Production (computer science) explained

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)

\Sigma

of terminal symbols that is disjoint from

and a distinguished symbol

S\inN

that is the start symbol.

In an unrestricted grammar, a production is of the form

u\tov

, where

and

are arbitrary strings of terminals and nonterminals, and

may not be the empty string. If

is the empty string, this is denoted by the symbol

\epsilon

, or

(rather than leaving the right-hand side blank). So productions are members of the cartesian product

V^*NV^* x V^*=(V^{*\setminus\Sigma}^*) x V^*

where

V:=N\cup\Sigma

is the vocabulary,

{}^*

is the Kleene star operator,

V^*NV^*

indicates concatenation,

\cup

denotes set union, and

\setminus

denotes set minus or set difference. If we do not allow the start symbol to occur in

(the word on the right side), we have to replace

V^*

(V\setminus\{S\})^*

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:

N\to(N\cup\Sigma)^*

Grammar generation

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

and

, with the start symbol

, and we have the following rules:

S → aSb

S → ba

then we start with

, and can choose a rule to apply to it. If we choose rule 1, we replace

with

aSb

and obtain the string

aSb

. If we choose rule 1 again, we replace

with

aSb

and obtain the string

aaSbb

. This process is repeated until we only have symbols from the alphabet (i.e.,

and

). If we now choose rule 2, we replace

with

and obtain the string

aababb

, and are done. We can write this series of choices more briefly, using symbols:

S ⇒ aSb ⇒ aaSbb ⇒ aababb

. The language of the grammar is the set of all the strings that can be generated using this process:

\{ba,abab,aababb,aaababbb,...c\}

References

See Klaus Reinhardt: Prioritatszahlerautomaten und die Synchronisation von Halbspursprachen ; Fakultät Informatik der Universität Stuttgart; 1994 (German)

Production (computer science) explained

Grammar generation

See also

References