What is the difference between a recursively enumerable grammar and a context-sensitive grammar?

A recursively enumerable grammar and a context-sensitive grammar are both types of formal grammars used in the field of formal languages. However, there are some key differences between the two.

1. Definition:
- Recursively Enumerable Grammar: A recursively enumerable grammar is a type of formal grammar where the production rules allow for the generation of any string that can be recognized by a Turing machine. In other words, it can generate any language that can be accepted by a Turing machine.
- Context-Sensitive Grammar: A context-sensitive grammar is a type of formal grammar where the production rules are more restricted compared to recursively enumerable grammars. In a context-sensitive grammar, the left-hand side of the production rule can have at most the same number of symbols as the right-hand side, and the production rules can be applied in any context.

2. Rule Application:
- Recursively Enumerable Grammar: In a recursively enumerable grammar, the production rules can be applied in any order and any number of times. This allows for a more flexible generation of strings, but it also means that the grammar may generate non-terminating or ambiguous strings.
- Context-Sensitive Grammar: In a context-sensitive grammar, the production rules must be applied in a specific order and a limited number of times. The context in which a rule can be applied is determined by the surrounding symbols. This ensures that the generated strings have a specific structure and avoids non-terminating or ambiguous strings.

3. Language Generation:
- Recursively Enumerable Grammar: A recursively enumerable grammar can generate any language that can be recognized by a Turing machine. This includes both regular languages, context-free languages, and even some non-context-free languages.
- Context-Sensitive Grammar: A context-sensitive grammar can generate context-sensitive languages, which are a proper subset of recursively enumerable languages. Context-sensitive languages are more powerful than context-free languages but less powerful than recursively enumerable languages.

In summary, the main difference between a recursively enumerable grammar and a context-sensitive grammar lies in the generative power and the restrictions on rule application. Recursively enumerable grammars can generate any language recognized by a Turing machine, while context-sensitive grammars have more limited generative power and stricter rules for rule application.