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In
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from prem ...
,
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
,
computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
, and
linguistics Linguistics is the scientific study of human language. It is called a scientific study because it entails a comprehensive, systematic, objective, and precise analysis of all aspects of language, particularly its nature and structure. Ling ...
, a formal language consists of words whose
letters Letter, letters, or literature may refer to: Characters typeface * Letter (alphabet), a character representing one or more of the sounds used in speech; any of the symbols of an alphabet. * Letterform, the graphic form of a letter of the alpha ...
are taken from an
alphabet An alphabet is a standardized set of basic written graphemes (called letters) that represent the phonemes of certain spoken languages. Not all writing systems represent language in this way; in a syllabary, each character represents a syllab ...
and are well-formed according to a specific set of rules. The alphabet of a formal language consists of symbols, letters, or tokens that concatenate into strings of the language. Each string concatenated from symbols of this alphabet is called a word, and the words that belong to a particular formal language are sometimes called ''well-formed words'' or '' well-formed formulas''. A formal language is often defined by means of a formal grammar such as a
regular grammar In theoretical computer science and formal language theory, a regular grammar is a grammar that is ''right-regular'' or ''left-regular''. While their exact definition varies from textbook to textbook, they all require that * all production rule ...
or context-free grammar, which consists of its
formation rule In mathematical logic, formation rules are rules for describing which strings of symbols formed from the alphabet of a formal language are syntactically valid within the language. These rules only address the location and manipulation of the str ...
s. In computer science, formal languages are used among others as the basis for defining the grammar of programming languages and formalized versions of subsets of natural languages in which the words of the language represent concepts that are associated with particular meanings or
semantics Semantics (from grc, σημαντικός ''sēmantikós'', "significant") is the study of reference, meaning, or truth. The term can be used to refer to subfields of several distinct disciplines, including philosophy, linguistics and comput ...
. In computational complexity theory, decision problems are typically defined as formal languages, and complexity classes are defined as the sets of the formal languages that can be parsed by machines with limited computational power. In
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from prem ...
and the foundations of mathematics, formal languages are used to represent the syntax of axiomatic systems, and mathematical formalism is the philosophy that all of mathematics can be reduced to the syntactic manipulation of formal languages in this way. The field of formal language theory studies primarily the purely
syntactical In linguistics, syntax () is the study of how words and morphemes combine to form larger units such as phrases and sentences. Central concerns of syntax include word order, grammatical relations, hierarchical sentence structure (constituency ...
aspects of such languages—that is, their internal structural patterns. Formal language theory sprang out of linguistics, as a way of understanding the syntactic regularities of natural languages.


History

In the 17th Century, Gottfried Leibniz imagined and described the characteristica universalis, a universal and formal language which utilised pictographs. During this period, Carl Friedrich Gauss also investigated the problem of Gauss codes. Gottlob Frege attempted to realize Leibniz’s ideas, through a notational system first outlined in ''
Begriffsschrift ''Begriffsschrift'' (German for, roughly, "concept-script") is a book on logic by Gottlob Frege, published in 1879, and the formal system set out in that book. ''Begriffsschrift'' is usually translated as ''concept writing'' or ''concept nota ...
'' (1879) and more fully developed in his 2-volume Grundgesetze der Arithmetik (1893/1903). This described a "formal language of pure language." In the first half of the 20th Century, several developments were made with relevance to formal languages. Axel Thue published four papers relating to words and language between 1906 and 1914. The last of these introduced what Emil Post later termed ‘Thue Systems’, and gave an early example of an undecidable problem. Post would later use this paper as the basis for a 1947 proof “that the word problem for semigroups was recursively insoluble”, and later devised the canonical system for the creation of formal languages. Noam Chomsky devised an abstract representation of formal and natural languages, known as the Chomsky hierarchy. In 1959 John Backus developed the Backus-Naur form to describe the syntax of a high level programming language, following his work in the creation of FORTRAN. Peter Naur invented a similar scheme in 1960.


