In
computer science, terminal and nonterminal symbols are the lexical elements used in specifying the
production rules constituting a
formal grammar
In formal language theory, a grammar (when the context is not given, often called a formal grammar for clarity) describes how to form strings from a language's alphabet that are valid according to the language's syntax. A grammar does not describe ...
. ''Terminal symbols'' are the elementary symbols of the
language defined by a formal grammar. ''Nonterminal symbols'' (or ''syntactic variables'') are replaced by groups of terminal symbols according to the production rules.
The terminals and nonterminals of a particular grammar are two
disjoint sets.
Terminal symbols
Terminal symbols are literal symbols that may appear in the outputs of the production rules of a formal grammar and which cannot be changed using the rules of the grammar. Applying the rules recursively to a source string of symbols will usually terminate in a final output string consisting only of terminal symbols.
Consider a grammar defined by two rules. Using pictoric marks interacting with each other:
# The symbol
ר
can become
ди
# The symbol
ר
can become
д
Here
д
is a terminal symbol because no rule exists which would change it into something else. On the other hand,
ר
has two rules that can change it, thus it is nonterminal. A
formal language defined or ''generated'' by a particular grammar is the set of strings that can be produced by the grammar ''and that consist only of terminal symbols''.
Nonterminal symbols
Nonterminal symbols are those symbols that can be replaced. They may also be called simply ''syntactic variables''. A formal grammar includes a ''start symbol'', a designated member of the set of nonterminals from which all the strings in the language may be derived by successive applications of the production rules. In fact, the language defined by a grammar is precisely the set of ''terminal'' strings that can be so derived.
Context-free grammar
In formal language theory, a context-free grammar (CFG) is a formal grammar whose production rules are of the form
:A\ \to\ \alpha
with A a ''single'' nonterminal symbol, and \alpha a string of terminals and/or nonterminals (\alpha can be emp ...
s are those grammars in which the left-hand side of each production rule consists of only a single nonterminal symbol. This restriction is non-trivial; not all languages can be generated by context-free grammars. Those that can are called context-free languages.
These are exactly the languages that can be recognized by a non-deterministic
push down automaton
In the theory of computation, a branch of theoretical computer science, a pushdown automaton (PDA) is
a type of automaton that employs a stack.
Pushdown automata are used in theories about what can be computed by machines. They are more capa ...
. Context-free languages are the theoretical basis for the syntax of most
programming languages.
Production rules
A grammar is defined by
production rules (or just 'productions') that specify which symbols may replace which other symbols; these rules may be used to
generate strings, or to parse them. Each such rule has a ''head'', or left-hand side, which consists of the string that may be replaced, and a ''body'', or right-hand side, which consists of a string that may replace it. Rules are often written in the form ''head'' → ''body''; e.g., the rule ''a'' → ''b'' specifies that ''a'' can be replaced by ''b''.
In the classic formalization of generative grammars first proposed by
Noam Chomsky in the 1950s,
a grammar ''G'' consists of the following components:
* A finite set of ''nonterminal symbols''.
* A finite set of ''terminal symbols'' that is
disjoint from .
* A finite set of ''production rules'', each rule of the form
::
:where
is the
Kleene star
In mathematical logic and computer science, the Kleene star (or Kleene operator or Kleene closure) is a unary operation, either on sets of strings or on sets of symbols or characters. In mathematics,
it is more commonly known as the free monoid ...
operator and denotes
set union, so
represents zero or more symbols, and means one ''nonterminal'' symbol. That is, each production rule maps from one string of symbols to another, where the first string contains at least one nonterminal symbol. In the case that the body consists solely of the
empty string, it may be denoted with a special notation (often , or ) in order to avoid confusion.
* A distinguished symbol
that is the ''start symbol''.
A grammar is formally defined as the ordered quadruple
. Such a formal grammar is often called a
rewriting system or a
phrase structure grammar in the literature.
Example
For instance, the following represents an integer (which may be signed) expressed in a variant of
Backus–Naur form
In computer science, Backus–Naur form () or Backus normal form (BNF) is a metasyntax notation for context-free grammars, often used to describe the syntax of languages used in computing, such as computer programming languages, document formats ...
:
::= '0' , '1' , '2' , '3' , '4' , '5' , '6' , '7' , '8' , '9'
::= -'
In this example, the symbols () are terminal symbols and and are nonterminal symbols.
Another example is:
:
S -> cAd
:
A -> a , ab
In this example, the symbols are terminal symbols and are nonterminal symbols.
See also
*
Recursive grammar
Notes
References
{{reflist
Formal languages
Pattern matching