Question

Let \(L \subseteq \{0,1\}^*\) be an arbitrary regular language accepted by a minimal DFA with \(k\) states. Which one of the following languages must necessarily be accepted by a minimal DFA with \(k\) states?

• A. \(L - \{01\}\)
• B. \(L \cup \{01\}\)
• C. \(\{0,1\}^* - L\)
• D. \(L \cdot L\)

Hint:

Complementing a DFA preserves the number of states—only accepting and rejecting states are swapped.

Answer 1

The Correct Option is C

To solve this problem, we need to identify which of the provided language operations will be accepted by a minimal DFA with exactly \(k\) states, given that \(L\) is a regular language already accepted by a minimal DFA with \(k\) states. Let's analyze each option one by one: 

1. Option 1: \(L - \{01\}\)
   • This operation describes the language formed by taking all strings in \(L\) and removing the string "01." Although \(L\) is accepted by a DFA with \(k\) states, removing one string can potentially reduce the number of states needed to recognize the language if that specific string is the cause of a state transition branch. Thus, the number of states required may be less than \(k\). Hence, a DFA representing \(L - \{01\}\) may not necessarily have \(k\) states.
2. Option 2: \(L \cup \{01\}\)
   • This represents the union of the language \(L\) with the set containing the single string "01." The addition of this string to \(L\) can result in requiring more states if "01" is not already in \(L\). Thus, the DFA for this union may have more than \(k\) states in order to include the additional string "01". Therefore, it may not necessarily be accepted by a minimal DFA with \(k\) states.
3. Option 3: \(\{0,1\}^* - L\)
   • \(\{0,1\}^*\) represents the set of all binary strings, and the operation \(\{0,1\}^* - L\) gives the complement of the language \(L\). One of the fundamental properties of DFAs is that if a language \(L\) is recognized by a DFA with \(k\) states, its complement can also be recognized by a DFA with the same number of states (by exchanging accepting and non-accepting states). Thus, the complement language \(\{0,1\}^* - L\) can indeed be accepted by a DFA with exactly \(k\) states.
4. Option 4: \(L \cdot L\)
   • This denotes the concatenation of \(L\) with itself. For concatenated languages, the minimum number of states needed to recognize them is not simply the sum of the states of the individual languages. In fact, this operation may require more than \(k\) states, as adding language complexity through concatenation generally increases the state count. Therefore, it likely won't be accepted by a minimal DFA with exactly \(k\) states.

Based on the analysis above, the correct choice is Option 3: \(\{0,1\}^* - L\). This option is correct because the complement of any regular language accepted by a DFA with \(k\) states can also be accepted by a DFA with \(k\) states.