The Chomsky hierarchy is a classification of formal languages based on the complexity of the grammars (or rules) used to generate them. It consists of four levels, each of which is characterized by the type of grammar that is used to define the language. The levels are:
The set of all regular languages is known as Type-3 (or Regular) languages. Regular languages can be recognized by finite automata, which are simple machines that read input strings and determine whether they belong to the language. Regular languages are defined by regular expressions, which are concise notation for specifying sets of strings.
The set of all context-free languages is known as Type-2 (or Context-Free) languages. Context-free languages can be recognized by pushdown automata, which are machines that use a stack to store and manipulate symbols as they read an input string. Context-free languages are defined by context-free grammars, which consist of a set of rules for generating strings.
The set of all context-sensitive languages is known as Type-1 (or Context-Sensitive) languages. Context-sensitive languages can be recognized by linear-bounded automata, which are machines that have a bounded amount of memory in addition to a read-only input tape and a write-only output tape. Context-sensitive languages are defined by context-sensitive grammars, which are similar to context-free grammars, but with stricter rules for generating strings.
The set of all recursive languages is known as Type-0 (or Recursive) languages. Recursive languages can be recognized by Turing machines, which are theoretical machines that can perform any computation that is algorithmically decidable. Recursive languages are defined by recursive definitions, which are mathematical definitions that specify how to build a string in a language.
The Chomsky hierarchy is often depicted as a pyramid, with the regular languages at the bottom and the recursive languages at the top. Each level of the hierarchy is a proper subset of the level above it, meaning that any language that can be generated by a grammar at a higher level can also be generated by a grammar at a lower level.
For example, all regular languages are also context-free, because regular languages can be defined using context-free grammars. However, not all context-free languages are regular, because there are context-free languages that cannot be defined using regular expressions. Similarly, all context-free languages are also context-sensitive, but not all context-sensitive languages are context-free, and all recursive languages are context-sensitive, but not all context-sensitive languages are recursive.
The Chomsky hierarchy is a useful way of organizing and classifying formal languages based on their complexity, and it has played a significant role in the development of theoretical computer science and the study of formal languages and automata.
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