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48-409: A code is a rule for converting a piece of information into another object or action, not necessarily of the same sort. Code may also refer to: 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

96-427: A function . In computability theory , a general recursive function is a partial function from the integers to the integers; no algorithm can exist for deciding whether an arbitrary such function is in fact total. When arrow notation is used for functions, a partial function f {\displaystyle f} from X {\displaystyle X} to Y {\displaystyle Y}

144-456: 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

192-426: A not-a-number value which is returned when a floating point operation is undefined and exceptions are suppressed, e.g. when the square root of a negative number is requested. In a programming language where function parameters are statically typed , a function may be defined as a partial function because the language's type system cannot express the exact domain of the function, so the programmer instead gives it

240-432: 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

288-598: 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

336-402: A function since every element on the left-hand set is associated with exactly one element in the right hand set. Consider the natural logarithm function mapping the real numbers to themselves. The logarithm of a non-positive real is not a real number, so the natural logarithm function doesn't associate any real number in the codomain with any non-positive real number in the domain. Therefore,

384-412: 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

432-402: A partial function f : X ⇀ Y , {\displaystyle f:X\rightharpoonup Y,} and any x ∈ X , {\displaystyle x\in X,} one has either: For example, if f {\displaystyle f} is the square root function restricted to the integers then f ( n ) {\displaystyle f(n)}

480-407: A partial function is a binary relation over two sets that associates to every element of the first set at most one element of the second set; it is thus a univalent relation . This generalizes the concept of a (total) function by not requiring every element of the first set to be associated to an element of the second set. A partial function is often used when its exact domain of definition

528-430: A partial function is the subset S of X on which the partial function is defined; in this case, the partial function may also be viewed as a function from S to Y . In the example of the square root operation, the set S consists of the nonnegative real numbers [ 0 , + ∞ ) . {\displaystyle [0,+\infty ).} The notion of partial function is particularly convenient when

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576-621: A partial function which is both injective and surjective has an injective function as inverse. Furthermore, a function which is injective may be inverted to a bijective partial function. The notion of transformation can be generalized to partial functions as well. A partial transformation is a function f : A ⇀ B , {\displaystyle f:A\rightharpoonup B,} where both A {\displaystyle A} and B {\displaystyle B} are subsets of some set X . {\displaystyle X.} For convenience, denote

624-410: 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

672-659: 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

720-411: 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

768-439: 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

816-526: Is a total function if and only if ob ⁡ ( C ) {\displaystyle \operatorname {ob} (C)} has one element. The reason for this is that two morphisms f : X → Y {\displaystyle f:X\to Y} and g : U → V {\displaystyle g:U\to V} can only be composed as g ∘ f {\displaystyle g\circ f} if Y = U , {\displaystyle Y=U,} that is,

864-558: 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

912-529: 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

960-404: Is not known, or is difficult to specify. However, even when the exact domain of definition is known, partial functions are often used for simplicity or brevity. This is the case in calculus , where, for example, the quotient of two functions is a partial function whose domain of definition cannot contain the zeros of the denominator; in this context, a partial function is generally simply called

1008-418: Is only defined if n {\displaystyle n} is a perfect square (that is, 0 , 1 , 4 , 9 , 16 , … {\displaystyle 0,1,4,9,16,\ldots } ). So f ( 25 ) = 5 {\displaystyle f(25)=5} but f ( 26 ) {\displaystyle f(26)} is undefined. A partial function arises from

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1056-438: Is said to be injective , surjective , or bijective when the function given by the restriction of the partial function to its domain of definition is injective, surjective, bijective respectively. Because a function is trivially surjective when restricted to its image, the term partial bijection denotes a partial function which is injective. An injective partial function may be inverted to an injective partial function, and

1104-466: Is sometimes written as f : X ⇀ Y , {\displaystyle f:X\rightharpoonup Y,} f : X ↛ Y , {\displaystyle f:X\nrightarrow Y,} or f : X ↪ Y . {\displaystyle f:X\hookrightarrow Y.} However, there is no general convention, and the latter notation is more commonly used for inclusion maps or embeddings . Specifically, for

