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A formal language is a set of words, i.e. finite strings of letters, symbols, or tokens. The set from which these letters are taken is called the alphabet over which the language is defined. A formal language is often defined by means of a formal grammar (also called its formation rules); accordingly, words that belong to a formal language are sometimes called well-formed words (or well-formed formulas). Formal languages are studied in computer science and linguistics; the field of formal language theory studies the purely syntactical aspects of such languages (that is, their internal structural patterns).

Formal languages are often used as the basis for richer constructs endowed with semantics. In computer science they are used, among other things, for the precise definition of data formats and the syntax of programming languages. Formal languages play a crucial role in the development of compilers, typically produced by means of a compiler compiler, which may be a single program or may be separated in tools like lexical analyzer generators (e.g. lex), and parser generators (e.g. yacc). Since formal languages alone do not have a semantics, other formal constructs are needed for the formal specification of program semantics. Formal languages are also used in logic and in foundations of mathematics to represent the syntax of formal theories. Logical systems can be seen as a formal language with additional constructs, like proof calculi, which define a consequence relation.^[1] "Tarski's definition of truth" in terms of a T-schema for first-order logic is an example of fully interpreted formal language; all its sentences have meanings that make them either true or false.

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 in the usual sense of the word, or more generally a character set such as ASCII. Alphabets can also be infinite; e.g. first-order logic is often expressed using an alphabet which, besides symbols such as ∧, ¬, ∀ and parentheses, contains infinitely many elements x₀, x₁, x ₂, … that play the role of variables. The elements of an alphabet are called its letters.

A word over an alphabet can be any finite sequence, or string, of letters. The set of all words over an alphabet Σ is usually denoted by Σ^* (using the Kleene star). For any alphabet there is only one word of length 0, the empty word, which is often denoted by e, ε or 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.

Definition

A formal language L over an alphabet Σ is just a subset of Σ^*, that is, a set of words over that alphabet.

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, no more nor 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 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 in L.
• A string containing + but not = is in L if and only if every + in the string separates two valid strings in 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 addition statements, 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, or that + means addition.

For finite languages one can simply enumerate all well-formed words. For example, we can describe a language L as just L = {"a", "b", "ab", "cba"}.

However, even over a finite (non-empty) alphabet such as Σ = {a, b} there are infinitely many words: "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}^* = {aⁿ}, where n ranges over the natural numbers and aⁿ 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"}).

Language-specification formalisms

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 precise characterization of how expensive exactly). 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 L₁ and L₂ are languages over some common alphabet.

• The concatenation L₁L₂ consists of all strings of the form vw where v is a string from L₁ and w is a string from L₂.
• The intersection L₁ ∩ L₂ of L₁ and L₂ consists of all strings which are contained in both languages
• The complement ¬L of a language with respect to a given alphabet consists of all strings over the alphabet that are not in the language.
• The Kleene star: the language consisting of all words that are concatenations of 0 or more words in the original language;
• Reversal:
  • Let e be the empty word, then e^R = e, and
  • for each non-empty word w = x₁…x_n over some alphabet, let w^R = x_n…x ₁,
  • then for a formal language L, L^R = {w^R | w ∈ L}.
• 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.

Closure properties of language families (L₁ Op L₂ where both L₁ and L₂ are in the language family given by the column). After Hopcroft and Ullman.

Operation		regular	DCFL	CFL	CSL	recursive	r.e.
Union		Yes	No	Yes	Yes	Yes	Yes
Intersection		Yes	No	No	Yes	Yes	Yes
Complement		Yes	Yes	No	Yes	Yes	No
Concatenation		Yes	No	Yes	Yes	Yes	Yes
Kleene star		Yes	No	Yes	Yes	Yes	Yes
Homomorphism		Yes	No	Yes	No	No	Yes
e-free Homomorphism		Yes	No	Yes	Yes	Yes	Yes
Substitution		Yes	No	Yes	Yes	No	Yes
Inverse Homomorphism		Yes	Yes	Yes	Yes	Yes	Yes
Reverse		Yes	No	Yes	Yes	Yes	Yes
Intersection with a Regular Language		Yes	Yes	Yes	Yes	Yes	Yes

Applications

Programming languages

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 in some executable format. Because of this, a parser usually outputs more than a yes/no answer, typically an abstract syntax tree, which 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

This diagram shows the syntactic divisions within a formal system. Symbols and strings of symbols may be broadly divided into nonsense and well-formed formulas. The set of well-formed formulas is divided into theorems and non-theorems. However, quite often, a formal system will simply define all of its well-formed formula as theorems.^[2]

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 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. Formal languages are useful in logic because they have formulas that can be interpreted as expressing logical truths. An interpretation of a formal language is the assignment of meanings to its symbols and formulas.

Model theory is the theory of interpretations of formal languages. The study of interpretations is called formal semantics. Giving an interpretation is synonymous with constructing a model. A model of a formula of a formal language is an interpretation of the language for which the formula comes out to be true.

Example

A possible interpretation of would be to take "" as meaning the same as the decimal digit "1", "" as meaning the same as the digit "0", and each formula as meaning the same as a decimal numeral composed exclusively of "1"s and "0"s. Therefore " " means "101" under this interpretation of .^[3]

See also

• Combinatorics on words
• Grammar framework
• Formal method
• Formal science
• Mathematical notation
• Rational language
• Descriptive complexity theory and Kolmogorov complexity

References

External links

• Alphabet on PlanetMath
• Language on 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):t
  • 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