

Formal language
A formal language is a set of words, i.e. finite strings of letters, or symbols. The inventory 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. Formal languages are a purely syntactical notion, so there is not necessarily any meaning associated with them. To distinguish the words that belong to a language from arbitrary words over its alphabet, the former are sometimes called wellformed words (or, in their application in logic, wellformed formulas). Formal languages are studied in the fields of logic, computer science and linguistics. Their most important practical application is for the precise definition of syntactically correct programs for a programming language. The branch of mathematics and computer science that is concerned only with the purely syntactical aspects of such languages, i.e. their internal structural patterns, is known as formal language theory. Although it is not formally part of the language, the words of a formal language often have a semantical dimension as well. In practice this is always tied very closely to the structure of the language, and a formal grammar (a set of formation rules that recursively defines the language) can help to deal with the meaning of (wellformed) words. Wellknown examples for this are "Tarski's definition of truth" in terms of a Tschema for firstorder logic, and compiler generators like lex and yacc. Contents
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. firstorder logic is often expressed using an alphabet which, besides symbols such as ∧, ¬, ∀ and parentheses, contains infinitely many elements x_{0}, x_{1}, 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. 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 deal with natural languages, the adjective "formal" is usually omitted as redundant. While formal language theory usually concerns itself with formal languages that are defined by some syntactical rules, the actual definition of a formal language is only as above: a (possibly infinite) set of finitelength strings, no more nor less. In practice, there are many languages that can be defined by rules, such as regular languages or contextfree languages. The notion of a formal grammar may be closer to the intuitive concept of a "language," one defined 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 defines it. The following rules define a formal language L over the alphabet Σ = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, +, =}:
Under these rules, the string "23+4=555" is in L, but the string "=234=+" is not. This formal language expresses natural numbers, wellformed addition statements, and wellformed 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 wellformed words. For example, we can define a language L as just L = {"a", "b", "ab", "cba"}. However, even over a finite (nonempty) alphabet such as Σ = {a, b} there are infinitely many words: "a", "abb", "ababba", "aaababbbbaab", …. Therefore formal languages are typically infinite, and defining an infinite formal language is not as simple as writing L = {"a", "b", "ab", "cba"}. Here are some examples of formal languages:
Languagespecification 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
Typical questions asked about such formalisms include:
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. Certain operations on languages are common. This includes the standard set operations, such as union, intersection, and complementation. Another class of operation is the elementwise application of string operations. Examples: suppose L_{1} and L_{2} are languages over some common alphabet.
Such 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 contextfree languages are known to be closed under union, concatenation, and intersection with regular languages, but not closed under intersection or complementation. Main article: String operations Some other operations frequently used in the study of formal languages are the following:
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