A formal grammar of this type consists of:
a finite set of production rules (left-hand side right-hand side) where each side consists of a sequence of these symbols
a finite set of nonterminal symbols (indicating that some production rule can yet be applied)
a finite set of terminal symbols (indicating that no production rule can be applied)
a start symbol (a distinguished nonterminal symbol)
The Chomsky hierarchy consists of the following levels:
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 with a nonterminal and , and strings of terminals and nonterminals. The strings and may be empty, but must be nonempty. The rule is allowed if 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 nondeterministic 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 with a nonterminal and a string of terminals and nonterminals. These languages are exactly all languages that can be recognized by a non-deterministic pushdown automaton. Context-free languages– or rather the subset of deterministic context-free language – are the theoretical basis for the phrase structure of most programming languages, though their syntax also includes context-sensitive name resolution due to declarations and scope. Often a subset of grammars are used to make parsing easier, such as by an LL parser.
Type-3 grammars (regular grammars) generate the regular languages. Such a grammar restricts its rules to a single nonterminal on the left-hand side and a right-hand side consisting of a single terminal, possibly followed by a single nonterminal (right regular). Alternatively, the right-hand side of the grammar can consist of a single terminal, possibly preceded by a single nonterminal (left regular); these generate the same languages – however, if left-regular rules and right-regular rules are combined, the language need no longer be regular. The rule is also allowed here if 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.