# Formal language

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In logic, 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.

## Contents

The alphabet of a formal language consists 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.

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.

## History

The first use of formal language is thought to be Gottlob Frege's 1879 Begriffsschrift , meaning "concept writing", which described a "formal language of pure language." [2]

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

## 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 or Unicode. The elements of an alphabet are called its letters. An alphabet may contain an infinite number of elements; [note 1] 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, 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 languageL 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 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.

### Constructions

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}* = {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 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

• 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 ${\displaystyle L_{1}}$ and ${\displaystyle L_{2}}$ are languages over some common alphabet ${\displaystyle \Sigma }$.

• The concatenation ${\displaystyle L_{1}\cdot L_{2}}$ consists of all strings of the form ${\displaystyle vw}$ where ${\displaystyle v}$ is a string from ${\displaystyle L_{1}}$ and ${\displaystyle w}$ is a string from ${\displaystyle L_{2}}$.
• The intersection${\displaystyle L_{1}\cap L_{2}}$ of ${\displaystyle L_{1}}$ and ${\displaystyle L_{2}}$ consists of all strings that are contained in both languages
• The complement${\displaystyle \neg L_{1}}$ of ${\displaystyle L_{1}}$ with respect to ${\displaystyle \Sigma }$ consists of all strings over ${\displaystyle \Sigma }$ that are not in ${\displaystyle L_{1}}$.
• 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 ${\displaystyle \varepsilon ^{R}=\varepsilon }$, and
• for each non-empty word ${\displaystyle w=\sigma _{1}\cdots \sigma _{n}}$ (where ${\displaystyle \sigma _{1},\ldots ,\sigma _{n}}$are elements of some alphabet), let ${\displaystyle w^{R}=\sigma _{n}\cdots \sigma _{1}}$,
• then for a formal language ${\displaystyle L}$, ${\displaystyle L^{R}=\{w^{R}\mid w\in L\}}$.
• 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. The theory of trios and abstract families of languages studies the most common closure properties of language families in their own right. [3]

Closure properties of language families (${\displaystyle L_{1}}$ Op ${\displaystyle L_{2}}$ where both ${\displaystyle L_{1}}$ and ${\displaystyle L_{2}}$ are in the language family given by the column). After Hopcroft and Ullman.
Operation Regular DCFL CFL IND CSL recursive RE
Union ${\displaystyle L_{1}\cup L_{2}=\{w\mid w\in L_{1}\lor w\in L_{2}\}}$YesNoYesYesYesYesYes
Intersection ${\displaystyle L_{1}\cap L_{2}=\{w\mid w\in L_{1}\land w\in L_{2}\}}$YesNoNoNoYesYesYes
Complement ${\displaystyle \neg L_{1}=\{w\mid w\not \in L_{1}\}}$YesYesNoNoYesYesNo
Concatenation ${\displaystyle L_{1}\cdot L_{2}=\{wz\mid w\in L_{1}\land z\in L_{2}\}}$YesNoYesYesYesYesYes
Kleene star${\displaystyle L_{1}^{*}=\{\varepsilon \}\cup \{wz\mid w\in L_{1}\land z\in L_{1}^{*}\}}$YesNoYesYesYesYesYes
(String) homomorphism ${\displaystyle h}$${\displaystyle h(L_{1})=\{h(w)\mid w\in L_{1}\}}$YesNoYesYesNoNoYes
ε-free (string) homomorphism ${\displaystyle h}$${\displaystyle h(L_{1})=\{h(w)\mid w\in L_{1}\}}$YesNoYesYesYesYesYes
Substitution ${\displaystyle \varphi }$${\displaystyle \varphi (L_{1})=\bigcup _{\sigma _{1}\cdots \sigma _{n}\in L_{1}}\varphi (\sigma _{1})\cdot \ldots \cdot \varphi (\sigma _{n})}$YesNoYesYesYesNoYes
Inverse homomorphism ${\displaystyle h^{-1}}$${\displaystyle h^{-1}(L_{1})=\bigcup _{w\in L_{1}}h^{-1}(w)}$YesYesYesYesYesYesYes
Reverse${\displaystyle L^{R}=\{w^{R}\mid w\in L\}}$YesNoYesYesYesYesYes
Intersection with a regular language ${\displaystyle R}$${\displaystyle L\cap R=\{w\mid w\in L\land w\in R\}}$YesYesYesYesYesYesYes

## 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 ${\displaystyle {\mathcal {FS}}}$ and ${\displaystyle {\mathcal {FS'}}}$ 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, 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.

