Chomsky Hierarchy | Cognitive Science | BSc.CSIT (TU) | Fourth Semester

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Chomsky HierarchyChomsky Hierarchy : Type 0, Type 1, Type 2, Type 3 grammars
Subject: Cognitive Science | BSc.CSIT (TU)
Fourth Semester | Tribhuvan University

Chomsky Hierarchy
When Noam Chomsky first formalized generative grammars in 1956, he classified them into types now known as the Chomsky hierarchy. The difference between these types is that they have increasingly strict production rules and can express fewer formal languages. The Chomsky hierarchy consists of the following levels:
Chomsky Hierarchy
Type-0 grammars (unrestricted grammars) include all formal grammars. They generate exactly all languages that can be recognized by a Turing machine. These languages are also known as the recursively enumerable languages. Note that this is different from the recursive languages which can be decided by an always-halting Turing machine.

Type-1 grammars (context-sensitive grammars) generate the context-sensitive languages. These grammars have rules of the form αAβ –> αγβ with A a non-terminal and α, β and γ strings of terminals and non-terminals. The strings α and β may be empty, but γ must be nonempty. The rule S –> Ɛ is allowed if S does not appear on the right side of any rule. The languages described by these grammars are exactly all languages that can be recognized by a linear bounded automaton (a non-deterministic Turing machine whose tape is bounded by a constant times the length of the input.)

Type-2 grammars (context-free grammars) generate the context-free languages. These are defined by rules of the form A –> γ with A a non-terminal and γ a string of terminals and non-terminals. These languages are exactly all languages that can be recognized by a non-deterministic pushdown automaton. Context-free languages are the theoretical basis for the syntax of most programming languages.

Type-3 grammars (regular grammars) generate the regular languages. Such a grammar restricts its rules to a single non-terminal on the left-hand side and a right-hand side consisting of a single terminal, possibly followed (or preceded, but not both in the same grammar) by a single non-terminal. The rule S –> Ɛ is also allowed here if S does not appear on the right side of any rule. These languages are exactly all languages that can be decided by a finite state automaton. Additionally, this family of formal languages can be obtained by regular expressions. Regular languages are commonly used to define search patterns and the lexical structure of programming languages.

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