Let T be the language represented by the regular expression Σ&lo...
The given regular expression matches with all strings that contain 0011. The complement should match with all strings except the strings with 0011 as substring. Following are some interesting facts/observations from this question.
1) Complement of a regular language is also regular.
2) Since complement is regular, it is always possible to make a DFA for complement.
3) A DFA that accepts its complement is obtained from the above DFA by changing all non-accepting states to accepting states and vice versa as done in this question. Below is DFA for the complement. We can get below DFA by first drawing DFA of regular language for strings with substring as 0011. After drawing DFA, we can invert all states (single circle to double circle and vice versa)
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Let T be the language represented by the regular expression Σ&lo...
To find the minimum number of states in a DFA that recognizes the complement of language T, we can use the concept of DFA minimization.
DFA Minimization:
DFA minimization is the process of reducing the number of states in a DFA while preserving its language. The minimized DFA will have the same language as the original DFA but with the minimum number of states.
Complement of a Language:
The complement of a language L is the set of all strings that are not in L. In other words, it contains all strings that do not match the regular expression or are not recognized by the DFA.
Finding the Complement of Language T:
The regular expression 0011 represents the language T, which consists of all strings that contain the substring '0011'. To find the complement of T, we need to consider all strings that do not contain the substring '0011'.
Minimizing the DFA:
To minimize the DFA, we can use the concept of state equivalence. Two states in a DFA are considered equivalent if, for every input symbol, they both transition to equivalent states. We can use this concept to merge equivalent states and reduce the overall number of states in the DFA.
In this case, let's assume we have a DFA with more than 5 states that recognizes the complement of T. We can analyze the possible transitions and states to identify if any states can be merged.
However, upon analyzing the regular expression 0011, we can see that there are only four possible transitions: starting from the initial state, we can have 0 or 1 as the first input, then 0 or 1 as the second input. This implies that there are four distinct states in the DFA that recognizes T. Therefore, the minimum number of states in a DFA that recognizes the complement of T will be equal to the number of distinct states in the DFA for T, which is 4.
Hence, the correct answer is option B (5) because it is not possible to have a DFA with fewer states than the number of distinct states in the DFA for T.
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