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In computer sciencefunctional 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 [1] or declarations [2] instead of statements.

In functional code, 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 statewhich may produce different results at different times when called with the same arguments but a different program state.

Eliminating side effectsi. Functional programming has its origins in lambda calculusa formal system developed in the s to investigate computabilitythe Entscheidungsproblemfunction definition, function applicationand recursion. Many functional programming languages can be viewed as elaborations on the lambda calculus.

Another well-known declarative programming paradigm, logic programmingis based on relations. In contrast, imperative programming changes state with commands in the source codethe simplest example being assignment.

Imperative programming does have subroutine functions, but these are not functions in the mathematical sense. They can have side effects that may change the value of program state. Functions without return values therefore make sense. Because of this, they lack referential transparencyi. Functional programming languages have largely been emphasized in academia rather than in commercial software development. However, prominent programming languages that support functional programming such marvinthegreat about binary matrix programming Common LispScheme[4] [5] [6] [7] Clojure[8] [9] Wolfram Language [10] also known as Marvinthegreat about binary matrix programmingRacket[11] Erlang[12] [13] [14] OCaml[15] [16] Haskell[17] [18] and F [19] [20] have been used in industrial and commercial applications by a wide variety of organizations.

JavaScriptone of the world's most widely distributed languages, [21] [22] has the properties of an untyped functional language, [23] in addition to imperative and object-oriented paradigms. Programming in a functional style can also be accomplished in languages that are not specifically designed for functional programming. For example, the imperative Perl programming language has been the subject of a book describing marvinthegreat about binary matrix programming to marvinthegreat about binary matrix programming functional programming concepts.

The Julia language also offers functional programming abilities. An interesting case is that of Scala [31] — it is frequently written in a functional style, but the presence of side effects and mutable state place it in a grey area between imperative and functional languages. Lambda calculus provides a theoretical framework for describing functions and their evaluation. It is a mathematical abstraction rather than a programming marvinthegreat about binary matrix programming it forms the marvinthegreat about binary matrix programming of almost all current functional programming languages.

An equivalent theoretical formulation, combinatory logicis commonly perceived as more abstract than lambda calculus and preceded it in invention.

Combinatory logic and lambda calculus were both originally developed to achieve a clearer approach to the foundations of mathematics. Later dialects, such as Scheme and Clojureand offshoots such as Dylan and Juliasought to simplify and rationalise Lisp around a cleanly functional core, while Common Lisp was designed to preserve and update the paradigmatic features of the numerous older dialects it replaced.

Information Processing Language IPL is sometimes cited as the first computer-based functional programming language. It does have a notion of generatorwhich amounts to a function that accepts a function as an argument, and, since it is an assembly-level language, code can be data, so IPL can be regarded as having higher-order functions.

However, it relies heavily on mutating list structure and similar imperative features. In the early s, Iverson and Roger Hui marvinthegreat about binary matrix programming J.

In the mids, Arthur Whitneywho had previously worked with Iverson, created Kwhich is used commercially in financial industries along with its descendant Q. A Functional Style and its Algebra of Programs". Backus's paper popularized research into functional programming, though it emphasized function-level programming rather than the lambda-calculus style now associated with functional programming. Meanwhile, the development of Schemea simple lexically scoped and impurely functional dialect of Lisp, as described in the influential Lambda Papers and the classic textbook Structure and Interpretation of Computer Programsbrought awareness of the power of functional programming to the wider programming-languages community.

This led to powerful new approaches to interactive theorem proving and has influenced the development of many subsequent functional programming languages. The Haskell language began with a consensus in to form an open standard for functional programming research; implementation releases have been ongoing since 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 are functions that can either take other functions as arguments or return them as results.

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 functions enable partial application or curryinga technique that applies a function to its arguments one at a time, with each application returning a new function that accepts the next argument. This lets a programmer succinctly express, for example, the successor function as the addition operator partially applied to the natural number one. This means that pure functions have several useful properties, many of which can be used to optimize the code:.

While most compilers for imperative programming languages detect pure functions and marvinthegreat about binary matrix programming common-subexpression elimination for pure function calls, they cannot always do this for pre-compiled libraries, which generally marvinthegreat about binary matrix programming not expose this information, thus preventing optimizations that involve those external functions.

Some compilers, such as gccadd extra keywords for a programmer to explicitly mark external functions as pure, to enable such optimizations. Fortran 95 also lets functions be designated pure.

Iteration looping in functional languages is usually accomplished via recursion. Recursive functions invoke themselves, letting an operation be repeated until it reaches the base case. Marvinthegreat about binary matrix programming some recursion requires maintaining a stack, tail recursion can be recognized and optimized by a compiler into the same code used to implement marvinthegreat about binary matrix programming in imperative languages.

The Scheme language standard requires implementations to recognize and optimize tail recursion. Tail recursion optimization can be implemented by transforming the program into continuation passing style during compiling, among other approaches. Common patterns of recursion can be factored out using higher order functions, with 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.

