Formal Languages Questions
The main difference between a recursively enumerable language and a context-free language lies in their respective levels of computational power.
A recursively enumerable language is a language for which there exists a Turing machine that can accept any string belonging to the language, but may either reject or loop indefinitely for strings not belonging to the language. In other words, a language is recursively enumerable if there exists an algorithm that can recognize it, but this algorithm may not always halt for strings not in the language.
On the other hand, a context-free language is a language that can be generated by a context-free grammar. A context-free grammar consists of a set of production rules that define how symbols can be rewritten. These production rules are applied in a context-free manner, meaning that the left-hand side of a production rule can be replaced by the right-hand side regardless of the context in which it appears. Context-free languages can be recognized by pushdown automata, which are computational devices with a finite control and an unbounded stack.
In summary, the key difference is that recursively enumerable languages can be recognized by Turing machines that may not always halt, while context-free languages can be generated by context-free grammars and recognized by pushdown automata.