Theory of Computation: Question Set – 13

What is the pumping lemma for context-free languages, and how is it used to prove that a language is not context-free?

The pumping lemma for context-free languages is a property that all context-free languages must satisfy. It states that for any context-free language L, there exists a pumping length p such that any string s in L with length greater than or equal to p can be decomposed as s = uvxyz, where |vy| > 0, |vxy| ≤ p, and the strings uvⁱxyⁱz are also in L for all i ≥ 0. If a language fails the pumping lemma, then it cannot be a context-free language.

Can a finite automaton recognize the language {anbncm | n, m ≥ 0}? If so, how can it be constructed?

No, a finite automaton cannot recognize the language {anbncm | n, m ≥ 0}. This language requires a context-free grammar or a pushdown automaton for recognition.

Can an NFA recognize a language that a DFA cannot, or vice versa?

Yes, an NFA can recognize a language that a DFA cannot, and vice versa. This is because NFAs have more expressive power than DFAs due to their ability to be in multiple states at once. However, DFAs have certain advantages over NFAs, such as being easier to construct and analyze.

What is the subset construction algorithm, and how is it used to convert an NFA to an equivalent DFA?

The subset construction algorithm is a method for converting an NFA to an equivalent DFA. The algorithm works by creating a new state in the DFA for each possible set of states that the NFA can be in at a given time. The initial state of the DFA is the set of states that the NFA can be in when starting from its initial state, and the final states are any sets of states that contain a final state of the NFA. Transitions in the DFA are determined by taking the union of all possible transitions from the states in the current set for each input symbol.

Can every regular language be recognized by a DFA?

Yes, every regular language can be recognized by a DFA. This is because the set of languages recognized by DFAs is equivalent to the set of languages recognized by regular expressions, and vice versa.

Can a regular language have an infinite number of strings?

Yes, a regular language can have an infinite number of strings. For example, the language (0 + 1)* contains an infinite number of strings consisting of any combination of 0s and 1s.

Can a regular language have an infinite number of distinct subwords?

No, a regular language cannot have an infinite number of distinct subwords. This is because a regular language is a finite set of strings, and any subword of a string in the language must also be in the language.

Can a regular language have an infinite number of distinct prefixes or suffixes?

No, a regular language cannot have an infinite number of distinct prefixes or suffixes. This is because a regular language is a finite set of strings, and any prefix or suffix of a string in the language must also be in the language.