In mathematics, computer science, and linguistics, a formal language is a set of strings of symbols.

The alphabet of a formal language is the set of symbols, letters, or tokens from which the strings of the language may be formed; frequently it is required to be finite. The strings formed from this alphabet are called words, 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, also called its formation rule.

The field of formal language theory studies 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 often used as the basis for defining programming languages and other systems in which the words of the language 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[link]

This section requires expansion.

The first formal language is thought 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."^[1]

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

Words over an alphabet[link]

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, x₂, … 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[link]

A formal language L over an alphabet Σ is 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[link]

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 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.

Constructions[link]

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 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"}^* = {"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"}).

Language-specification formalisms[link]

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[link]

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…x₁,
  • 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. The theory of trios and abstract families of languages studies the most common closure properties of language families in their own right.^[2]

Closure properties of language families (Failed to parse (Missing texvc executable; please see math/README to configure.): L_1 Op Failed to parse (Missing texvc executable; please see math/README to configure.): L_2 where both Failed to parse (Missing texvc executable; please see math/README to configure.): L_1 and Failed to parse (Missing texvc executable; please see math/README to configure.): L_2 are in the language family given by the column). After Hopcroft and Ullman.

Operation		Regular	DCFL	CFL	IND	CSL	recursive	RE
Union	Failed to parse (Missing texvc executable; please see math/README to configure.): \{w | w \in L_1 \lor w \in L_2\}	Yes	No	Yes	Yes	Yes	Yes	Yes
Intersection	Failed to parse (Missing texvc executable; please see math/README to configure.): \{w | w \in L_1 \land w \in L_2\}	Yes	No	No	No	Yes	Yes	Yes
Complement	Failed to parse (Missing texvc executable; please see math/README to configure.): \{w | w \not\in L_1\}	Yes	Yes	No	No	Yes	Yes	No
Concatenation	Failed to parse (Missing texvc executable; please see math/README to configure.): L_1\cdot L_2 = \{w\cdot z | w \in L_1 \land z \in L_2\}	Yes	No	Yes	Yes	Yes	Yes	Yes
Kleene star	Failed to parse (Missing texvc executable; please see math/README to configure.): L_1^{*} = \{\epsilon\} \cup \{w \cdot z | w \in L_1 \land z \in L_1^{*}\}	Yes	No	Yes	Yes	Yes	Yes	Yes
Homomorphism		Yes	No	Yes	Yes	No	No	Yes
e-free 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	Failed to parse (Missing texvc executable; please see math/README to configure.): \{w^R | w \in L\}	Yes	No	Yes	Yes	Yes	Yes	Yes
Intersection with a regular language	Failed to parse (Missing texvc executable; please see math/README to configure.): \{w | w \in L_1 \land w \in R\}, R \text{ regular}	Yes	Yes	Yes	Yes	Yes	Yes	Yes

Applications[link]

Programming languages[link]

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, 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[link]

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 Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{FS}

and Failed to parse (Missing texvc executable; please see math/README to configure.): \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 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[link]

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 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[link]

• Combinatorics on words
• Grammar framework
• Formal method
• Mathematical notation

References[link]

Notes[link]

General references[link]

External links[link]

• Alphabet, PlanetMath.org.
• Language, PlanetMath.org.
• 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

Mary Jo Putney (born in New York) is a best-selling American author of over twenty-five historical and contemporary romance novels. She has also been published fantasy romance novels as M.J. Putney. Her books are known for their unusual subject matter, including alcoholism, death, and domestic abuse.

Biography[link]

Putney was born and raised in New York. She attended Syracuse University, earning degrees in English literature and Industrial design.^[1] She served as the art editor of The New Internationalist magazine in London and worked as a designer in California^[2] before settling in Baltimore, Maryland in 1980^[2] to run her own freelance graphic design business.^[1]

After purchasing her first computer for her business, Putney realized that it would make writing very easy. She began work on her first novel, a traditional Regency romance, which sold in one week. Signet liked the novel so much that it offered Putney a three-book contract immediately.^[2] In 1987 that first novel, The Diabolical Baron, was published. Since then, she has published twenty-nine books (as of January 2007). Her books have been ranked on the national bestseller lists of the New York Times, USAToday, and Publishers Weekly. ^[1] The vast majority of her works have been historical romance, although she did write three contemporary romances. She has also begun writing fantasy romance and romantic fantasy as M.J. Putney.

