Terminal and nonterminal symbols

Nonterminal symbols

Nonterminal symbols are those symbols that cannot appear in the formal language defined by a formal grammar. A formal grammar includes a start symbol, which is a designated member of the set of nonterminal symbols. We can derive a set of strings of only terminal symbols by successively applying the production rules. The generated set is a formal language over the set of terminal symbols.

Context-free grammars 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. 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 can replace which other symbols; these rules can 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 can be replaced, and a body, or right-hand side, which consists of a string that can 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 N of nonterminal symbols.
• A finite set Σ of terminal symbols that is disjoint from N.
• A finite set P of production rules, each rule of the form

${\displaystyle V^{\ast }N{V}^{\ast }\to V^{\ast }}$

where ${\displaystyle V^{\ast }}$ denotes the set of all possible finite-length strings over the vocabulary ${\displaystyle V=\mathrm{\Sigma }\cup N}$ using Kleene star. That is, each production rule replaces one string of symbols that contains at least one nonterminal symbol with another. In the case that the body consists solely of the empty string, it can be denoted with a special notation (often Λ, e or ε) to avoid confusion.

• A distinguished symbol ${\displaystyle S\in N}$ that is the start symbol.

A grammar is formally defined as the ordered quadruple ${\displaystyle ⟨N,\mathrm{\Sigma },P,S⟩}$. Such a formal grammar is often called a rewriting system or a phrase structure grammar in the literature.

Example

Backus–Naur form is a notation for expressing certain grammars. For instance, the following production rules in Backus-Naur form are used to represent an integer (which can be signed):

<span id="mwiw"></span><span class="p" id="mwjA"><</span><span class="nc" id="mwjQ">digit</span><span class="p" id="mwjg">></span> <span class="o" id="mwjw">::=</span> '0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9' <span class="p" id="mwkA"><</span><span class="nc" id="mwkQ">integer</span><span class="p" id="mwkg">></span> <span class="o" id="mwkw">::=</span> ['-'] <span class="p" id="mwlA"><</span><span class="nc" id="mwlQ">digit</span><span class="p" id="mwlg">></span> {<span class="p" id="mwlw"><</span><span class="nc" id="mwmA">digit</span><span class="p" id="mwmQ">></span>}

In this example, terminal symbols are ${\displaystyle \{-,0,1,2,3,4,5,6,7,8,9\}}$, and nonterminal symbols are ${\displaystyle \{}$<digit>, <integer>${\displaystyle \}}$.

Another example is:

${\displaystyle \text{S}⟶\text{cAd}}$

${\displaystyle \text{A}⟶\text{a}|\text{ab}}$

In this example, terminal symbols are ${\displaystyle \{\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}\}}$, and nonterminal symbols are ${\displaystyle \{\mathrm{S},\mathrm{A}\}}$.