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53-447: The central thesis of the book argues that of these two technologies, it was the telegraph that was the more significant, since the ability to communicate globally at all in real-time was a qualitative shift, while according to Standage the change brought on by the modern Internet was merely a quantitative shift. The book describes to general readers how some of the uses of telegraph in commercial, military, and social communication were, in

106-456: A communication channel or storage in a storage medium . An early example is an invention of language , which enabled a person, through speech , to communicate what they thought, saw, heard, or felt to others. But speech limits the range of communication to the distance a voice can carry and limits the audience to those present when the speech is uttered. The invention of writing , which converted spoken language into visual symbols , extended

159-634: 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 F S {\displaystyle {\mathcal {FS}}} and F S ′ {\displaystyle {\mathcal {FS'}}} may have all

212-399: 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 accompanied with a formal grammar that describes it. The following rules describe a formal language  L over the alphabet Σ = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, +, =}: Under these rules,

265-433: A formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules called a formal grammar . The alphabet of a formal language consists of symbols, letters, or tokens that concatenate into strings called words. Words that belong to a particular formal language are sometimes called well-formed words or well-formed formulas . A formal language

318-432: A corresponding sequence of amino acids that form a protein molecule; a type of codon called a stop codon signals the end of the sequence. In mathematics , a Gödel code was the basis for the proof of Gödel 's incompleteness theorem . Here, the idea was to map mathematical notation to a natural number (using a Gödel numbering ). There are codes using colors, like traffic lights , the color code employed to mark

371-461: A finite number of elements, and many results apply only to them. It often makes sense to use an alphabet in the usual sense of the word, or more generally any finite character encoding such as ASCII or Unicode . 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

424-598: A front for the American Black Chamber run by Herbert Yardley between the First and Second World Wars. The purpose of most of these codes was to save on cable costs. The use of data coding for data compression predates the computer era; an early example is the telegraph Morse code where more-frequently used characters have shorter representations. Techniques such as Huffman coding are now used by computer-based algorithms to compress large data files into

477-692: A good compromise between expressivity and ease of parsing , and are widely used in practical applications. 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 L 1 {\displaystyle L_{1}} and L 2 {\displaystyle L_{2}} are languages over some common alphabet Σ {\displaystyle \Sigma } . Such string operations are used to investigate closure properties of classes of languages. A class of languages

530-578: A high level programming language, following his work in the creation of FORTRAN . Peter Naur was the secretary/editor for the ALGOL60 Report in which he used Backus–Naur form to describe the Formal part of ALGOL60. An alphabet , in the context of formal languages, can be any set ; its elements are called letters . An alphabet may contain an infinite number of elements; however, most definitions in formal language theory specify alphabets with

583-498: 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

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636-400: 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

689-412: A more compact form for storage or transmission. Character encodings are representations of textual data. A given character encoding may be associated with a specific character set (the collection of characters which it can represent), though some character sets have multiple character encodings and vice versa. Character encodings may be broadly grouped according to the number of bytes required to represent

742-515: A sense, analogous to modern uses of the internet. A few rather unusual stories are related, about couples who fell in love and even married over the wires, criminals who were caught through the telegraph, and so on. The culture which developed between telegraph operators also had some rather unexpected affinities with the modern Internet. Both cultures made or make use of complex text coding and abbreviated language slang , both required network security experts, and both attracted criminals who used

795-410: A sequence of target symbols. In this section, we consider codes that encode each source (clear text) character by a code word from some dictionary, and concatenation of such code words give us an encoded string. Variable-length codes are especially useful when clear text characters have different probabilities; see also entropy encoding . A prefix code is a code with the "prefix property": there

848-659: A single character: there are single-byte encodings, multibyte (also called wide) encodings, and variable-width (also called variable-length) encodings. The earliest character encodings were single-byte, the best-known example of which is ASCII . ASCII remains in use today, for example in HTTP headers . However, single-byte encodings cannot model character sets with more than 256 characters. Scripts that require large character sets such as Chinese, Japanese and Korean must be represented with multibyte encodings. Early multibyte encodings were fixed-length, meaning that although each character

