Formal Languages Questions
The main difference between a context-free grammar and a recursively enumerable grammar lies in their generative power and the complexity of their parsing.
A context-free grammar (CFG) is a formal grammar where each production rule has a single nonterminal symbol on the left-hand side and a sequence of terminals and/or nonterminals on the right-hand side. CFGs are used to describe context-free languages, which can be recognized by pushdown automata. These languages have a relatively simple structure and can be parsed efficiently using algorithms like the CYK or Earley parsers.
On the other hand, a recursively enumerable grammar (also known as a type-0 grammar or unrestricted grammar) is a formal grammar that allows more flexibility in its production rules. In a recursively enumerable grammar, the left-hand side of a production rule can be any string of symbols, including both terminals and nonterminals. This makes recursively enumerable grammars more expressive and capable of generating more complex languages, including non-context-free languages. However, the parsing of recursively enumerable languages is undecidable, meaning that there is no algorithm that can determine whether a given string belongs to the language generated by a recursively enumerable grammar.
In summary, the key difference between a context-free grammar and a recursively enumerable grammar is the complexity of the languages they can generate and the parsing algorithms that can be applied to them. Context-free grammars generate context-free languages, which have a simpler structure and can be parsed efficiently. Recursively enumerable grammars, on the other hand, can generate more complex languages, including non-context-free languages, but their parsing is undecidable.