Function composition (computer science)In computer science, function composition is an act or mechanism to combine simple functions to build more complicated ones. Like the usual composition of functions in mathematics, the result of each function is passed as the argument of the next, and the result of the last one is the result of the whole. Programmers frequently apply functions to results of other functions, and almost all programming languages allow it. In some cases, the composition of functions is interesting as a function in its own right, to be used later.
Loop spaceIn topology, a branch of mathematics, the loop space ΩX of a pointed topological space X is the space of (based) loops in X, i.e. continuous pointed maps from the pointed circle S1 to X, equipped with the compact-open topology. Two loops can be multiplied by concatenation. With this operation, the loop space is an A∞-space. That is, the multiplication is homotopy-coherently associative. The set of path components of ΩX, i.e. the set of based-homotopy equivalence classes of based loops in X, is a group, the fundamental group π1(X).
Type constructorIn the area of mathematical logic and computer science known as type theory, a type constructor is a feature of a typed formal language that builds new types from old ones. Basic types are considered to be built using nullary type constructors. Some type constructors take another type as an argument, e.g., the constructors for product types, function types, power types and list types. New types can be defined by recursively composing type constructors.
Partial applicationIn computer science, partial application (or partial function application) refers to the process of fixing a number of arguments to a function, producing another function of smaller arity. Given a function , we might fix (or 'bind') the first argument, producing a function of type . Evaluation of this function might be represented as . Note that the result of partial function application in this case is a function that takes two arguments. Partial application is sometimes incorrectly called currying, which is a related, but distinct concept.
Clean (programming language)Clean is a general-purpose purely functional computer programming language. It was called the Concurrent Clean System, then the Clean System, later just Clean. Clean has been developed by a group of researchers from the Radboud University in Nijmegen since 1987. The language Clean first appeared in 1987. Although development of the language has slowed, some researchers are still working in the language. In 2018, a spin-off company was founded that uses Clean.
Categorical abstract machineThe categorical abstract machine (CAM) is a model of computation for programs that preserves the abilities of applicative, functional, or compositional style. It is based on the techniques of applicative computing. The notion of the categorical abstract machine arose in the mid-1980s. It took its place in computer science as a kind of theory of computation for programmers, represented by and embedded into the combinatory logic. CAM is a transparent and sound mathematical representation for the languages of functional programming.
Function applicationIn mathematics, function application is the act of applying a function to an argument from its domain so as to obtain the corresponding value from its range. In this sense, function application can be thought of as the opposite of function abstraction. Function application is usually depicted by juxtaposing the variable representing the function with its argument encompassed in parentheses. For example, the following expression represents the application of the function ƒ to its argument x.
Operator associativityIn programming language theory, the associativity of an operator is a property that determines how operators of the same precedence are grouped in the absence of parentheses. If an operand is both preceded and followed by operators (for example, ^ 3 ^), and those operators have equal precedence, then the operand may be used as input to two different operations (i.e. the two operations indicated by the two operators). The choice of which operations to apply the operand to, is determined by the associativity of the operators.
Scott continuityIn mathematics, given two partially ordered sets P and Q, a function f: P → Q between them is Scott-continuous (named after the mathematician Dana Scott) if it preserves all directed suprema. That is, for every directed subset D of P with supremum in P, its has a supremum in Q, and that supremum is the image of the supremum of D, i.e. , where is the directed join. When is the poset of truth values, i.e. Sierpiński space, then Scott-continuous functions are characteristic functions of open sets, and thus Sierpiński space is the classifying space for open sets.
Puppe sequenceIn mathematics, the Puppe sequence is a construction of homotopy theory, so named after Dieter Puppe. It comes in two forms: a long exact sequence, built from the mapping fibre (a fibration), and a long coexact sequence, built from the mapping cone (which is a cofibration). Intuitively, the Puppe sequence allows us to think of homology theory as a functor that takes spaces to long-exact sequences of groups. It is also useful as a tool to build long exact sequences of relative homotopy groups.
