Final answer:
To prove that L₁ = {0ᶦ1ʲ0ᵏ1ᵏ ∣ i, j, and k are positive integers} is regular, we can construct a deterministic finite automaton (DFA) that recognizes L₁.
Step-by-step explanation:
The language L₁ is defined as {0ᶦ1ʲ0ᵏ1ᵏ ∣ i, j, and k are positive integers}. To prove that L₁ is regular, we can construct a deterministic finite automaton (DFA) that recognizes L₁.
Let's start by defining the states of the DFA. The DFA will have three states: q₀, q₁, and q₂. The start state will be q₀, and the accepting state will be q₁.
Next, let's define the transitions of the DFA. For each transition, we will consider the next symbol in the input string and the current state to determine the next state. The transitions are as follows:
- From q₀, if the next symbol is 0, the DFA moves to q₀.
- From q₀, if the next symbol is 1, the DFA moves to q₁.
- From q₁, if the next symbol is 0, the DFA moves to q₂.
- From q₂, if the next symbol is 1, the DFA moves to q₂.
Finally, let's define the final state of the DFA. The DFA reaches the final state q₁ whenever it reads a 1 after the initial 0s. The DFA rejects the input if it reaches any other state. Thus, the language L₁ is regular.