In computer science
, a grammar
is informally called a recursive grammar if it contains production rules
that are recursive
, meaning that expanding a non-terminal according to these rules can eventually lead to a string that includes the same non-terminal again. Otherwise it is called a non-recursive grammar.
For example, a grammar for a context-free language
is left recursive
if there exists a non-terminal symbol ''A'' that can be put through the production rules to produce a string with ''A'' (as the leftmost symbol).
All types of grammars in the Chomsky hierarchy
can be recursive and it is recursion that allows the production of infinite sets of words.
A non-recursive grammar can produce only a finite language; and each finite language can be produced by a non-recursive grammar.
For example, a straight-line grammar
produces just a single word.
A recursive context-free grammar that contains no useless rules
necessarily produces an infinite language. This property forms the basis for an algorithm
that can test efficiently whether a context-free grammar produces a finite or infinite language.