Functional programming
2008/9 Schools Wikipedia Selection. Related subjects: Computer Programming
In computer science, functional programming is a programming paradigm that treats computation as the evaluation of mathematical functions and avoids state and mutable data. It emphasizes the application of functions, in contrast with the imperative programming style that emphasizes changes in state.
The lambda calculus provides the model for functional programming. Modern functional languages can be viewed as embellishments to the lambda calculus.
While not fully functional, both the original Lisp and APL were important in the development of functional programming. Later versions of Lisp such as Scheme and variants of APL did provide full functional support. Other important functional languages include Erlang, Haskell, and ML.
Functional programming languages, especially purely functional ones, have largely been emphasized in academia rather than in commercial software development. However, notable functional programming languages used in industry and commercial applications include Erlang (concurrent applications), R (statistics), Mathematica (symbolic math), ML, J and K (financial analysis), and domain-specific programming languages like XSLT.
History
Lambda calculus provides a theoretical framework for describing functions and their evaluation. Though it is a mathematical abstraction rather than a programming language, it forms the basis of almost all functional programming languages today.
Combinatory logic is an equivalent theoretical foundation, developed by Moses Schönfinkel and Haskell Curry. It was originally developed to achieve a clearer approach to the foundations of mathematics. Combinatory logic is commonly perceived as more abstract than lambda calculus and preceded it in invention.
An early functional-flavored language was LISP, developed by John McCarthy while at MIT for the IBM 700/7000 series scientific computers in the late 1950s. LISP introduced many features now found in functional languages, though LISP is technically a multi-paradigm language. Scheme and Dylan were later attempts to simplify and improve LISP.
Information Processing Language (IPL) is sometimes cited as the first computer-based functional programming language. It is an assembly-style language for manipulating lists of symbols. It does have a notion of "generator", which amounts to a function accepting a function as an argument, and, since it is an assembly-level language, code can be used as data, so IPL can be regarded as having higher-order functions. However, it relies heavily on mutating list structure and similar imperative features.
Kenneth E. Iverson developed the APL programming language in the early 1960s, described in his 1962 book A Programming Language ( ISBN 9780471430148). APL was the primary influence on John Backus's FP programming language. In the early 1990s, Iverson and Roger Hui created a successor to APL, the J programming language. In the mid 1990s, Arthur Whitney, who had previously worked with Iverson, created the K programming language, which is used commercially in financial industries.
John Backus presented the FP programming language in his 1977 Turing Award lecture Can Programming Be Liberated From the von Neumann Style? A Functional Style and its Algebra of Programs. He defines functional programs as being built up in a hierarchical way by means of "combining forms" that allow an "algebra of programs"; in modern language, this means that functional programs follow the principle of compositionality. Backus's paper popularized research into functional programming, though it emphasized function-level programming rather than the lambda-calculus style which has come to be associated with functional programming.
In the 1970s the ML programming language was created by Robin Milner at the University of Edinburgh, and David Turner developed initially the language SASL at the University of St. Andrews and later the language Miranda at the University of Kent. ML eventually developed into several dialects, the most common of which are now Objective Caml and Standard ML. The Haskell programming language was released in the late 1980s in an attempt to gather together many ideas in functional programming research.
Concepts
A number of concepts and paradigms are specific to functional programming, and generally foreign to imperative programming (including object oriented programming). However, programming languages are often hybrids of several programming paradigms so programmers using "mostly imperative" languages may have utilized some of these concepts.
Higher-order functions
Functions are higher-order when they can take other functions as arguments, and return them as results. (The derivative and antiderivative in calculus are examples of this.)
Higher-order functions are closely related to first-class functions, in that higher-order functions and first-class functions both allow functions as arguments and results of other functions. The distinction between the two is subtle: "higher-order" describes a mathematical concept of functions that operate on other functions, while "first-class" is a computer science term that describes programming language entities that have no restriction on their use (thus first-class functions can appear anywhere in the program that other first-class entities like numbers can, including as arguments to other functions and as their return values).
Higher-order functions enable currying, a technique in which a function is applied to its arguments one at a time, with each application returning a new (higher-order) function that accepts the next argument.
