Question

Find context-free grammars for the following languages.

(a) L={aⁿbⁿ∣n is not a multiple of 3}.
(b) L={aⁿbᵐcᵏ∣k=n+m,n,m,k≥0}

Answer 1

The student's question pertains to creating context-free grammars for two different languages with certain constraints. The first language requires avoiding multiples of three, and the second requires the number of c's to be the sum of a's and b's. CFGs have been provided for both languages, adhering to their respective specifications.

Step-by-step explanation:

The student is asking to construct context-free grammars (CFGs) for two different languages. The first language L= n is not a multiple of 3 requires a grammar that generates an equal number of a's and b's where the number of these symbols is not divisible by 3. The second language L= k=n+m, n, m, k≥0 requires a grammar that generates strings with a sequence of a's followed by b's and then c's where the total number of c's is equal to the sum of the number of a's and b's.

For language L= n is not a multiple of 3:

A possible context-free grammar is:

S → aSb | aab | abb | ε

For language L= k=n+m, n, m, k≥0:

A possible context-free grammar is:

• S → aSc | bSc | ε

These grammars ensure that for the first language, there cannot be a multiple of three pairs since the production rules are constructed to avoid this pattern. While for the second language, the CFG ensures the necessary condition for the number of c's by introducing them at each addition of an a or a b.