1152-432: Is the non-negative integers ) is a partial function: It is defined only when x ≥ y . {\displaystyle x\geq y.} In denotational semantics a partial function is considered as returning the bottom element when it is undefined. In computer science a partial function corresponds to a subroutine that raises an exception or loops forever. The IEEE floating point standard defines

1200-599: Is the only proper partial operation (because division by zero is not defined). The set of all partial functions (partial transformations ) on a given base set, X , {\displaystyle X,} forms a regular semigroup called the semigroup of all partial transformations (or the partial transformation semigroup on X {\displaystyle X} ), typically denoted by P T X . {\displaystyle {\mathcal {PT}}_{X}.} The set of all partial bijections on X {\displaystyle X} forms

1248-511: 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

1296-621: 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

1344-415: The symmetric inverse semigroup . Charts in the atlases which specify the structure of manifolds and fiber bundles are partial functions. In the case of manifolds, the domain is the point set of the manifold. In the case of fiber bundles, the domain is the space of the fiber bundle. In these applications, the most important construction is the transition map , which is the composite of one chart with

1392-457: 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

1440-416: The codomain is Y ∪ { c } , {\displaystyle Y\cup \{c\},} an operation which is injective (unique and invertible by restriction). The first diagram at the top of the article represents a partial function that is not a function since the element 1 in the left-hand set is not associated with anything in the right-hand set. Whereas, the second diagram represents

1488-402: The codomain of f {\displaystyle f} must equal the domain of g . {\displaystyle g.} The category of sets and partial functions is equivalent to but not isomorphic with the category of pointed sets and point-preserving maps. One textbook notes that "This formal completion of sets and partial maps by adding “improper,” “infinite” elements

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1536-404: 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

1584-508: The consideration of maps between two sets X and Y that may not be defined on the entire set X . A common example is the square root operation on the real numbers R {\displaystyle \mathbb {R} } : because negative real numbers do not have real square roots, the operation can be viewed as a partial function from R {\displaystyle \mathbb {R} } to R . {\displaystyle \mathbb {R} .} The domain of definition of

1632-447: The exact domain of definition is unknown or even unknowable. For a computer-science example of the latter, see Halting problem . In case the domain of definition S is equal to the whole set X , the partial function is said to be total . Thus, total partial functions from X to Y coincide with functions from X to Y . Many properties of functions can be extended in an appropriate sense of partial functions. A partial function

1680-484: 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^{*}}

1728-425: 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

1776-403: The latter also written as ⋃ D ⊆ X Y D . {\textstyle \bigcup _{D\subseteq {X}}Y^{D}.} In finite case, its cardinality is because any partial function can be extended to a function by any fixed value c {\displaystyle c} not contained in Y , {\displaystyle Y,} so that

1824-472: 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

1872-446: The natural logarithm function is not a function when viewed as a function from the reals to themselves, but it is a partial function. If the domain is restricted to only include the positive reals (that is, if the natural logarithm function is viewed as a function from the positive reals to the reals), then the natural logarithm is a function. Subtraction of natural numbers (in which N {\displaystyle \mathbb {N} }

1920-415: 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

1968-412: 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,

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2016-464: 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". Total function More technically,

2064-530: 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

2112-521: The set of all partial functions f : X ⇀ Y {\displaystyle f:X\rightharpoonup Y} from a set X {\displaystyle X} to a set Y {\displaystyle Y} by [ X ⇀ Y ] . {\displaystyle [X\rightharpoonup Y].} This set is the union of the sets of functions defined on subsets of X {\displaystyle X} with same codomain Y {\displaystyle Y} :

2160-466: The smallest domain which is expressible as a type and contains the domain of definition of the function. In category theory , when considering the operation of morphism composition in concrete categories , the composition operation ∘ : hom ⁡ ( C ) × hom ⁡ ( C ) → hom ⁡ ( C ) {\displaystyle \circ \;:\;\hom(C)\times \hom(C)\to \hom(C)}

2208-422: 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

2256-405: Was reinvented many times, in particular, in topology ( one-point compactification ) and in theoretical computer science ." The category of sets and partial bijections is equivalent to its dual . It is the prototypical inverse category . Partial algebra generalizes the notion of universal algebra to partial operations . An example would be a field , in which the multiplicative inversion

2304-500: 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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