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Hylomorphism (computer Science)
In computer science, and in particular functional programming, a hylomorphism is a recursive function, corresponding to the composition of an anamorphism (which first builds a set of results; also known as 'unfolding') followed by a catamorphism (which then folds these results into a final return value). Fusion of these two recursive computations into a single recursive pattern then avoids building the intermediate data structure. This is an example of deforestation, a program optimization strategy. A related type of function is a metamorphism, which is a catamorphism followed by an anamorphism. Formal definition A hylomorphism h : A \rightarrow C can be defined in terms of its separate anamorphic and catamorphic parts. The anamorphic part can be defined in terms of a unary function g : A \rightarrow B \times A defining the list of elements in B by repeated application (''"unfolding"''), and a predicate p : A \rightarrow \text providing the terminating condition. The catamorph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computer Science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical disciplines (including the design and implementation of Computer architecture, hardware and Computer programming, software). Computer science is generally considered an area of research, academic research and distinct from computer programming. Algorithms and data structures are central to computer science. The theory of computation concerns abstract models of computation and general classes of computational problem, problems that can be solved using them. The fields of cryptography and computer security involve studying the means for secure communication and for preventing Vulnerability (computing), security vulnerabilities. Computer graphics (computer science), Computer graphics and computational geometry address the generation of images. Progr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word isomorphism is derived from the Ancient Greek: ἴσος ''isos'' "equal", and μορφή ''morphe'' "form" or "shape". The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may be identified. In mathematical jargon, one says that two objects are . An automorphism is an isomorphism from a structure to itself. An isomorphism between two structures is a canonical isomorphism (a canonical map that is an isomorphism) if there is only one isomorphism between the two structures (as it is the case for solutions of a univer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Apomorphism
In formal methods of computer science, an apomorphism (from '' ἀπό'' — Greek for "apart") is the categorical dual of a paramorphism and an extension of the concept of anamorphism (coinduction). Whereas a paramorphism models primitive recursion over an inductive data type, an apomorphism models primitive corecursion over a coinductive data type. Origins The term "apomorphism" was introduced in ''Functional Programming with Apomorphisms (Corecursion)''. See also * Morphism * Morphisms of F-algebras ** From an initial algebra to an algebra: Catamorphism ** From a coalgebra to a final coalgebra: Anamorphism ** An anamorphism followed by an catamorphism: Hylomorphism Hylomorphism (also hylemorphism) is a philosophical theory developed by Aristotle, which conceives every physical entity or being (''ousia'') as a compound of matter (potency) and immaterial form (act), with the generic form as immanently real ... ** Extension of the idea of catamorphisms: Paramorphism R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paramorphism
In formal methods of computer science, a paramorphism (from Greek '' παρά'', meaning "close together") is an extension of the concept of catamorphism first introduced by Lambert Meertens to deal with a form which “eats its argument and keeps it too”, as exemplified by the factorial function. Its categorical dual is the apomorphism. It is a more convenient version of catamorphism in that it gives the combining step function immediate access not only to the result value recursively computed from each recursive subobject, but the original subobject itself as well. Example Haskell implementation, for lists: cata :: (a -> b -> b) -> b -> -> b para :: (a -> ( b) -> b) -> b -> -> b ana :: (b -> (a, b)) -> b -> apo :: (b -> (a, Either b)) -> b -> cata f b (a:as) = f a (cata f b as) cata _ b [] = b para f b (a:as) = f a (as, para f b as) para _ b [] = b ana u b = case u b of (a, b') -> a : ana u b' apo u b = case u b of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anamorphism
In computer programming, an anamorphism is a function that generates a sequence by repeated application of the function to its previous result. You begin with some value A and apply a function f to it to get B. Then you apply f to B to get C, and so on until some terminating condition is reached. The anamorphism is the function that generates the list of A, B, C, etc. You can think of the anamorphism as unfolding the initial value into a sequence. The above layman's description can be stated more formally in category theory: the anamorphism of a coinductive type denotes the assignment of a coalgebra to its unique morphism to the final coalgebra of an endofunctor. These objects are used in functional programming as '' unfolds''. The categorical dual (aka opposite) of the anamorphism is the catamorphism. Anamorphisms in functional programming In functional programming, an anamorphism is a generalization of the concept of '' unfolds'' on coinductive lists. Formally, anamorph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Catamorphism
In category theory, the concept of catamorphism (from the Ancient Greek: "downwards" and "form, shape") denotes the unique homomorphism from an initial algebra into some other algebra. In functional programming, catamorphisms provide generalizations of '' folds'' of lists to arbitrary algebraic data types, which can be described as initial algebras. The dual concept is that of anamorphism that generalize ''unfolds''. A hylomorphism is the composition of an anamorphism followed by a catamorphism. Definition Consider an initial F-algebra (A, in) for some endofunctor F of some category into itself. Here in is a morphism from FA to A. Since it is initial, we know that whenever (X, f) is another F-algebra, i.e. a morphism f from FX to X, there is a unique homomorphism h from (A, in) to (X, f). By the definition of the category of F-algebra, this h corresponds to a morphism from A to X, conventionally also denoted h, such that h \circ in = f \circ Fh. In the context of F-algebra, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
