A production or production rule in computer science is a rewrite rule specifying a symbol substitution that can be recursively performed to generate new symbol sequences.
A finite set of productions
is the main component in the specification of a formal grammar (specifically a generative grammar).
The other components are a finite set
of nonterminal symbols, a finite set (known as an alphabet)
of terminal symbols that is disjoint from
In an unrestricted grammar, a production is of the form
are arbitrary strings of terminals and nonterminals, and
is the empty string, this is denoted by the symbol
(rather than leaving the right-hand side blank).
is the Kleene star operator,
denotes set union, and
If we do not allow the start symbol to occur in
(the word on the right side), we have to replace
on the right side of the cartesian product symbol.
[1] The other types of formal grammar in the Chomsky hierarchy impose additional restrictions on what constitutes a production.
Notably in a context-free grammar, the left-hand side of a production must be a single nonterminal symbol.
So productions are of the form: To generate a string in the language, one begins with a string consisting of only a single start symbol, and then successively applies the rules (any number of times, in any order) to rewrite this string.
This stops when a string containing only terminals is obtained.
The language consists of all the strings that can be generated in this manner.
Any particular sequence of legal choices taken during this rewriting process yields one particular string in the language.
If there are multiple different ways of generating this single string, then the grammar is said to be ambiguous.
For example, assume the alphabet consists of
This process is repeated until we only have symbols from the alphabet (i.e.,
We can write this series of choices more briefly, using symbols:
The language of the grammar is the set of all the strings that can be generated using this process:
{\displaystyle \{ba,abab,aababb,aaababbb,\dotsc \}}