Putney has won the Romance Writers of America RITA Award twice, for Dancing on the Wind and The Rake and the Reformer and has been a RITA finalist nine times. She is on the Romance Writers of America Honor Roll for bestselling authors, and has been awarded two Romantic Times Career Achievement Awards and four Golden Leaf Awards.^[1]

Putney lives in Baltimore with her cats. She is not married.^[2]

Bibliography[link]

As M.J. Putney[link]

Dark Mirror[link]

• Dark Mirror (2011)
• Dark Passage (2011)
• Dark Destiny (2012)

As Mary Jo Putney[link]

Single novels[link]

• Lady of Fortune (1988)
• Uncommon Vows (1991)
• Carousel of Hearts (1992)
• Dearly Beloved (1990)
• The Bargain (1999) (revised from The Would-be Widow (1988))

Bride Trilogy[link]

1. The Wild Child (1998)
2. The China Bride (2000)
3. The Bartered Bride (2002)

Circle of Friends Series[link]

1. The Burning Point (2000)
2. The Spiral Path (2002)
3. "A Holiday Fling" in Christmas Revels (2002)
4. Twist of Fate (2003)

Davenport Series[link]

1. The Diabolical Baron (1987)
2. The Rake (1998) (revised from The Rake and the Reformer (1989))
3. "Sunshine for Christmas" in A Regency Christmas II (1990)

Fallen Angels Series[link]

1. Thunder and Roses (1993) (reissued in Great Britain as Fallen Angel (2009))
2. Petals in the Storm (1993) (revised from The Controversial Countess (1989))
3. Dancing on the Wind (1994)
4. Angel Rogue (1995) (revised from The Rogue and the Runaway (1990))
5. Shattered Rainbows (1996)
6. River of Fire (1996)
7. One Perfect Rose (1997)

Guardian Series[link]

1. "The Alchemical Marriage" in Irresistible Forces (2004)
2. A Kiss of Fate (2004)
3. Stolen Magic (2005) (as M.J. Putney)
4. A Distant Magic (2006)
5. "The Tuesday Enchantress" in The Mammoth Book of Paranormal Romance (2009)
6. "The White Rose of Scotland" in Chalice of Roses (2010)

Lost Lords Series[link]

1. Loving a Lost Lord (2009)
2. Never Less than a Lady (2010)
3. Nowhere Near Respectable (2011)

Silk Trilogy[link]

1. Silk and Shadows (1991)
2. Silk and Secrets (1992)
3. Veils of Silk (1992)

Stone Saint Series[link]

1. The Marriage Spell (2006)

Collections[link]

• Christmas Revels (2002) includes 5 novellas: Sunshine for Christmas, The Christmas Cuckoo, The Christmas Tart, The Black Beast of Belleterre, and A Holiday Fling
• Dangerous to Know (2007) includes the novel The Diabolical Baron (1987) and the novella Mad, Bad, and Dangerous to Know
• Christmas Mischief (2010) includes 3 novellas: Sunshine for Christmas, The Christmas Cuckoo, and The Christmas Tart

Omnibus in Collaboration[link]

• "The Christmas Cuckoo" in A Regency Christmas (1991) (with Mary Balogh, Sandra Heath, Melinda McRae and Edith Layton)
• "The Christmas Tart" in A Regency Christmas IV (1992) (with Mary Balogh, Marjorie Farrell, Sandra Heath and Emma Lange)
• "The Black Beast of Belleterre" in A Victorian Christmas (1992)(with Edith Layton, Betina Krahn, Patricia Gaffney and Patricia Rice)
• "Mad, Bad and Dangerous to Know" in Rakes and Rogues (1993) (with Mary Balogh, Melinda McRae, Anita Mills and Maura Seger) reissued in Captured Hearts (1999) (with Mary Balogh, Edith Layton, Patricia Rice and Joan Wolf)
• "The Wedding of the Century" in Promised Brides (1994) (with Kristin James and Julie Tetel) reissued in Bride by Arrangement (2000) (with Merline Lovelace and Gayle Wilson)
• "The Devil's Spawn" in Dashing and Dangerous (1995) (with Mary Balogh, Edith Layton, Melinda McRae and Anita Mills)
• "Goldspun Promises" / "No Brighter Dream" / "Dancing on the Wind" (1995) (with Elizabeth Gregg, Katherine Kingsley)
• "Dangerous Gifts" in Faery Magic (1996) (with Jo Beverley, Karen Harbaugh and Barbara Samuel)
• "The Best Husband Money Can Buy" in Stocking Full of Joy (1997) (with Jill Barnett, Justine Dare, Susan King)
• "Avalon" in In Our Dreams (1998) (with Ruth Glick, Barbara Cummings, Courtney Henke, and Corey McFadden)
• "The Stargazer's Familiar" in A Constellation of Cats (2002) (with Andre Norton, Elizabeth Ann Scarborough, Jody Lynn Nye, Kristine Kathryn Rusch, Mickey Zucker Reichert, and others) reissued in The Journey Home (2005)(with Catherine Asaro, Diane Chamberlain, Mallory Kane, Mary Kirk, Candice Kohl, Lucy Grijalva, Linda Madl, Patricia Rice, CB Scott, and Rebecca York)
• "A Dragon's Tale" in Bewitched, Bothered and BeVampyred (2005) (with Gena Showalter, Mary Janice Davidson, Alesia Holliday, Susan Grant, Patricia Rice and Vicki Lewis Thompson)
• "The Dragon and the Dark Knight" in Dragon Lovers (2007) (with Jo Beverley, Karen Harbaugh and Barbara Samuel)

References and sources[link]

External links[link]

• Mary Jo Putney's Official Website & M.J. Putney's Official Website
• Mary Jo Putney at Word Wenches