901-411: A skunk!"), or AYYLU ("Not clearly coded, repeat more clearly."). Code words were chosen for various reasons: length , pronounceability , etc. Meanings were chosen to fit perceived needs: commercial negotiations, military terms for military codes, diplomatic terms for diplomatic codes, any and all of the preceding for espionage codes. Codebooks and codebook publishers proliferated, including one run as

954-516: A system of notations and symbols intended to facilitate the description of machines"). Heinz Zemanek rated it as an equivalent to a programming language for the numerical control of machine tools. 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

1007-413: 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

1060-418: 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 ,

1113-439: Is a total function mapping each symbol from S to a sequence of symbols over T. The extension C ′ {\displaystyle C'} of C {\displaystyle C} , is a homomorphism of S ∗ {\displaystyle S^{*}} into T ∗ {\displaystyle T^{*}} , which naturally maps each sequence of source symbols to

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1166-570: 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. A compiler usually has two distinct components. A lexical analyzer , sometimes generated by

1219-558: Is difficult or impossible. For example, semaphore , where the configuration of flags held by a signaler or the arms of a semaphore tower encodes parts of the message, typically individual letters, and numbers. Another person standing a great distance away can interpret the flags and reproduce the words sent. In information theory and computer science , a code is usually considered as an algorithm that uniquely represents symbols from some source alphabet , by encoded strings, which may be in some other target alphabet. An extension of

1272-529: Is no valid code word in the system that is a prefix (start) of any other valid code word in the set. Huffman coding is the most known algorithm for deriving prefix codes. Prefix codes are widely referred to as "Huffman codes" even when the code was not produced by a Huffman algorithm. Other examples of prefix codes are country calling codes , the country and publisher parts of ISBNs , and the Secondary Synchronization Codes used in

1325-482: Is not as simple as writing L  = {a, b, ab, cba}. Here are some examples of formal languages: 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 Typical questions asked about such formalisms include: Surprisingly often,

1378-574: 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 languages and formalized versions of subsets of natural languages, in which the words of the language represent concepts that are associated with meanings or semantics . In computational complexity theory , decision problems are typically defined as formal languages, and complexity classes are defined as

1431-511: The UMTS WCDMA 3G Wireless Standard. Kraft's inequality characterizes the sets of codeword lengths that are possible in a prefix code. Virtually any uniquely decodable one-to-many code, not necessarily a prefix one, must satisfy Kraft's inequality. Codes may also be used to represent data in a way more resistant to errors in transmission or storage. This so-called error-correcting code works by including carefully crafted redundancy with

1484-621: The Unicode character set; UTF-8 is the most common encoding of text media on the Internet. Biological organisms contain genetic material that is used to control their function and development. This is DNA , which contains units named genes from which messenger RNA is derived. This in turn produces proteins through a genetic code in which a series of triplets ( codons ) of four possible nucleotides can be translated into one of twenty possible amino acids . A sequence of codons results in

1537-492: The adjective "formal" is often omitted as redundant. While formal language theory usually concerns itself with formal languages that are described by some syntactic 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

1590-530: 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 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 ,

1643-491: 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

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1696-484: 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. In 1907, Leonardo Torres Quevedo introduced a formal language for the description of mechanical drawings (mechanical devices), in Vienna . He published "Sobre un sistema de notaciones y símbolos destinados a facilitar la descripción de las máquinas" ("On

1749-457: The code for representing sequences of symbols over the source alphabet is obtained by concatenating the encoded strings. Before giving a mathematically precise definition, this is a brief example. The mapping is a code, whose source alphabet is the set { a , b , c } {\displaystyle \{a,b,c\}} and whose target alphabet is the set { 0 , 1 } {\displaystyle \{0,1\}} . Using

1802-410: 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. 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

1855-404: The confidentiality of communications, although ciphers are now used instead. Secret codes intended to obscure the real messages, ranging from serious (mainly espionage in military, diplomacy, business, etc.) to trivial (romance, games) can be any kind of imaginative encoding: flowers , game cards, clothes, fans, hats, melodies, birds, etc., in which the sole requirement is the pre-agreement on

1908-409: The development of semaphore systems and undersea cables". The Los Angeles Times had some criticisms of the book but gave a mostly positive review. Code In communications and information processing , code is a system of rules to convert information —such as a letter , word , sound, image, or gesture —into another form, sometimes shortened or secret , for communication through

