Terminal and nonterminal symbols

Informal languages,terminal and nonterminal symbolsare thelexical elementsused in specifying theproduction rulesconstituting aformal grammar.Terminal symbolsare the elementary symbols of thelanguagedefined as part of a formal grammar.Nonterminal symbols(orsyntactic variables) are replaced by groups of terminal symbols according to the production rules.

The terminals and nonterminals of a particular grammar are in twocompletely separate sets.

Terminal symbols[edit]

Terminal symbols are 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. In this grammar, the symbolБis a terminal symbol andΨis both a non-terminal symbol and the start symbol. The production rules for creating strings are as follows:

The symbolΨcan becomeБΨ
The symbolΨcan becomeБ

HereБis a terminal symbol because no rule exists which would change it into something else. On the other hand,Ψhas two rules that can change it, thus it is nonterminal. Aformal languagedefined orgeneratedby a particular grammar is the set of strings that can be produced by the grammarand that consist only of terminal symbols.Diagram 1 illustrates a string that can be produced with this grammar.

Nonterminal symbols[edit]

Nonterminal symbols are those symbols that can be replaced. They may also be called simplysyntactic variables.A formal grammar includes astart 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 ofterminalstrings that can be so derived.

Context-free grammarsare 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-deterministicpush down automaton.Context-free languages are the theoretical basis for the syntax of mostprogramming languages.

Production rules[edit]

A grammar is defined byproduction 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 ahead,or left-hand side, which consists of the string that may be replaced, and abody,or right-hand side, which consists of a string that may replace it. Rules are often written in the formhead→body;e.g., the rulea→bspecifies thatacan be replaced byb.

In the classic formalization of generative grammars first proposed byNoam Chomskyin the 1950s,^[2]^[3]a grammarGconsists of the following components:

A finite set $N$ ofnonterminal symbols.
A finite set $Σ$ ofterminal symbolsthat isdisjointfrom $N$ .
A finite set $P$ ofproduction rules,each rule of the form

(\Sigma \cup N)^{*}N(\Sigma \cup N)^{*}\rightarrow (\Sigma \cup N)^{*}

where

{}^{*}

is theKleene staroperator and

\cup

denotesset union,so

(\Sigma \cup N)^{*}

represents zero or more symbols, and

N

means onenonterminalsymbol. 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 theempty string^{[note 1]},it may be denoted with a special notation (often

Λ

,

e

or

ε

) in order to avoid confusion.

A distinguished symbol $S\in N$ that is thestart symbol.

A grammar is formally defined as the ordered quadruple $\langle N,\Sigma,P,S\rangle$ .Such a formal grammar is often called arewriting systemor aphrase structure grammarin the literature.^[4]^[5]

Example[edit]

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

<digit>::='0' | '1' | '2' | '3' | '4' | '5' | '6' | '7' | '8' | '9'
<integer>::=['-']<digit>{<digit>}

In this example, the symbols (-,0,1,2,3,4,5,6,7,8,9) are terminal symbols and<digit>and<integer>are nonterminal symbols. ^{[note 2]}

Another example is:

{\ce {S -> cAd}}

{\ce {A -> a | ab}}

In this example, the symbols $a,b,c,d$ are terminal symbols and $S,A$ are nonterminal symbols.

Notes[edit]

^It contains no symbols at all.
^This example supports strings with leading zeroes like "0056" or "0000", as well as negative zero strings like "-0" and "-00000".

References[edit]

^Rosen, K. H. (2012). Discrete mathematics and its applications. McGraw-Hill. pages 847-851
^Chomsky, Noam (1956). "Three Models for the Description of Language".IRE Transactions on Information Theory.2(3): 113–123.doi:10.1109/TIT.1956.1056813.S2CID 19519474.
^Chomsky, Noam (1957).Syntactic Structures.The Hague:Mouton.
^Ginsburg, Seymour(1975).Algebraic and automata theoretic properties of formal languages.North-Holland. pp. 8–9.ISBN 0-7204-2506-9.
^Harrison, Michael A.(1978).Introduction to Formal Language Theory.Reading, Mass.: Addison-Wesley Publishing Company. pp.13.ISBN 0-201-02955-3.

[4] It contains no symbols at all.

[7] This example supports strings with leading zeroes like "0056" or "0000", as well as negative zero strings like "-0" and "-00000".

[1] Rosen, K. H. (2012). Discrete mathematics and its applications. McGraw-Hill. pages 847-851

[Chomsky1956-2] Chomsky, Noam (1956). "Three Models for the Description of Language".IRE Transactions on Information Theory.2(3): 113–123.doi:10.1109/TIT.1956.1056813.S2CID 19519474.

[Chomsky1957-3] Chomsky, Noam (1957).Syntactic Structures.The Hague:Mouton.

[5] Ginsburg, Seymour(1975).Algebraic and automata theoretic properties of formal languages.North-Holland. pp. 8–9.ISBN 0-7204-2506-9.

[6] Harrison, Michael A.(1978).Introduction to Formal Language Theory.Reading, Mass.: Addison-Wesley Publishing Company. pp.13.ISBN 0-201-02955-3.

[1]

[2]

[3]

[note 1]

[4]

[5]

[note 2]