Pure functions
Purely functional programs have no side effects. This makes it easier to reason about their behaviour. However, almost no programmers bother to write purely functional programs, since, by definition, a program with no side effects (one that accepts no input, produces no output, and interfaces with no external devices) is formally equivalent to a program that does nothing; typically, purity is used to enforce a separation of concerns where one clearly-delineated section of the program does impure operations like I/O, and calls pure functions and libraries as needed to compute answers.
For example, the result of applying a pure function to pure arguments does not depend on the order of evaluation. As a result, a language which has no impure functions (a "purely functional language," such as Haskell) may use call-by-need evaluation. However, not all functional languages are pure. The Lisp family of languages are not pure because they allow side-effects.
Since pure functions do not modify shared variables, pure functions can be executed in parallel without interfering with one another. Pure functions are therefore thread-safe, which allow interpreters and compilers to use call-by-future evaluation.
Pure functional programming languages typically enforce referential transparency, which is the notion that 'equals can be substituted for equals': if two expressions have "equal" values (for some notion of equality), then one can be substituted for the other in any larger expression without affecting the result of the computation. For example, in
y = f(x) * f(x);
a compiler can factor out f(x) if it is pure, transforming the program to
z = f(x); y = z * z;
and eliminating the second evaluation of the (possibly costly) call to f(x). This optimization is called common subexpression elimination.
However, if a function has side effects, the function call cannot be eliminated. Consider the following program fragment:
y = random() * random();
The second call to random cannot be eliminated, because its return value may be different from that of the first call. Similarly,
y = printf("x") * printf("x");
cannot be optimized away; even if printf returns the same value both times; failing to make the second call would result in different program output.
While most compilers for imperative programming languages detect pure functions, and perform common subexpression elimination for pure function calls, pre-compiled libraries generally do not expose this information, preventing calls to external functions from being optimized away. Some compilers, such as gcc, add extra keywords for a programmer to explicitly mark external functions as pure so that this optimization can be performed in the presence of precompiled libraries. Fortran 95 allows functions without side-effects to be designated "pure".
Recursion
Iteration (looping) in functional languages is usually accomplished via recursion. Recursive functions invoke themselves, allowing an operation to be performed over and over. Recursion may require maintaining a stack, but tail recursion can be recognized and optimized by a compiler into the same code used to implement iteration in imperative languages. The Scheme programming language standard requires implementations to recognize and optimize tail recursion.
Common patterns of recursion can be factored out using higher order functions, catamorphisms and anamorphisms (or "folds" and "unfolds") being the most obvious examples. Such higher order functions play a role analogous to built-in control structures such as loops in imperative languages.
Strict, non-strict and lazy evaluation
Functional languages can be categorized by whether they use strict or non-strict evaluation, concepts that refer to how function arguments are processed when an expression is being evaluated. To illustrate, consider the following two functions f and g:
f(x) := x^2+x+1
g(x, y) := x+y
The following expression can be evaluated in one of two ways.
f(g(1, 4))
Evaluate the innermost function g first:
f(g(1, 4)) → f(1+4) → f(5) → 5^2+5+1 → 31
Or evaluate the outermost function f first:
f(g(1, 4)) → g(1,4)^2+g(1,4)+1 → (1+4)^2+(1+4)+1 → 5^2+5+1 → 31
The first case is an instance of strict evaluation: arguments to a function are evaluated before the function call; while the second case is an instance of non-strict evaluation where arguments are passed to the function unevaluated and the calling function determines when the arguments are to be evaluated.
Strict evaluation has efficiency advantages. An argument is evaluated once with strict evaluation, while it may be evaluated multiple times with non-strict evaluation, as can be seen in the above example where g(1,4) is evaluated twice. Also, strict evaluation is easier to implement since the arguments passed to a function are data values, whereas with non-strict evaluation arguments may be expressions, requiring some notion of closure. For these reasons, the earliest functional languages, such as Lisp, ISWIM and ML use strict evaluation.