## Notes

1. For example, first-order logic is often expressed using an alphabet that, besides symbols such as ∧, ¬, ∀ and parentheses, contains infinitely many elements x0, x1, x2, … that play the role of variables.

## Related Research Articles

In formal language theory, a context-free grammar (CFG) is a formal grammar whose production rules are of the form

A finite-state machine (FSM) or finite-state automaton, finite automaton, or simply a state machine, is a mathematical model of computation. It is an abstract machine that can be in exactly one of a finite number of states at any given time. The FSM can change from one state to another in response to some inputs; the change from one state to another is called a transition. An FSM is defined by a list of its states, its initial state, and the inputs that trigger each transition. Finite-state machines are of two types—deterministic finite-state machines and non-deterministic finite-state machines. A deterministic finite-state machine can be constructed equivalent to any non-deterministic one.

In mathematical logic and computer science, the Kleene star is a unary operation, either on sets of strings or on sets of symbols or characters. In mathematics it is more commonly known as the free monoid construction. The application of the Kleene star to a set is written as . It is widely used for regular expressions, which is the context in which it was introduced by Stephen Kleene to characterize certain automata, where it means "zero or more repetitions".

1. If is a set of strings, then is defined as the smallest superset of that contains the empty string and is closed under the string concatenation operation.
2. If is a set of symbols or characters, then is the set of all strings over symbols in , including the empty string .

In theoretical computer science and formal language theory, a regular language is a formal language that can be defined by a regular expression, in the strict sense in theoretical computer science.

Automata theory is the study of abstract machines and automata, as well as the computational problems that can be solved using them. It is a theory in theoretical computer science. The word automata comes from the Greek word αὐτόματα, which means "self-making". An automaton is an abstract self-propelled computing device which follows a predetermined sequence of operations automatically. An automaton with a finite number of states is called a Finite Automaton (FA) or Finite State Machine (FSM).

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 for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A formal system is essentially an "axiomatic system".

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 abstract algebra, the free monoid on a set is the monoid whose elements are all the finite sequences of zero or more elements from that set, with string concatenation as the monoid operation and with the unique sequence of zero elements, often called the empty string and denoted by ε or λ, as the identity element. The free monoid on a set A is usually denoted A. The free semigroup on A is the subsemigroup of A containing all elements except the empty string. It is usually denoted A+.

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 family of formalisms in natural language syntax which share the central assumption that syntactic constituents combine as functions and arguments. Categorial grammar posits a close relationship between the syntax and semantic composition, since it typically treats syntactic categories as corresponding to semantic types. Categorial grammars were developed in the 1930s by Kazimierz Ajdukiewicz, Yehoshua Bar-Hillel, and Joachim Lambek. It saw a surge of interest in the 1970s following the work of Richard Montague, whose Montague grammar assumed a similar view of syntax. It continues to be a major paradigm, particularly within formal semantics.

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.

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.

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, an alphabet is a non-empty set of symbols/glyphs, typically thought of as representing letters, characters, or digits but among other possibilities the "symbols" could also be a set of phonemes. Alphabets in this technical sense of a set are used in a diverse range of fields including logic, mathematics, computer science, and linguistics. An alphabet may have any cardinality ("size") and depending on its purpose maybe be finite, countable, or even uncountable.

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.

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.

In formal language theory, a grammar describes how to form strings from a 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. A formal grammar is defined as a set of production rules for strings in a formal language.

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.

## References

### Citations

1. 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.
2. 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.
3. Hopcroft & Ullman (1979), Chapter 11: Closure properties of families of languages.

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