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In mathematics, computer science, and linguistics, a **formal language** consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules.

**Mathematics** includes the study of such topics as quantity, structure (algebra), space (geometry), and change. It has no generally accepted definition.

**Computer science** is the study of processes that interact with data and that can be represented as data in the form of programs. It enables the use of algorithms to manipulate, store, and communicate digital information. A computer scientist studies the theory of computation and the practice of designing software systems.

**Linguistics** is the scientific study of language. It involves analysing language form, language meaning, and language in context. The earliest activities in the documentation and description of language have been attributed to the 6th-century-BC Indian grammarian Pāṇini who wrote a formal description of the Sanskrit language in his * Aṣṭādhyāyī*.

- History
- Words over an alphabet
- Definition
- Examples
- Constructions
- Language-specification formalisms
- Operations on languages
- Applications
- Programming languages
- Formal theories, systems and proofs
- See also
- References
- Citation footnotes
- General references
- External links

The alphabet of a formal language consist of symbols, letters, or tokens that concatenate into strings of the language.^{ [1] } 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 or context-free grammar, which consists of its formation rules.

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 be identified with the set of formulas in the language.

In formal language theory, a **grammar** is a set of production rules for strings in a formal language. The rules describe how to form strings from the language's alphabet that are valid according to the language's syntax. A grammar does not describe the meaning of the strings or what can be done with them in whatever context—only their form.

In theoretical computer science and formal language theory, a **regular grammar** is a formal grammar that is right-regular or left-regular. Every regular grammar describes a regular language.

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 languages. 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. 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 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.

In linguistics, **syntax** is the set of rules, principles, and processes that govern the structure of sentences in a given language, usually including word order. The term *syntax* is also used to refer to the study of such principles and processes. The goal of many syntacticians is to discover the syntactic rules common to all languages.

In neuropsychology, linguistics, and the 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 can take different forms, such as speech or signing. They are distinguished from constructed and formal languages such as those used to program computers or to study logic.

A **programming language** is a formal language, which comprises a set of instructions that produce various kinds of output. Programming languages are used in computer programming to implement algorithms.

The first formal language is thought to be the one used by Gottlob Frege in his * Begriffsschrift * (1879), literally meaning "concept writing", and which Frege described as a "formal language of pure thought."^{ [2] }

**Friedrich Ludwig Gottlob Frege** was a German philosopher, logician, and mathematician. He worked as a mathematics professor at the University of Jena, and is understood by many to be the father of analytic philosophy, concentrating on the philosophy of language, logic, and mathematics. Though largely ignored during his lifetime, Giuseppe Peano (1858–1932) and Bertrand Russell (1872–1970) introduced his work to later generations of philosophers.

* Begriffsschrift* is a book on logic by Gottlob Frege, published in 1879, and the formal system set out in that book.

Axel Thue's early semi-Thue system, which can be used for rewriting strings, was influential on formal grammars.

**Axel Thue**, was a Norwegian mathematician, known for his original work in diophantine approximation and combinatorics.

In theoretical computer science and mathematical logic a **string rewriting system** (**SRS**), historically called a **semi-Thue system**, is a rewriting system over strings from a alphabet. Given a binary relation between fixed strings over the alphabet, called **rewrite rules**, denoted by , an SRS extends the rewriting relation to all strings in which the left- and right-hand side of the rules appear as substrings, that is , where , , , and are strings.

An **alphabet**, in the context of formal languages, can be any set, although it often makes sense to use an alphabet in the usual sense of the word, or more generally a character set such as ASCII or Unicode. The elements of an alphabet are called its **letters**. An alphabet may contain an infinite number of elements;^{ [3] } however, most definitions in formal language theory specify alphabets with a finite number of elements, and most results apply only to them.

In mathematics, a **set** is a collection of distinct objects, considered as an object in its own right. For example, the numbers 2, 4, and 6 are distinct objects when considered separately, but when they are considered collectively they form a single set of size three, written {2, 4, 6}. The concept of a set is one of the most fundamental in mathematics. Developed at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived. In mathematics education, elementary topics from set theory such as Venn diagrams are taught at a young age, while more advanced concepts are taught as part of a university degree.

An **alphabet** is a standard set of letters that represent the phonemes of any spoken language it is used to write. This is in contrast to other types of writing systems, such as syllabaries and logographic systems.

**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. Most modern character-encoding schemes are based on ASCII, although they support many additional characters.

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 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, 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.

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.

The following rules describe a formal language L over the alphabet Σ = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, +, =}:

- Every nonempty string that does not contain "+" or "=" and does not start with "0" is in L.
- The string "0" is in L.
- A string containing "=" is in L if and only if there is exactly one "=", and it separates two valid strings of L.
- A string containing "+" but not "=" is in L if and only if every "+" in the string separates two valid strings of L.
- No string is in L other than those implied by the previous rules.

Under these rules, the string "23+4=555" is in L, but the string "=234=+" is not. This formal language expresses natural numbers, well-formed additions, and well-formed addition equalities, but it expresses only what they look like (their syntax), not what they mean (semantics). For instance, nowhere in these rules is there any indication that "0" means the number zero, "+" means addition, "23+4=555" is false, etc.

