In

^{*} (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

^{*}, 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

^{*}, 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

the set .

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 premi ...

, the set of possible formulas of a particular logic is a formal language, and an interpretation 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

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.),

"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

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 premi ...

, 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 practical disciplines (includ ...

, 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. Lingui ...

, a formal language consists of words
A word is a basic element of language that carries an objective or practical meaning, can be used on its own, and is uninterruptible. Despite the fact that language speakers often have an intuitive grasp of what a word is, there is no cons ...

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 alphab ...

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 sylla ...

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 formula
In mathematical logic, propositional logic and predicate logic, a well-formed formula, abbreviated WFF or wff, often simply formula, is a finite sequence of symbols from a given alphabet that is part of a formal language. A formal language can ...

s''. A formal language is often defined by means of 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 ...

such as a regular grammar or 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 em ...

, which consists of its formation rules.
In computer science, formal languages are used among others as the basis for defining the grammar of programming language
A programming language is a system of notation for writing computer programs. Most programming languages are text-based formal languages, but they may also be graphical. They are a kind of computer language.
The description of a programming ...

s 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
In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and relating these classes to each other. A computational problem is a task solved b ...

, decision problem
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 whe ...

s are typically defined as formal languages, and complexity class
In computational complexity theory, a complexity class is a set of computational problems of related resource-based complexity. The two most commonly analyzed resources are time and memory.
In general, a complexity class is defined in terms of ...

es 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 premi ...

and the foundations of mathematics
Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathem ...

, formal languages are used to represent the syntax of axiomatic system
In mathematics and logic, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A theory is a consistent, relatively-self-contained body of knowledge which usually contain ...

s, 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 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 language
In neuropsychology, linguistics, and philosophy of language, a natural language or ordinary language is any language that has evolved naturally in humans through use and repetition without conscious planning or premeditation. Natural languages ...

s.
History

In the 17th Century,Gottfried Leibniz
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathem ...

imagined and described the characteristica universalis
The Latin term ''characteristica universalis'', commonly interpreted as ''universal characteristic'', or ''universal character'' in English, is a universal and formal language imagined by Gottfried Leibniz able to express mathematical, scienti ...

, a universal and formal language which utilised pictographs
A pictogram, also called a pictogramme, pictograph, or simply picto, and in computer usage an icon, is a graphic symbol that conveys its meaning through its pictorial resemblance to a physical object. Pictographs are often used in writing and gr ...

. During this period, Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...

also investigated the problem of Gauss codes.
Gottlob Frege
Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic phil ...

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 notat ...

'' (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
Emil Leon Post (; February 11, 1897 – April 21, 1954) was an American mathematician and logician. He is best known for his work in the field that eventually became known as computability theory.
Life
Post was born in Augustów, Suwałki Go ...

later termed ‘Thue Systems’, and gave an early example of an undecidable problem
In computability theory and computational complexity theory, an undecidable problem is a decision problem for which it is proved to be impossible to construct an algorithm that always leads to a correct yes-or-no answer. The halting problem is an ...

. 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
Avram Noam Chomsky (born December 7, 1928) is an American public intellectual: a linguist, philosopher, cognitive scientist, historian, social critic, and political activist. Sometimes called "the father of modern linguistics", Chomsky ...

devised an abstract representation of formal and natural languages, known as the Chomsky hierarchy
In formal language theory, computer science and linguistics, the Chomsky hierarchy (also referred to as the Chomsky–Schützenberger hierarchy) is a containment hierarchy of classes of formal grammars.
This hierarchy of grammars was described by ...

. In 1959 John Backus
John Warner Backus (December 3, 1924 – March 17, 2007) was an American computer scientist. He directed the team that invented and implemented FORTRAN, the first widely used high-level programming language, and was the inventor of the Back ...

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, although it often makes sense to use analphabet
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 sylla ...

in the usual sense of the word, or more generally any finite character encoding
Character encoding is the process of assigning numbers to graphical characters, especially the written characters of human language, allowing them to be stored, transmitted, and transformed using digital computers. The numerical values that ...

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 of ...

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, w ...

. 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
String or strings may refer to:
* String (structure), a long flexible structure made from threads twisted together, which is used to tie, bind, or hang other objects
Arts, entertainment, and media Films
* ''Strings'' (1991 film), a Canadian ani ...

) of letters. The set of all words over an alphabet Σ is usually denoted by Σconcatenation
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 premi ...

, 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 asubset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...

of Σnatural language
In neuropsychology, linguistics, and philosophy of language, a natural language or ordinary language is any language that has evolved naturally in humans through use and repetition without conscious planning or premeditation. Natural languages ...

s, 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 language
In theoretical computer science and formal language theory, a regular language (also called a rational language) is a formal language that can be defined by a regular expression, in the strict sense in theoretical computer science (as opposed to ...

s or context-free languages. The notion of 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 ...

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 expressesnatural 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 (constituency), ...

), 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 ( = ∅). 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: * = Σcontext-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 em ...

);
* the set of inputs upon which a certain Turing machine
A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer algor ...

halts; or
* the set of maximal strings of alphanumeric
Alphanumericals or alphanumeric characters are a combination of alphabetical and numerical characters. More specifically, they are the collection of Latin letters and Arabic digits. An alphanumeric code is an identifier made 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 of ...

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 someformal 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 ...