Words over an alphabet

An alphabet, in the context of formal languages, can be any
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
, although it often makes sense to use an
alphabet An alphabet is a standardized set of basic written graphemes (called letters) that represent the phonemes of certain spoken languages. Not all writing systems represent language in this way; in a syllabary, each character represents a syllab ...
in the usual sense of the word, or more generally any finite character encoding such as
ASCII ASCII ( ), abbreviated from American Standard Code for Information Interchange, is a character encoding standard for electronic communication. ASCII codes represent text in computers, telecommunications equipment, and other devices. Because ...
or
Unicode Unicode, formally The Unicode Standard,The formal version reference is is an information technology standard for the consistent encoding, representation, and handling of text expressed in most of the world's writing systems. The standard, ...
. The elements of an alphabet are called its letters. An alphabet may contain an infinite number of elements; however, most definitions in formal language theory specify alphabets with a finite number of elements, and most results apply only to them. A word over an alphabet can be any finite sequence (i.e., string) of letters. The set of all words over an alphabet Σ is usually denoted by Σ* (using the Kleene star). The length of a word is the number of letters it is composed of. For any alphabet, there is only one word of length 0, the ''empty word'', which is often denoted by e, ε, λ or even Λ. By
concatenation In formal language theory and computer programming, string concatenation is the operation of joining character strings end-to-end. For example, the concatenation of "snow" and "ball" is "snowball". In certain formalisations of concatenat ...
one can combine two words to form a new word, whose length is the sum of the lengths of the original words. The result of concatenating a word with the empty word is the original word. In some applications, especially in
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from prem ...
, the alphabet is also known as the ''vocabulary'' and words are known as ''formulas'' or ''sentences''; this breaks the letter/word metaphor and replaces it by a word/sentence metaphor.


Definition

A formal language ''L'' over an alphabet Σ is a subset of Σ*, that is, a set of words over that alphabet. Sometimes the sets of words are grouped into expressions, whereas rules and constraints may be formulated for the creation of 'well-formed expressions'. In computer science and mathematics, which do not usually deal with natural languages, the adjective "formal" is often omitted as redundant. While formal language theory usually concerns itself with formal languages that are described by some syntactical rules, the actual definition of the concept "formal language" is only as above: a (possibly infinite) set of finite-length strings composed from a given alphabet, no more and no less. In practice, there are many languages that can be described by rules, such as regular languages or context-free languages. The notion of a formal grammar may be closer to the intuitive concept of a "language," one described by syntactic rules. By an abuse of the definition, a particular formal language is often thought of as being equipped with a formal grammar that describes it.


Examples

The following rules describe a formal language  over the alphabet Σ = : * Every nonempty string that does not contain "+" or "=" and does not start with "0" is in . * The string "0" is in . * A string containing "=" is in  if and only if there is exactly one "=", and it separates two valid strings of . * A string containing "+" but not "=" is in  if and only if every "+" in the string separates two valid strings of . * No string is in  other than those implied by the previous rules. Under these rules, the string "23+4=555" is in , but the string "=234=+" is not. This formal language expresses
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
s, well-formed additions, and well-formed addition equalities, but it expresses only what they look like (their
syntax In linguistics, syntax () is the study of how words and morphemes combine to form larger units such as phrases and sentences. Central concerns of syntax include word order, grammatical relations, hierarchical sentence structure ( constituenc ...
), not what they mean (
semantics Semantics (from grc, σημαντικός ''sēmantikós'', "significant") is the study of reference, meaning, or truth. The term can be used to refer to subfields of several distinct disciplines, including philosophy, linguistics and comput ...
). For instance, nowhere in these rules is there any indication that "0" means the number zero, "+" means addition, "23+4=555" is false, etc.


Constructions

For finite languages, one can explicitly enumerate all well-formed words. For example, we can describe a language  as just  = . The degenerate case of this construction is the empty language, which contains no words at all ( = 
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
). However, even over a finite (non-empty) alphabet such as Σ =  there are an infinite number of finite-length words that can potentially be expressed: "a", "abb", "ababba", "aaababbbbaab", .... Therefore, formal languages are typically infinite, and describing an infinite formal language is not as simple as writing ''L'' = . Here are some examples of formal languages: * = Σ*, the set of ''all'' words over Σ; * = * = , where ''n'' ranges over the natural numbers and "a''n''" means "a" repeated ''n'' times (this is the set of words consisting only of the symbol "a"); * the set of syntactically correct programs in a given programming language (the syntax of which is usually defined by a context-free grammar); * the set of inputs upon which a certain Turing machine halts; or * the set of maximal strings of alphanumeric
ASCII ASCII ( ), abbreviated from American Standard Code for Information Interchange, is a character encoding standard for electronic communication. ASCII codes represent text in computers, telecommunications equipment, and other devices. Because ...
characters on this line, i.e.,
the set .


