0no1” indicates an even number of 0s and no 1s;

• state “odd0no1” indicates an odd number of 0s and no 1s;

• state “even0one1” indicates an even number of 0s, followed by a 1;

• state “odd0one1” indicates an odd number of 0s, followed by a 1;

• state “dead” indicates the encounter of a second 1.

“even0no1” is the start state, since at the beginning of every string, the number of 0s encountered is zero (i.e. even) and no 1 has been encountered. It is not a final state since ε ∉ L₁. (i.e.an empty string is not part of the language). States “odd0no1” & “even0one1” are final, and “dead” is a dead state from which the DFA should never again emerge.

To answer Question 1, you will:

1. Draw/provide the state diagram for the DFA

Provide at least three test cases that prove your DFA accepts them (or rejects them). In other words, accepts the language L₁, i.e. prove that L(M) = L₁, where M is your DFA.

Answer 1