Formal language

In mathematics, computer science and linguistics, a formal language is one that has a particular set of symbols, and whose expressions are made according to a particular set of rules. The symbol ${\displaystyle {\mathcal {L}}}$ is often used as a variable for formal languages in logic.^[1]

Unlike natural languages, the symbols and formulas in formal languages are syntactically and semantically related to one another in a precise way.^[2] As a result, formal languages are completely (or almost completely) void of ambiguity.^[3]

Examples

Some examples of formal languages include:

• The set of all words over ${\displaystyle {a,b}\,}$
• The set ${\displaystyle \left\{a^{n}\right\}}$, where ${\displaystyle n\,}$ is a natural number and ${\displaystyle a^{n}\,}$ means ${\displaystyle a\,}$ repeated ${\displaystyle n}$ times
• Finite languages, such as ${\displaystyle \{\{a,b\},\{a,aa,bba\}\}\,}$
• The set of syntactically correct programs in a given programming language
• The set of inputs upon which a certain Turing machine halts

Specification

A formal language can be specified in a great variety of ways, such as:

• Strings produced by some formal grammar (see Chomsky hierarchy)
• Strings described or matched by a regular expression
• Strings accepted by some automaton, such as a Turing machine or finite state automaton
• Strings indicated by a decision procedure (a set of related yes/no questions) where the answer is 'yes'

Related pages

• Language for languages in general
• Syntax for the form of a language in general
• Semantics for the meanings in a language
• Natural language for languages that are not formal
• Computer language for application of formal languages in computing
• Programming language for the application of formal languages to program computers

References

1. "Comprehensive List of Logic Symbols". Math Vault. 2020-04-06. Retrieved 2020-10-09.
2. "Definition of formal language | Dictionary.com". www.dictionary.com. Retrieved 2020-10-09.
3. "1.11. Formal and Natural Languages — How to Think like a Computer Scientist: Interactive Edition". runestone.academy. Retrieved 2020-10-09.

Further reading

• Hopcroft, J.; Ullman, J. (1979). Introduction to Automata Theory, Languages, and Computation. Addison-Wesley. ISBN 0-201-02988-X.
• Rasiowa, Helena; Sikorski, Roman (1970). "Chapter 6. Algebra of formalized languages". The Mathematics of Metamathematics (3rd ed.). PWN.
• Rozemberg, G.; Salomaa, A., eds. (1979). Introduction to Automata Theory, Languages, and Computation. Addison-Wesley. ISBN 978-3-540-61486-9.

Other websites

• http://icalp06.dsi.unive.it/ Archived 2006-06-09 at the Wayback Machine ICALP 2006 33rd International Colloquium on Automata, Languages and Programming.
• http://www.cs.auckland.ac.nz/CDMTCS/conferences/dlt/DLTConfSeries.html International Conferences on Developments in Language Theory