Formal Languages Questions Medium
The Chomsky hierarchy of formal languages is a classification system that categorizes formal languages based on their generative power and the types of grammars that can generate them. It was proposed by Noam Chomsky, a linguist and computer scientist, in the 1950s.
The Chomsky hierarchy consists of four levels, each representing a different type of formal language and the corresponding grammar that can generate it. These levels are:
1. Type 0: Unrestricted Grammar (Recursively Enumerable Languages)
At this level, the most powerful type of formal language is represented. These languages can be generated by unrestricted grammars, which have no restrictions on the production rules. They can generate any language that can be recognized by a Turing machine, making them equivalent to the class of recursively enumerable languages.
2. Type 1: Context-Sensitive Grammar (Context-Sensitive Languages)
Context-sensitive languages can be generated by context-sensitive grammars, where the production rules have certain restrictions. The left-hand side of the production rule can be replaced by a string of symbols, but the length of the replacement string cannot be shorter than the length of the original string. These languages can be recognized by linear-bounded automata, which are Turing machines with a linear amount of working memory.
3. Type 2: Context-Free Grammar (Context-Free Languages)
Context-free languages can be generated by context-free grammars, where the left-hand side of the production rule consists of a single non-terminal symbol. The replacement string on the right-hand side can be any combination of terminal and non-terminal symbols. These languages can be recognized by pushdown automata, which are finite-state machines with an additional stack memory.
4. Type 3: Regular Grammar (Regular Languages)
Regular languages can be generated by regular grammars, which have the simplest form of production rules. The left-hand side of the production rule consists of a single non-terminal symbol, and the right-hand side can only be a single terminal symbol or a terminal symbol followed by a non-terminal symbol. These languages can be recognized by finite-state automata, which are the simplest form of automata.
The Chomsky hierarchy provides a framework for understanding the complexity and expressive power of different types of formal languages. It also helps in the design and analysis of programming languages, compilers, and formal language processing algorithms.