The Chomsky Hierarchy is a classification of formal languages in terms of their generative power, ranging from regular languages to recursively enumerable languages. It provides a framework to understand the computational complexity and capabilities of different types of grammars and automata in theoretical computer science and linguistics.
Non-context-free languages are languages that cannot be generated by any context-free grammar and require more computational power to be recognized, such as a pushdown automaton with two stacks. These languages often exhibit patterns or dependencies that cannot be captured by the limited memory of a single-stack automaton, making them more complex than context-free languages.
Context-free languages are a class of formal languages that can be generated by context-free grammars, which are used to describe the syntax of programming languages and natural languages. They are recognized by pushdown automata, making them more powerful than regular languages but less powerful than context-sensitive languages.