Regular grammar and context-free grammar

Context-free grammar (CFG) and regular grammar are two different types of grammars used in formal language theory. In order to understand which statement is correct, let's first define these two types of grammars.

Regular Grammar:
A regular grammar is a type of formal grammar that is used to define regular languages. A regular grammar consists of a set of production rules, where each rule is of the form A → αB or A → α, where A and B are non-terminals, α is a string of terminals and non-terminals, and ε represents the empty string. The left-hand side of the rule (A) can only have one non-terminal symbol.

Context-Free Grammar:
A context-free grammar is a type of formal grammar that is used to define context-free languages. A context-free grammar consists of a set of production rules, where each rule is of the form A → α, where A is a non-terminal and α is a string of terminals and non-terminals. The left-hand side of the rule (A) can have multiple non-terminal symbols.

Which statement is correct?

Now let's analyze each of the given statements to determine which one is correct:

a) All regular grammars are context-free but not vice versa.
This statement is correct. Every regular grammar is also a context-free grammar. Regular languages are a subset of context-free languages. Regular grammars can be represented by both finite state machines and regular expressions, whereas context-free grammars cannot.

b) All context-free grammars are regular grammars but not vice versa.
This statement is incorrect. Context-free grammars are more expressive than regular grammars. There exist context-free languages that cannot be generated by regular grammars.

c) Regular grammar and context-free grammar are the same entity.
This statement is incorrect. Regular grammar and context-free grammar are two different types of grammars. Regular grammars define regular languages, while context-free grammars define context-free languages.

d) None of the mentioned.
This statement is incorrect. As explained above, statement a) is correct.

Therefore, the correct statement is option a) "All regular grammars are context-free but not vice versa."