1961-484: The extension of the code, the encoded string 0011001 can be grouped into codewords as 0 011 0 01, and these in turn can be decoded to the sequence of source symbols acab . Using terms from formal language theory , the precise mathematical definition of this concept is as follows: let S and T be two finite sets, called the source and target alphabets , respectively. A code C : S → T ∗ {\displaystyle C:\,S\to T^{*}}

2014-425: The infantry on the battlefield, etc. Communication systems for sensory impairments, such as sign language for deaf people and braille for blind people, are based on movement or tactile codes. Musical scores are the most common way to encode music . Specific games have their own code systems to record the matches, e.g. chess notation . In the history of cryptography , codes were once common for ensuring

2067-472: The meaning by both the sender and the receiver. Other examples of encoding include: Other examples of decoding include: Acronyms and abbreviations can be considered codes, and in a sense, all languages and writing systems are codes for human thought. International Air Transport Association airport codes are three-letter codes used to designate airports and used for bag tags . Station codes are similarly used on railways but are usually national, so

2120-483: The networks to commit fraud , hack private communications, and send unwanted messages. Critical reception of the book was mostly positive. Smithsonian magazine gave a positive review for The Victorian Internet , but stated that it was "not the book for readers who want in-depth accounts of the lives of scientist-inventors like Thomas Edison or Charles Wheatstone, detailed financial histories of companies like Western Union , or technical treatments of subjects like

2173-415: The nominal value of the electrical resistors or that of the trashcans devoted to specific types of garbage (paper, glass, organic, etc.). In marketing , coupon codes can be used for a financial discount or rebate when purchasing a product from a (usual internet) retailer. In military environments, specific sounds with the cornet are used for different uses: to mark some moments of the day, to command

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2226-721: 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

2279-426: The purely syntactic 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 the 17th century, Gottfried Leibniz imagined and described the characteristica universalis , a universal and formal language which utilised pictographs . Later, Carl Friedrich Gauss investigated

2332-412: The range of communication across space and time . The process of encoding converts information from a source into symbols for communication or storage. Decoding is the reverse process, converting code symbols back into a form that the recipient understands, such as English or/and Spanish. One reason for coding is to enable communication in places where ordinary plain language , spoken or written,

2385-523: The same code can be used for different stations if they are in different countries. Occasionally, a code word achieves an independent existence (and meaning) while the original equivalent phrase is forgotten or at least no longer has the precise meaning attributed to the code word. For example, '30' was widely used in journalism to mean "end of story", and has been used in other contexts to signify "the end". Formal language theory In logic , mathematics , computer science , and linguistics ,

2438-530: The same information to be sent with fewer characters , more quickly, and less expensively. Codes can be used for brevity. When telegraph messages were the state of the art in rapid long-distance communication, elaborate systems of commercial codes that encoded complete phrases into single mouths (commonly five-minute groups) were developed, so that telegraphers became conversant with such "words" as BYOXO ("Are you trying to weasel out of our deal?"), LIOUY ("Why do you not answer my question?"), BMULD ("You're

2491-436: 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

2544-411: The sequence is a theorem of a formal system. Formal proofs are useful because their theorems can be interpreted as true propositions. 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

2597-429: 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. The field of formal language theory studies primarily

2650-417: 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

2703-422: The stored (or transmitted) data. Examples include Hamming codes , Reed–Solomon , Reed–Muller , Walsh–Hadamard , Bose–Chaudhuri–Hochquenghem , Turbo , Golay , algebraic geometry codes , low-density parity-check codes , and space–time codes . Error detecting codes can be optimised to detect burst errors , or random errors . A cable code replaces words (e.g. ship or invoice ) with shorter words, allowing

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2756-517: 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. For finite languages, one can explicitly enumerate all well-formed words. For example, we can describe

2809-500: Was represented by more than one byte, all characters used the same number of bytes ("word length"), making them suitable for decoding with a lookup table. The final group, variable-width encodings, is a subset of multibyte encodings. These use more complex encoding and decoding logic to efficiently represent large character sets while keeping the representations of more commonly used characters shorter or maintaining backward compatibility properties. This group includes UTF-8 , an encoding of

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