Language-specification formalisms

Formal languages are used as tools in multiple disciplines. However, formal language theory rarely concerns itself with particular languages (except as examples), but is mainly concerned with the study of various types of formalisms to describe languages. For instance, a language can be given as * those strings generated by some formal grammar; * those strings described or matched by a particular regular expression; * those strings accepted by some
automaton An automaton (; plural: automata or automatons) is a relatively self-operating machine, or control mechanism designed to automatically follow a sequence of operations, or respond to predetermined instructions.Automaton – Definition and More ...
, such as a Turing machine or finite-state automaton; * those strings for which some
decision procedure In computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question of the input values. An example of a decision problem is deciding by means of an algorithm wheth ...
(an
algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
that asks a sequence of related YES/NO questions) produces the answer YES. Typical questions asked about such formalisms include: * What is their expressive power? (Can formalism ''X'' describe every language that formalism ''Y'' can describe? Can it describe other languages?) * What is their recognizability? (How difficult is it to decide whether a given word belongs to a language described by formalism ''X''?) * What is their comparability? (How difficult is it to decide whether two languages, one described in formalism ''X'' and one in formalism ''Y'', or in ''X'' again, are actually the same language?). Surprisingly often, the answer to these decision problems is "it cannot be done at all", or "it is extremely expensive" (with a characterization of how expensive). Therefore, formal language theory is a major application area of computability theory and complexity theory. Formal languages may be classified in the Chomsky hierarchy based on the expressive power of their generative grammar as well as the complexity of their recognizing
automaton An automaton (; plural: automata or automatons) is a relatively self-operating machine, or control mechanism designed to automatically follow a sequence of operations, or respond to predetermined instructions.Automaton – Definition and More ...
. Context-free grammars and
regular grammar In theoretical computer science and formal language theory, a regular grammar is a grammar that is ''right-regular'' or ''left-regular''. While their exact definition varies from textbook to textbook, they all require that * all production rule ...
s provide a good compromise between expressivity and ease of parsing, and are widely used in practical applications.


Operations on languages

Certain operations on languages are common. This includes the standard set operations, such as union, intersection, and complement. Another class of operation is the element-wise application of string operations. Examples: suppose L_1 and L_2 are languages over some common alphabet \Sigma. * The ''
concatenation In formal language theory and computer programming, string concatenation is the operation of joining character strings end-to-end. For example, the concatenation of "snow" and "ball" is "snowball". In certain formalisations of concatenat ...
'' L_1 \cdot L_2 consists of all strings of the form vw where v is a string from L_1 and w is a string from L_2. * The ''intersection'' L_1 \cap L_2 of L_1 and L_2 consists of all strings that are contained in both languages * The ''complement'' \neg L_1 of L_1 with respect to \Sigma consists of all strings over \Sigma that are not in L_1. * The Kleene star: the language consisting of all words that are concatenations of zero or more words in the original language; * ''Reversal'': ** Let ''ε'' be the empty word, then \varepsilon^R = \varepsilon, and ** for each non-empty word w = \sigma_1 \cdots \sigma_n (where \sigma_1, \ldots, \sigma_nare elements of some alphabet), let w^R = \sigma_n \cdots \sigma_1, ** then for a formal language L, L^R = \. *
String homomorphism In computer science, in the area of formal language theory, frequent use is made of a variety of string functions; however, the notation used is different from that used for computer programming, and some commonly used functions in the theoretica ...
Such string operations are used to investigate
closure properties Closure may refer to: Conceptual Psychology * Closure (psychology), the state of experiencing an emotional conclusion to a difficult life event Computer science * Closure (computer programming), an abstraction binding a function to its scope * ...
of classes of languages. A class of languages is closed under a particular operation when the operation, applied to languages in the class, always produces a language in the same class again. For instance, the context-free languages are known to be closed under union, concatenation, and intersection with regular languages, but not closed under intersection or complement. The theory of trios and abstract families of languages studies the most common closure properties of language families in their own right., Chapter 11: Closure properties of families of languages. :


Applications


Programming languages

A compiler usually has two distinct components. A
lexical analyzer In computer science, lexical analysis, lexing or tokenization is the process of converting a sequence of characters (such as in a computer program or web page) into a sequence of ''lexical tokens'' ( strings with an assigned and thus identified ...
, sometimes generated by a tool like lex, identifies the tokens of the programming language grammar, e.g. identifiers or keywords, numeric and string literals, punctuation and operator symbols, which are themselves specified by a simpler formal language, usually by means of regular expressions. At the most basic conceptual level, a parser, sometimes generated by a parser generator like yacc, attempts to decide if the source program is syntactically valid, that is if it is well formed with respect to the programming language grammar for which the compiler was built. Of course, compilers do more than just parse the source code – they usually translate it into some executable format. Because of this, a parser usually outputs more than a yes/no answer, typically an abstract syntax tree. This is used by subsequent stages of the compiler to eventually generate an executable containing machine code that runs directly on the hardware, or some intermediate code that requires a virtual machine to execute.


