Final answer:
To find context-free grammars for the given languages, we must establish production rules that adhere to the language's specifications. For the first language where n is not a multiple of 3, a CFG uses an intermediate state to control the count. For the second language, where k is the sum of n and m, the CFG directly associates each a or b with a c.
Step-by-step explanation:
Context-Free Grammar for Given Languages
To construct a context-free grammar (CFG) for the languages specified, we need to create production rules that generate the strings of the language.
a) L=anbn
A context-free grammar for this language can be constructed by ensuring the grammar generates an equal number of a's and b's while avoiding multiples of three. One possible CFG is:
S -> aSb | aabXbb | ε
X -> aXbb | aabbXbb | ε
In the above CFG, 'S' is the start symbol, and ε represents the empty string. The rule S -> aSb generates equal numbers of a's and b's, while X is used to ensure that the count is not a multiple of 3 by adding two a's and two b's at a time.
b) L=anbmck
For language b, a CFG can be created which ensures that the number of c's is equal to the sum of the number of a's and b's. A plausible CFG is:
S -> aSc | bSc | ε
This CFG starts with S and substitutes each a or b with the corresponding c, allowing for the creation of strings where k equals the sum of n and m.