A formal grammar provides an axiom schema for a formal language, which is a set of finite-length sequences of symbols that may be constructed by applying production rules to another sequence of symbols. A rule may be applied by replacing an occurrence of the symbols on its left-hand side with those that appear on its right-hand side. A sequence of rule applications is called a derivation. Such a grammar defines the formal language: all words consisting solely of terminal symbols which can be reached by a derivation from the start symbol. Nonterminals are often represented by uppercase letters, terminals by lowercase letters, and the start symbol by. For example, the grammar with terminals, nonterminals, production rules and start symbol, defines the language of all words of the form . The following is a simpler grammar that defines the same language: Terminals, Nonterminals, Start symbol, Production rules As another example, a grammar for a toy subset of English language is given by: ;terminals: ;nonterminals: ;production rules and start symbol. An example derivation is Other sequences that can be derived from this grammar are: "ideas hate great linguists", and "ideas generate". While these sentences are nonsensical, they are syntactically correct. A syntactically incorrect sentence cannot be derived from this grammar. See "Colorless green ideas sleep furiously" for a similar example given by Chomsky in 1957; see Phrase structure grammar and Phrase structure rules for more natural language examples and the problems of formal grammar in that area.
Type-0 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 or Turing-recognizable languages. Note that this is different from the recursive languages, which can be decided by an always-halting Turing machine.
Type-1 grammars
Type-1 grammars generate context-sensitive languages. These grammars have rules of the form with a nonterminal and, and strings of terminals and/or 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
Type-2 grammars
Type-2 grammars generate the context-free languages. These are defined by rules of the form with being a nonterminal and being a string of terminals and/or nonterminals. These languages are exactly all languages that can be recognized by a non-deterministic pushdown automaton. Context-free languages—or rather its 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 is used to make parsing easier, such as by an LL parser.
Type-3 grammars
Type-3 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. Alternatively, the right-hand side of the grammar can consist of a single terminal, possibly preceded by a single nonterminal. 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.