For finite languages, one can explicitly enumerate all well-formed words. For example, we can describe a language L as just L = {a, b, ab, cba}. The degenerate case of this construction is the **empty language**, which contains no words at all (L = ∅ ).

However, even over a finite (non-empty) alphabet such as Σ = {a, b} 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* = {a, b, ab, cba}. Here are some examples of formal languages:

- L = Σ
^{*}, the set of*all*words over Σ; - L = {a}
^{*}= {a^{n}}, 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 characters on this line, i.e.,

the set {the, set, of, maximal, strings, alphanumeric, ASCII, characters, on, this, line, i, e}.

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, such as a Turing machine or finite state automaton;
- those strings for which some decision procedure (an algorithm 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. Context-free grammars and regular grammars provide a good compromise between expressivity and ease of parsing, and are widely used in practical applications.

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 and are languages over some common alphabet .

- The
*concatenation*consists of all strings of the form where is a string from and is a string from . - The
*intersection*of and consists of all strings that are contained in both languages - The
*complement*of with respect to consists of all strings over that are not in . - 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 , and - for each non-empty word (where are elements of some alphabet), let ,
- then for a formal language , .

- Let
- 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 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.^{ [4] }

Closure properties of language families ( Op where both and are in the language family given by the column). After Hopcroft and Ullman. Operation Regular DCFL CFL IND CSL recursive RE Union Yes No Yes Yes Yes Yes Yes Intersection Yes No No No Yes Yes Yes Complement Yes Yes No No Yes Yes No Concatenation Yes No Yes Yes Yes Yes Yes Kleene star Yes No Yes Yes Yes Yes Yes (String) homomorphism Yes No Yes Yes No No Yes ε-free (string) homomorphism Yes No Yes Yes Yes Yes Yes Substitution Yes No Yes Yes Yes No Yes Inverse homomorphism Yes Yes Yes Yes Yes Yes Yes Reverse Yes No Yes Yes Yes Yes Yes Intersection with a regular language Yes Yes Yes Yes Yes Yes Yes

A compiler usually has two distinct components. A lexical analyzer, generated by a tool like `lex`

, identifies the tokens of the programming language grammar, e.g. identifiers or keywords, which are themselves expressed in a simpler formal language, usually by means of regular expressions. At the most basic conceptual level, a parser, usually generated by a parser generator like ` yacc `

, attempts to decide if the source program is valid, that is if it belongs to the programming language 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.

In mathematical logic, 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 rules, 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 and 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.

Formal languages are entirely syntactic in nature but may be given semantics that give meaning to the elements of the language. For instance, in mathematical logic, 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 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.

In formal language theory, a **context-free grammar** (**CFG**) is a certain type of formal grammar: a set of production rules that describe all possible strings in a given formal language. Production rules are simple replacements. For example, the rule

In theoretical computer science and formal language theory, a **regular language** is a formal language that can be expressed using a regular expression, in the strict sense of the latter notion used in theoretical computer science.

**Metalogic** is the study of the metatheory of logic. Whereas *logic* studies how logical systems can be used to construct valid and sound arguments, metalogic studies the properties of logical systems. Logic concerns the truths that may be derived using a logical system; metalogic concerns the truths that may be derived *about* the languages and systems that are used to express truths.

A **formal system** is used to infer theorems from axioms according to a set of rules. These rules used to carry out the inference of theorems from axioms are known as the **logical calculus** of the formal system. A formal system is essentially an "axiomatic system". In 1921, David Hilbert proposed to use such system as the foundation for the knowledge in mathematics. A formal system may represent a well-defined system of abstract thought.

In logic, **syntax** is anything having to do with formal languages or formal systems without regard to any interpretation or meaning given to them. Syntax is concerned with the rules used for constructing, or transforming the symbols and words of a language, as contrasted with the semantics of a language which is concerned with its meaning.

**Categorial grammar** is a term used for a family of formalisms in natural language syntax motivated by the principle of compositionality and organized according to the view that syntactic constituents should generally combine as functions or according to a function-argument relationship. Most versions of categorial grammar analyze sentence structure in terms of constituencies and are therefore phrase structure grammars.

A **finite-state transducer** (**FST**) is a finite-state machine with two memory *tapes*, following the terminology for Turing machines: an input tape and an output tape. This contrasts with an ordinary finite-state automaton, which has a single tape. An FST is a type of finite-state automaton that maps between two sets of symbols. An FST is more general than a finite-state automaton (FSA). An FSA defines a formal language by defining a set of accepted strings, while an FST defines relations between sets of strings.

In the theory of formal languages of computer science, mathematics, and linguistics, a **Dyck word** is a balanced string of square brackets [ and ]. The set of Dyck words forms the **Dyck language**.

**Conjunctive grammars** are a class of formal grammars studied in formal language theory. They extend the basic type of grammars, the context-free grammars, with a conjunction operation. Besides explicit conjunction, conjunctive grammars allow implicit disjunction represented by multiple rules for a single nonterminal symbol, which is the only logical connective expressible in context-free grammars. Conjunction can be used, in particular, to specify intersection of languages. A further extension of conjunctive grammars known as Boolean grammars additionally allows explicit negation.

