In computer science, functional programming is a programming paradigm—a style of building the structure and elements of computer programs—that treats computation as the evaluation of mathematical functions and avoids changing-state and mutable data. It is a declarative programming paradigm, which means programming is done with expressions or declarations instead of statements. Functional code is idempotent, the output value of a function depends only on the arguments that are passed to the function, so calling a function f twice with the same value for an argument x produces the same result f(x) each time; this is in contrast to procedures depending on a local or global state, which may produce different results at different times when called with the same arguments but a different program state. Eliminating side effects, i.e., changes in state that do not depend on the function inputs, can make it much easier to understand and predict the behavior of a program, which is one of the key motivations for the development of functional programming.
Lisp is a family of computer programming languages with a long history and a distinctive, fully parenthesized prefix notation.
Originally specified in 1958, Lisp is the second-oldest high-level programming language in widespread use today. Only Fortran is older, by one year. Lisp has changed since its early days, and many dialects have existed over its history. Today, the best known general-purpose Lisp dialects are Clojure, Common Lisp, and Scheme.
Lambda calculus is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution. It is a universal model of computation that can be used to simulate any Turing machine. It was first introduced by mathematician Alonzo Church in the 1930s as part of his research of the foundations of mathematics.
In theoretical computer science and formal language theory, a regular language is a formal language that can be expressed using a regular expression, in the strict sense of the latter notion used in theoretical computer science.
Scheme is a programming language that supports multiple paradigms, including functional and imperative programming. It is one of the three main dialects of Lisp, alongside Common Lisp and Clojure. Unlike Common Lisp, Scheme follows a minimalist design philosophy, specifying a small standard core with powerful tools for language extension.
In theoretical computer science and mathematics, the theory of computation is the branch that deals with how efficiently problems can be solved on a model of computation, using an algorithm. The field is divided into three major branches: automata theory and languages, computability theory, and computational complexity theory, which are linked by the question: "What are the fundamental capabilities and limitations of computers?".
In programming languages, a closure, also lexical closure or function closure, is a technique for implementing lexically scoped name binding in a language with first-class functions. Operationally, a closure is a record storing a function together with an environment. The environment is a mapping associating each free variable of the function with the value or reference to which the name was bound when the closure was created. Unlike a plain function, a closure allows the function to access those captured variables through the closure's copies of their values or references, even when the function is invoked outside their scope.
In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation (symbol) that specifies places in an expression where substitution may take place and is not a parameter of this or any container expression. Some older books use the terms real variable and apparent variable for free variable and bound variable. The idea is related to a placeholder, or a wildcard character that stands for an unspecified symbol.
Combinatory logic is a notation to eliminate the need for quantified variables in mathematical logic. It was introduced by Moses Schönfinkel and Haskell Curry, and has more recently been used in computer science as a theoretical model of computation and also as a basis for the design of functional programming languages. It is based on combinators which were introduced by Schönfinkel in 1920 with the idea of providing an analogous way to build up functions - and to remove any mention of variables - particularly in predicate logic. A combinator is a higher-order function that uses only function application and earlier defined combinators to define a result from its arguments.
In computer science's combinatory logic, a fixed-point combinator is a higher-order function fix that, for any function f that has an attractive fixed point, returns a fixed point x of that function. A fixed point of a function is a value that, when applied as the input of the function, returns the same value as its output.
In computer programming, cons
is a fundamental function in most dialects of the Lisp programming language. cons
constructs memory objects which hold two values or pointers to values. These objects are referred to as (cons) cells, conses, non-atomic s-expressions ("NATSes"), or (cons) pairs. In Lisp jargon, the expression "to cons x onto y" means to construct a new object with (cons x y). The resulting pair has a left half, referred to as the car
, and a right half, referred to as the cdr
.
A typed lambda calculus is a typed formalism that uses the lambda-symbol to denote anonymous function abstraction. In this context, types are usually objects of a syntactic nature that are assigned to lambda terms; the exact nature of a type depends on the calculus considered. From a certain point of view, typed lambda calculi can be seen as refinements of the untyped lambda calculus but from another point of view, they can also be considered the more fundamental theory and untyped lambda calculus a special case with only one type.
In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context. Mathematical symbols can designate numbers (constants), variables, operations, functions, brackets, punctuation, and grouping to help determine order of operations, and other aspects of logical syntax.
In functional programming, continuation-passing style (CPS) is a style of programming in which control is passed explicitly in the form of a continuation. This is contrasted with direct style, which is the usual style of programming. Gerald Jay Sussman and Guy L. Steele, Jr. coined the phrase in AI Memo 349 (1975), which sets out the first version of the Scheme programming language.
John C. Reynolds gives a detailed account of the numerous discoveries of continuations.
System F, also known as the (Girard–Reynolds) polymorphic lambda calculus or the second-order lambda calculus, is a typed lambda calculus that differs from the simply typed lambda calculus by the introduction of a mechanism of universal quantification over types. System F thus formalizes the notion of parametric polymorphism in programming languages, and forms a theoretical basis for languages such as Haskell and ML. System F was discovered independently by logician Jean-Yves Girard (1972) and computer scientist John C. Reynolds (1974).
Lambda lifting is a meta-process that restructures a computer program so that functions are defined independently of each other in a global scope. An individual "lift" transforms a local function into a global function. It is a two step process, consisting of;
Evaluation strategies are used by programming languages to determine when to evaluate the argument(s) of a function call and what kind of value to pass to the function. For example, call by value/call by reference specifies that a function application evaluates the argument before it proceeds to the evaluation of the function's body and that it passes two capabilities to the function, namely, the ability to look up the current value of the argument and to modify it via an assignment statement. The notion of reduction strategy in lambda calculus is similar but distinct.
In mathematics and computer science, apply is a function that applies functions to arguments. It is central to programming languages derived from lambda calculus, such as LISP and Scheme, and also in functional languages. It has a role in the study of the denotational semantics of computer programs, because it is a continuous function on complete partial orders. Apply is also a continuous function in homotopy theory, and, indeed underpins the entire theory: it allows a homotopy deformation to be viewed as a continuous path in the space of functions. Likewise, valid mutations (refactorings) of computer programs can be seen as those that are "continuous" in the Scott topology.
In lambda calculus, a branch of mathematical logic concerned with the formal study of functions, a reduction strategy is how a complex expression is reduced to a simple expression by successive reduction steps. It is similar to but subtly different from the notion of evaluation strategy in computer science.
