
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 practical disciplines (inclu ...
, and
linguistics
Linguistics is the scientific study of human
Humans (''Homo sapiens'') are the most abundant and widespread species of primate, characterized by bipedalism and exceptional cognitive skills due to a large and complex brain. This ...
, a formal language consists of
words whose
letters 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 ...
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 or
context-free grammar, 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 programmin ...
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. 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 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'' (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, 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 ...
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. 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 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 number
A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in lang ...
s, 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.
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:
* = 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, 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.
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
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
,
.
*
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.
[, 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.
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, 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.
Interpretations and models
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
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 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)
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 Henry VIII of England, King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer.
Cambr ...
, 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 MarylandFormal 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