retagged by
421 views
0 votes
0 votes

Consider the following language $L$ over the alphabet $\Sigma=\{a, b, c\}$. $$ L=\left\{a^{i} b^{j} c^{k} \mid i, j, k \geq 0 \text {, and if } i=1 \text { then } j=k\right\} $$

  1. Show that $L$ is not regular.
  2. Show that $L$ is a context-free language.
retagged by

2 Answers

0 votes
0 votes
i) To show that L is not a regular language, we can use the pumping lemma for regular languages. Assume that L is regular and let p be the pumping length given by the lemma. Consider the string s = a^pb^pc^p. Since |s| = 3p ≥ p, there exists a decomposition s = xyz, where |xy| ≤ p and |y| ≥ 1, such that xy^iz ∈ L for all i ≥ 0.

Since |xy| ≤ p, the substring y consists of only a's, only b's, or a combination of a's and b's. If y consists of only a's or only b's, then pumping y will either increase the number of a's or the number of b's, breaking the condition that i=1 implies j=k. If y consists of a combination of a's and b's, then pumping y will cause the number of a's and the number of b's to become unequal, again breaking the condition that i=1 implies j=k.

Therefore, we have a contradiction, and L cannot be a regular language.

ii) To show that L is a context-free language, we can construct a context-free grammar (CFG) that generates it. One possible CFG is:

S → aSc | A A → aAb | B B → bBc | ε

The production rules generate strings of the form a^ib^jc^k where i, j, and k are non-negative integers. The rule S → aSc generates strings with any number of a's and c's between a prefix of a's and a suffix of c's, while the rule S → A generates strings with an equal number of b's and c's.

The rule A → aAb generates strings with any number of a's and b's between a prefix and suffix of a's, while the rule A → B generates strings with an equal number of a's and b's. Finally, the rule B → bBc generates strings with any number of b's and c's between a prefix of b's and a suffix of c's, while the rule B → ε generates the empty string.

Since L can be generated by a CFG, it is a context-free language.

Related questions

0 votes
0 votes
2 answers
2
admin asked Aug 8, 2022
535 views
Let $G$ be a simple undirected graph having $n$ vertices with the property that the average of the degrees of the vertices in $G$ is at least $4.$ Consider the following ...
0 votes
0 votes
1 answer
3