However there are reasons for preferring non-strict evaluation. Lambda calculus provides a stronger theoretic foundation for languages that employ non-strict evaluation. Also non-strict evaluation provides for a more expressive language. For example, it supports infinite data structures, such as a list of all prime numbers (such structures are of use when an indefinite but finite part of the structure is required).
The need for a more efficient form of non-strict evaluation led to the development of lazy evaluation, a type of non-strict evaluation, where the initial evaluation of an argument is shared throughout the evaluation sequence. Consequently an argument (such as g(1,4) in the above example) is never evaluated more than once. Lazy evaluation tends to be used by pure functional languages such as Miranda, Clean and Haskell; however many other more recent functional languages continue to use strict evaluation.
Functional programming in non-functional languages
It is possible to employ a functional style of programming in languages that are not traditionally considered functional languages. Some non-functional languages have borrowed features such as higher-order functions, and list comprehensions from functional programming languages. This makes it easier to adopt a functional style when using these languages. Functional constructs such as higher-order functions and lazy lists can be obtained in C++ via libraries. In C one can use function pointers to get some of the effects of higher-order functions, for example one can implement the common function map using function pointers. Widespread declarative domain specific languages like SQL and Lex/ Yacc, while not always Turing-complete, use some elements of functional programming, especially in eschewing mutable values.
Comparison of functional and imperative programming
Functional programming is very different from imperative programming. The most significant differences stem from the fact that functional programming avoids side effects, which are used in imperative programming to implement state and I/O. Pure functional programming disallows side effects completely. Disallowing side effects provides for referential transparency, which makes it easier to verify, optimize, and parallelize programs, and easier to write automated tools to perform those tasks.
Higher order functions are rarely used in older imperative programming. Where a traditional imperative program might use a loop to traverse a list, a functional style would often use a higher-order function, map, that takes as arguments a function and a list, applies the function to each element of the list, and returns a list of the results.
Simulating state
There are tasks—for example, maintaining a bank account balance—that often seem most naturally implemented with state. Pure functional programming performs these tasks, and I/O tasks such as accepting user input and printing to the screen, in a different way.
The pure functional programming language Haskell implements them using monads, derived from category theory. Monads are powerful and offer an intuitive way to model state (and other side effects such as IO) in an imperative manner without losing purity. While existing monads are easy to use, many find it difficult to understand how to define new monads (which is sometimes needed for certain types of libraries).
Alternative methods such as Hoare logic and uniqueness have been developed to track side effects in programs. Some modern research languages use effect systems to make explicit the presence of side effects.
Efficiency issues
Functional programming languages have automatic memory management with garbage collection, in contrast to older imperative languages like C and Pascal which use explicit memory management. Functional programming languages have been perceived as less efficient in their use of CPU and memory than those languages. However, many modern imperative languages such as Java, Perl, Python, and Ruby also perform automatic memory management.
Functional programming languages have become more efficient over the years. For programs that perform intensive numerical computations, functional languages such as OCaml and Clean are similar in speed to C. For programs that handle large matrices and multidimensional databases, array functional languages (such as J and K) were designed with speed optimization in mind.
Purely functional languages have a reputation for being slower than imperative languages. However, immutability of data can, in many cases, lead to execution efficiency in allowing the compiler to make assumptions that are unsafe in an imperative language. The worst-case slowdown was shown to be exponential. Situations where such slowdowns arise occur very rarely in practice. They do imply, however, that non-functional supersets of applicative languages are sometimes important.
Coding styles
Imperative programs tend to emphasize the series of steps taken by a program in carrying out an action, while functional programs tend to emphasize the composition and arrangement of functions, often without specifying explicit steps. A simple example of two solutions to the same programming goal (using the same multi-paradigm language Python) illustrates this.
# imperative style target = [] # create empty list for item in source_list: # iterate over each thing in source trans1 = G(item) # transform the item with the G() function trans2 = F(trans1) # second transform with the F() function target.append(trans2) # add transformed item to target
A functional version has a different feel to it:
# functional style # FP-oriented languages often have standard compose() compose2 = lambda F, G: lambda x: F(G(x)) target = map(compose2(F,G), source_list)
In contrast to the imperative style that describes the steps involved in building target, the functional style describes the mathematical relationship between source_list and target.