;
* those strings described or matched by a particular regular expression
A regular expression (shortened as regex or regexp; sometimes referred to as rational expression) is a sequence of characters that specifies a search pattern in text. Usually such patterns are used by string-searching algorithms for "find" ...

;
* those strings accepted by some automaton, such as a Turing machine
A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer algor ...

or finite-state automaton;
* those strings for which some decision procedure (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
In formal language theory, computer science and linguistics, the Chomsky hierarchy (also referred to as the Chomsky–Schützenberger hierarchy) is a containment hierarchy of classes of formal grammars.
This hierarchy of grammars was described by ...

based on the expressive power of their generative grammar as well as the complexity of their recognizing automaton. 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 em ...

s and regular grammars provide a good compromise between expressivity and ease of parsing
Parsing, syntax analysis, or syntactic analysis is the process of analyzing a string of symbols, either in natural language, computer languages or data structures, conforming to the rules of a formal grammar. The term ''parsing'' comes from Latin ...

, 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 $\backslash Sigma$. * 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 ...

'' $L\_1\; \backslash cdot\; L\_2$ 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'' $L\_1\; \backslash cap\; L\_2$ of $L\_1$ and $L\_2$ consists of all strings that are contained in both languages
* The ''complement'' $\backslash neg\; L\_1$ of $L\_1$ with respect to $\backslash Sigma$ consists of all strings over $\backslash Sigma$ 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 $\backslash varepsilon^R\; =\; \backslash varepsilon$, and
** for each non-empty word $w\; =\; \backslash sigma\_1\; \backslash cdots\; \backslash sigma\_n$ (where $\backslash sigma\_1,\; \backslash ldots,\; \backslash sigma\_n$are elements of some alphabet), let $w^R\; =\; \backslash sigma\_n\; \backslash cdots\; \backslash sigma\_1$,
** then for a formal language $L$, $L^R\; =\; \backslash $.
* String homomorphism
Such string operations are used to investigate closure properties 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 language
In theoretical computer science and formal language theory, a regular language (also called a rational language) is a formal language that can be defined by a regular expression, in the strict sense in theoretical computer science (as opposed to ...

s, 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, sometimes generated by a tool like`lex`

, identifies the tokens of the programming language grammar, e.g. identifier
An identifier is a name that identifies (that is, labels the identity of) either a unique object or a unique ''class'' of objects, where the "object" or class may be an idea, physical countable object (or class thereof), or physical noncountable ...

s 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
A regular expression (shortened as regex or regexp; sometimes referred to as rational expression) is a sequence of characters that specifies a search pattern in text. Usually such patterns are used by string-searching algorithms for "find" ...

. At the most basic conceptual level, a parser
Parsing, syntax analysis, or syntactic analysis is the process of analyzing a string of symbols, either in natural language, computer languages or data structures, conforming to the rules of a formal grammar. The term ''parsing'' comes from Lat ...

, sometimes generated by a parser generator
In computer science, a compiler-compiler or compiler generator is a programming tool that creates a parser, interpreter, or compiler from some form of formal description of a programming language and machine.
The most common type of compiler ...

like ```
yacc
Yacc (Yet Another Compiler-Compiler) is a computer program for the Unix operating system developed by Stephen C. Johnson. It is a Look Ahead Left-to-Right Rightmost Derivation (LALR) parser generator, generating a LALR parser (the part of a co ...
```

, 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
In computer science, an abstract syntax tree (AST), or just syntax tree, is a tree representation of the abstract syntactic structure of text (often source code) written in a formal language. Each node of the tree denotes a construct occurring ...

. This is used by subsequent stages of the compiler to eventually generate an executable containing machine code
In computer programming, machine code is any low-level programming language, consisting of machine language instructions, which are used to control a computer's central processing unit (CPU). Each instruction causes the CPU to perform a ve ...

that runs directly on the hardware, or some intermediate code that requires a virtual machine
In computing, a virtual machine (VM) is the virtualization/ emulation of a computer system. Virtual machines are based on computer architectures and provide functionality of a physical computer. Their implementations may involve specialized hard ...

to execute.
Formal theories, systems, and proofs

Inmathematical 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 formal ...

, 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 ...

s, which may be interpreted as valid rules of inference, or a set of axiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...

s, 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 $\backslash mathcal$ and $\backslash mathcal$ 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 proposition
In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...

s) 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 givensemantics
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 model theory
In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the ...

. In model theory, the terms that occur in a formula are interpreted as objects within mathematical structures
In mathematics, a structure is a set endowed with some additional features on the set (e.g. an operation, relation, metric, or topology). Often, the additional features are attached or related to the set, so as to provide it with some additi ...

, 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 In abstract algebra, the free monoid on a set is the monoid whose elements are all the finite sequences (or strings) of zero or more elements from that set, with string concatenation as the monoid operation and with the unique sequence of zero elem ...

* Formal method
* Grammar framework
* Mathematical notation
Mathematical notation consists of using symbols for representing operations, unspecified numbers, relations and any other mathematical objects, and assembling them into expressions and formulas. Mathematical notation is widely used in mathem ...

* Associative array
In computer science, an associative array, map, symbol table, or dictionary is an abstract data type that stores a collection of (key, value) pairs, such that each possible key appears at most once in the collection. In mathematical terms an a ...

* 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 Press i ...

, 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, 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 university, public Land-grant university, land-grant research university in College Park, Maryland. Founded in 1856, UMD is the Flagship un ...

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
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, (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