Formal theories, systems, and proofs

In
mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of forma ...
, a ''formal theory'' is a set of sentences expressed in a formal language. A ''formal system'' (also called a ''logical calculus'', or a ''logical system'') consists of a formal language together with a deductive apparatus (also called a ''deductive system''). The deductive apparatus may consist of a set of
transformation rule In the philosophy of logic, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax, and returns a conclusion (or conclusions). For example, the rule of in ...
s, which may be interpreted as valid rules of inference, or a set of axioms, or have both. A formal system is used to derive one expression from one or more other expressions. Although a formal language can be identified with its formulas, a formal system cannot be likewise identified by its theorems. Two formal systems \mathcal and \mathcal may have all the same theorems and yet differ in some significant proof-theoretic way (a formula A may be a syntactic consequence of a formula B in one but not another for instance). A ''formal proof'' or ''derivation'' is a finite sequence of well-formed formulas (which may be interpreted as sentences, or propositions) each of which is an axiom or follows from the preceding formulas in the sequence by a rule of inference. The last sentence in the sequence is a theorem of a formal system. Formal proofs are useful because their theorems can be interpreted as true propositions.


Interpretations and models

Formal languages are entirely syntactic in nature, but may be given
semantics Semantics (from grc, σημαντικός ''sēmantikós'', "significant") is the study of reference, meaning, or truth. The term can be used to refer to subfields of several distinct disciplines, including philosophy, linguistics and comput ...
that give meaning to the elements of the language. For instance, in mathematical
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from prem ...
, the set of possible formulas of a particular logic is a formal language, and an
interpretation Interpretation may refer to: Culture * Aesthetic interpretation, an explanation of the meaning of a work of art * Allegorical interpretation, an approach that assumes a text should not be interpreted literally * Dramatic Interpretation, an event ...
assigns a meaning to each of the formulas—usually, a truth value. The study of interpretations of formal languages is called formal semantics. In mathematical logic, this is often done in terms of model theory. In model theory, the terms that occur in a formula are interpreted as objects within mathematical structures, and fixed compositional interpretation rules determine how the truth value of the formula can be derived from the interpretation of its terms; a ''model'' for a formula is an interpretation of terms such that the formula becomes true.


See also

* Combinatorics on words * Free monoid * Formal method * Grammar framework * Mathematical notation * Associative array *
String (computer science) In computer programming, a string is traditionally a sequence of characters, either as a literal constant or as some kind of variable. The latter may allow its elements to be mutated and the length changed, or it may be fixed (after creation) ...


Notes


References


Citations


Sources

; Works cited * ; General references * A. G. Hamilton, ''Logic for Mathematicians'',
Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambridge University Pr ...
, 1978, . * Seymour Ginsburg, ''Algebraic and automata theoretic properties of formal languages'', North-Holland, 1975, . * Michael A. Harrison, ''Introduction to Formal Language Theory'', Addison-Wesley, 1978. * *
Grzegorz Rozenberg Grzegorz Rozenberg (born 14 March 1942, Warsaw) is a Polish and Dutch computer scientist. His primary research areas are natural computing, formal language and automata theory, graph transformations, and concurrent systems. He is referred ...
, Arto Salomaa, ''Handbook of Formal Languages: Volume I-III'', Springer, 1997, . * Patrick Suppes, ''Introduction to Logic'', D. Van Nostrand, 1957, .


External links

* *
University of Maryland The University of Maryland, College Park (University of Maryland, UMD, or simply Maryland) is a public land-grant research university in College Park, Maryland. Founded in 1856, UMD is the flagship institution of the University System of ...

Formal Language Definitions
* James Power
"Notes on Formal Language Theory and Parsing"
, 29 November 2002. * Drafts of some chapters in the "Handbook of Formal Language Theory", Vol. 1–3, G. Rozenberg and A. Salomaa (eds.), Springer Verlag, (1997): ** Alexandru Mateescu and Arto Salomaa
"Preface" in Vol.1, pp. v–viii, and "Formal Languages: An Introduction and a Synopsis", Chapter 1 in Vol. 1, pp.1–39
** Sheng Yu
"Regular Languages", Chapter 2 in Vol. 1
** Jean-Michel Autebert, Jean Berstel, Luc Boasson

** Christian Choffrut and Juhani Karhumäki
"Combinatorics of Words", Chapter 6 in Vol. 1
** Tero Harju and Juhani Karhumäki
"Morphisms", Chapter 7 in Vol. 1, pp. 439–510
** Jean-Eric Pin
"Syntactic semigroups", Chapter 10 in Vol. 1, pp. 679–746
** M. Crochemore and C. Hancart
"Automata for matching patterns", Chapter 9 in Vol. 2
** Dora Giammarresi, Antonio Restivo
"Two-dimensional Languages", Chapter 4 in Vol. 3, pp. 215–267
{{DEFAULTSORT:Formal Language Theoretical computer science Combinatorics on words