F-algebra
In mathematics, specifically in category theory, ''F''-algebras generalize the notion of algebraic structure. Rewriting the algebraic laws in terms of morphisms eliminates all references to quantified elements from the axioms, and these algebraic laws may then be glued together in terms of a single functor ''F'', the ''signature''. ''F''-algebras can also be used to represent data structures used in programming, such as lists and trees. The main related concepts are initial ''F''-algebras which may serve to encapsulate the induction principle, and the dual construction ''F''-coalgebras. Definition If C is a category, and F : C \rightarrow C is an endofunctor of C, then an F-algebra is a tuple (A, \alpha), where A is an object of C and \alpha is a C- morphism F(A) \rightarrow A. The object A is called the ''carrier'' of the algebra. When it is permissible from context, algebras are often referred to by their carrier only instead of the tuple. A homomorphism from an F-al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in topology, continuous functions, and so on. In category theory, ''morphism'' is a broadly similar idea: the mathematical objects involved need not be sets, and the relationships between them may be something other than maps, although the morphisms between the objects of a given category have to behave similarly to maps in that they have to admit an associative operation similar to function composition. A morphism in category theory is an abstraction of a homomorphism. The study of morphisms and of the structures (called "objects") over which they are defined is central to category theory. Much of the terminology of morphisms, as well as the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Summation
In mathematics, summation is the addition of a sequence of any kind of numbers, called ''addends'' or ''summands''; the result is their ''sum'' or ''total''. Beside numbers, other types of values can be summed as well: functions, vectors, matrices, polynomials and, in general, elements of any type of mathematical objects on which an operation denoted "+" is defined. Summations of infinite sequences are called series. They involve the concept of limit, and are not considered in this article. The summation of an explicit sequence is denoted as a succession of additions. For example, summation of is denoted , and results in 9, that is, . Because addition is associative and commutative, there is no need of parentheses, and the result is the same irrespective of the order of the summands. Summation of a sequence of only one element results in this element itself. Summation of an empty sequence (a sequence with no elements), by convention, results in 0. Very often, the elements ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Leaf Node
In computer science, a tree is a widely used abstract data type that represents a hierarchical tree structure with a set of connected nodes. Each node in the tree can be connected to many children (depending on the type of tree), but must be connected to exactly one parent, except for the ''root'' node, which has no parent. These constraints mean there are no cycles or "loops" (no node can be its own ancestor), and also that each child can be treated like the root node of its own subtree, making recursion a useful technique for tree traversal. In contrast to linear data structures, many trees cannot be represented by relationships between neighboring nodes in a single straight line. Binary trees are a commonly used type, which constrain the number of children for each parent to exactly two. When the order of the children is specified, this data structure corresponds to an ordered tree in graph theory. A value or pointer to other data may be associated with every node in the tree ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fib 4
Fib may refer to: Places * Kingdom of Fib, now Fife, Scotland Arts, entertainment, and media * Fib (poetry), a form of poetry * Festival Internacional de Benicàssim, a Spanish music festival * '' Folket i Bild'', a Swedish news magazine Enterprises * First International Bank (Liberia) * First Investment Bank, a Bulgarian bank Medicine * Fascia iliaca block * Fibrillarin * Fibrillation * Fibrinogen * Fibula Science and technology * FiB index, or Fisheries in Balance index * Flying inflatable boat * Focused ion beam * Forwarding information base * Fluoride-ion Battery Sport * Federation of International Bandy * Fédération Internationale de Boules, the highest international authority of bocce sport Other uses * Fib (lie), a form of bending the truth that is usually forgiven because it is not intended to deceive * Barcelona School of Informatics (Catalan: ') * Fires brigade, of the United States Army * International Federation for Structural Concrete (French: ) * Unit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fibonacci Sequence
In mathematics, the Fibonacci numbers, commonly denoted , form a integer sequence, sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from 1 and 1 or sometimes (as did Fibonacci) from 1 and 2. Starting from 0 and 1, the first few values in the sequence are: :0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144. The Fibonacci numbers were first described in Indian mathematics, as early as 200 BC in work by Pingala on enumerating possible patterns of Sanskrit poetry formed from syllables of two lengths. They are named after the Italian mathematician Leonardo of Pisa, later known as Fibonacci, who introduced the sequence to Western European mathematics in his 1202 book ''Liber Abaci''. Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the ''Fibonacci Quarterly''. Applications of Fibonacci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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