In formal language theory, a string is defined as a finite sequence of members of an underlying base set; this set is called the **alphabet** of a string or collection of strings. The members of the set are called *symbols*, and are typically thought of as representing letters, characters, or digits. For example, a common alphabet is {0,1}, the **binary alphabet**, and a binary string is a string drawn from the alphabet {0,1}. An infinite sequence of letters may be constructed from elements of an alphabet as well.

In computer science, **terminal and nonterminal symbols** are the lexical elements used in specifying the production rules constituting a formal grammar. *Terminal symbols* are the elementary symbols of the language defined by a formal grammar. *Nonterminal symbols* are replaced by groups of terminal symbols according to the production rules.

In logic, especially mathematical logic, a **signature** lists and describes the non-logical symbols of a formal language. In universal algebra, a signature lists the operations that characterize an algebraic structure. In model theory, signatures are used for both purposes.

**Indexed grammars** are a generalization of context-free grammars in that nonterminals are equipped with lists of *flags*, or *index symbols*. The language produced by an indexed grammar is called an **indexed language**.

A **production** or **production rule** in computer science is a *rewrite rule* specifying a symbol substitution that can be recursively performed to generate new symbol sequences. A finite set of productions is the main component in the specification of a formal grammar. The other components are a finite set of nonterminal symbols, a finite set of terminal symbols that is disjoint from and a distinguished symbol that is the *start symbol*.

An **embedded pushdown automaton** or **EPDA** is a computational model for parsing languages generated by tree-adjoining grammars (TAGs). It is similar to the context-free grammar-parsing pushdown automaton, except that instead of using a plain stack to store symbols, it has a stack of iterated stacks that store symbols, giving TAGs a generative capacity between context-free grammars and context-sensitive grammars, or a subset of the mildly context-sensitive grammars. Embedded pushdown automata should not be confused with nested stack automata which have more computational power.

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 strings of the language. It does not describe anything else about a language, such as its semantics. .

In computer science, more specifically in automata and formal language theory, **nested words** are a concept proposed by Alur and Madhusudan as a joint generalization of words, as traditionally used for modelling linearly ordered structures, and of ordered unranked trees, as traditionally used for modelling hierarchical structures. Finite-state acceptors for nested words, so-called **nested word automata**, then give a more expressive generalization of finite automata on words. The linear encodings of languages accepted by finite nested word automata gives the class of **visibly pushdown languages**. The latter language class lies properly between the regular languages and the deterministic context-free languages. Since their introduction in 2004, these concepts have triggered much research in that area.

- ↑ See e.g. Reghizzi, Stefano Crespi (2009),
*Formal Languages and Compilation*, Texts in Computer Science, Springer, p. 8, ISBN 9781848820500,An alphabet is a finite set

. - ↑ Martin Davis (1995). "Influences of Mathematical Logic on Computer Science". In Rolf Herken (ed.).
*The universal Turing machine: a half-century survey*. Springer. p. 290. ISBN 978-3-211-82637-9. - ↑ For example, first-order logic is often expressed using an alphabet that, besides symbols such as ∧, ¬, ∀ and parentheses, contains infinitely many elements
*x*_{0},*x*_{1},*x*_{2}, … that play the role of variables. - ↑ Hopcroft & Ullman (1979), Chapter 11: Closure properties of families of languages.

- A. G. Hamilton,
*Logic for Mathematicians*, Cambridge University Press, 1978, ISBN 0-521-21838-1. - Luis M. Augusto,
*Languages, machines, and classical computation*, London: College Publications, 2019. ISBN 978-1-84890-300-5.Web page - Seymour Ginsburg,
*Algebraic and automata theoretic properties of formal languages*, North-Holland, 1975, ISBN 0-7204-2506-9. - Michael A. Harrison,
*Introduction to Formal Language Theory*, Addison-Wesley, 1978. - John E. Hopcroft and Jeffrey D. Ullman,
*Introduction to Automata Theory, Languages, and Computation*, Addison-Wesley Publishing, Reading Massachusetts, 1979. ISBN 81-7808-347-7. - Rautenberg, Wolfgang (2010).
*A Concise Introduction to Mathematical Logic*(3rd ed.). New York: Springer Science+Business Media. doi:10.1007/978-1-4419-1221-3. ISBN 978-1-4419-1220-6 . - Grzegorz Rozenberg, Arto Salomaa,
*Handbook of Formal Languages: Volume I-III*, Springer, 1997, ISBN 3-540-61486-9. - Patrick Suppes,
*Introduction to Logic*, D. Van Nostrand, 1957, ISBN 0-442-08072-7.

Wikimedia Commons has media related to . Formal languages |

- Hazewinkel, Michiel, ed. (2001) [1994], "Formal language",
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 - "Alphabet".
*PlanetMath*. - "Language".
*PlanetMath*. - University of Maryland, 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, "Context-Free Languages and Push-Down Automata", Chapter 3 in Vol. 1
- 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

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