Computer Science Engineering (CSE) Exam > Computer Science Engineering (CSE) Questions > Let P be a regular language and Q be a contex...
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Let P be a regular language and Q be a context free language such that Q ⊆ P. (For example, let P be the language represented by the regular expression p*q* and Q be {pnqn | n ∈ N}).
Q. Then which of the following is ALWAYS regular?
- a)P ∩ Q
- b)P − Q
- c)∑ * − P
- d)∑ * − Q
Correct answer is option 'C'. Can you explain this answer?
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Verified Answer
Let P be a regular language and Q be a context free language such that...
Σ* − P is the complement of P so it is always regular,
since regular languages are closed under complementation
Most Upvoted Answer
Let P be a regular language and Q be a context free language such that...
Explanation:
To determine which of the given options is always regular, we need to analyze the properties of regular and context-free languages.
- A regular language can be represented by a regular expression or can be accepted by a finite automaton.
- A context-free language can be generated by a context-free grammar or accepted by a pushdown automaton.
P Q:
If we concatenate a regular language P with a context-free language Q, the resulting language may or may not be regular. It depends on the specific languages P and Q.
P Q:
If we take the intersection of a regular language P and a context-free language Q, the resulting language is always regular. This is because the intersection of a regular language and a context-free language can be recognized by a finite automaton, which proves that the resulting language is regular.
* P:
If we take the Kleene star (closure) of a regular language P, the resulting language is always regular. This is because the Kleene star operation can be implemented using a finite automaton, which proves that the resulting language is regular.
* Q:
If we take the Kleene star (closure) of a context-free language Q, the resulting language may or may not be regular. It depends on the specific language Q. The Kleene star of a context-free language can be recognized by a pushdown automaton, but it may not be possible to convert this pushdown automaton into an equivalent finite automaton in some cases.
Therefore, based on the properties of regular and context-free languages, option C (* P) is the only one that is always regular.
To determine which of the given options is always regular, we need to analyze the properties of regular and context-free languages.
- A regular language can be represented by a regular expression or can be accepted by a finite automaton.
- A context-free language can be generated by a context-free grammar or accepted by a pushdown automaton.
P Q:
If we concatenate a regular language P with a context-free language Q, the resulting language may or may not be regular. It depends on the specific languages P and Q.
P Q:
If we take the intersection of a regular language P and a context-free language Q, the resulting language is always regular. This is because the intersection of a regular language and a context-free language can be recognized by a finite automaton, which proves that the resulting language is regular.
* P:
If we take the Kleene star (closure) of a regular language P, the resulting language is always regular. This is because the Kleene star operation can be implemented using a finite automaton, which proves that the resulting language is regular.
* Q:
If we take the Kleene star (closure) of a context-free language Q, the resulting language may or may not be regular. It depends on the specific language Q. The Kleene star of a context-free language can be recognized by a pushdown automaton, but it may not be possible to convert this pushdown automaton into an equivalent finite automaton in some cases.
Therefore, based on the properties of regular and context-free languages, option C (* P) is the only one that is always regular.
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Let P be a regular language and Q be a context free language such that Q ⊆ P. (For example, let P be the language represented by the regular expression p*q* and Q be {pnqn | n ∈ N}).Q. Then which of the following is ALWAYS regular?a)P ∩ Qb)P − Qc)∑ * − Pd)∑ * − QCorrect answer is option 'C'. Can you explain this answer?
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