Most general purpose functional programming languages allow unrestricted recursion and are Turing completewhich makes marvinthegreat about binary matrix programming halting problem undecidablecan cause unsoundness of equational reasoningand generally requires the introduction of inconsistency into the logic expressed by the language's type system. Some special purpose languages such as Coq allow only well-founded recursion and are strongly normalizing nonterminating computations can be expressed only with infinite streams of values called codata.

As a consequence, these languages fail to be Turing complete and expressing certain functions marvinthegreat about binary matrix programming them is impossible, but they can still express a wide class of interesting computations while avoiding the problems introduced by unrestricted recursion. Functional programming limited to well-founded recursion with a few other constraints is called total functional programming. Functional languages can be categorized by whether they use strict eager or non-strict lazy evaluation, concepts that refer to how function arguments are processed when an expression is being evaluated.

The technical difference is in the denotational semantics of expressions containing failing or divergent computations. Under strict evaluation, the evaluation of any term containing a failing subterm fails. For example, the expression:. Under lazy evaluation, the length function returns the value 4 i. In brief, strict evaluation always fully evaluates function arguments before invoking the function. Lazy evaluation does not evaluate function arguments unless their values are required to evaluate the function call itself.

The usual implementation strategy for lazy evaluation in functional languages is graph reduction. Hughes argues for lazy evaluation as a mechanism for marvinthegreat about binary matrix programming program modularity through separation of marvinthegreat about binary matrix programmingby easing independent implementation of producers and consumers of data streams. Especially since the development of Hindley—Milner marvinthegreat about binary matrix programming inference in the s, functional programming languages have tended to use typed lambda calculusrejecting all invalid programs at compilation time and risking false positive errorsas opposed to the untyped lambda calculusthat accepts all valid programs at compilation time and risks false negative errorsused in Lisp and its variants such as Schemethough they reject all invalid programs at runtime, when the information is enough to not reject valid programs.

The use of algebraic datatypes makes manipulation of complex data structures convenient; the presence of strong marvinthegreat about binary matrix programming type checking makes programs more reliable in absence of other reliability techniques like test-driven developmentwhile type inference frees the programmer from the need to manually marvinthegreat about binary matrix programming types to the compiler in most cases.

Some research-oriented functional languages such as CoqAgdaCayenneand Epigram are based on intuitionistic type theorywhich lets types depend on terms. Such types are called dependent types. These type systems do not have decidable type inference and are difficult to understand and program with [ citation needed ].

But dependent types can express arbitrary propositions in predicate logic. Through the Curry—Howard isomorphismthen, well-typed programs in these languages become a means of writing formal mathematical proofs from which a compiler can generate certified code. While these languages are mainly of interest in academic research including in formalized mathematicsthey have begun to be used in engineering as well.

Compcert is a compiler for a subset of the C programming language that is written in Coq and formally verified. A limited form of dependent types called generalized algebraic data types GADT's can be implemented in a way that provides some of the benefits of dependently typed programming while avoiding most of its inconvenience.

Functional programs do not have assignment statements, that is, the value of a variable in a functional program never changes once defined. This eliminates any chances of side effects because any variable can be replaced with its actual value at any point of execution. So, functional marvinthegreat about binary matrix programming are referentially transparent. Let us say that the initial value marvinthegreat about binary matrix programming x was 1then two consecutive evaluations of the variable x yields 10 and respectively.

In fact, assignment statements are never referentially transparent. Functional programs exclusively use this type of function and are therefore referentially transparent. It is possible to use a functional style of programming in languages that are not traditionally considered functional languages.

JavaScriptLua [52] and Python had first class functions from their inception. In PHPanonymous classesclosures and lambdas are fully supported. Libraries and language extensions marvinthegreat about binary matrix programming immutable data structures are being developed to aid programming in the functional style. In Javaanonymous classes can sometimes be used to simulate closures ; [56] however, anonymous classes are not always proper replacements to closures because they have more limited capabilities.

In Canonymous classes are not necessary, because closures and lambdas are fully supported. Libraries and language extensions for immutable data structures are being developed to aid programming in the functional style in C. Many object-oriented design patterns are expressible in functional programming terms: Similarly, the idea of immutable data from functional programming is often included in imperative programming languages, [59] for example the tuple in Python, which is an immutable array.

Purely functional data structures are often represented in a different way than their imperative counterparts. Arrays can be replaced by maps or random access listswhich admit purely functional implementation, but have logarithmic access and update times. Therefore, purely functional data structures can be used in non-functional languages, but they may not be the most marvinthegreat about binary matrix programming tool, especially if persistence is not required. Functional programming is very different from imperative programming.

Pure functional programming completely prevents side-effects and provides referential transparency. Higher-order functions are rarely used in older imperative programming. A traditional imperative program might use a loop to traverse and modify a list. There are tasks for example, maintaining a bank account balance that often seem most naturally implemented marvinthegreat about binary matrix programming state. The pure functional programming language Haskell implements them using monadsderived from category theory.

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Nowadays the McCulloch-Pitts neuron tends to be overlooked in favour of simpler neuronal models but they were and are still important.

They proved that something that behaved like a biological neuron was capable of computation and early computer designers often thought in terms of them. Before the neural network algorithms in use today were devised, there was an alternative.

McCulloch-Pitts neurons are more like the sort of approach we see today in neuromorphic chips where neurons are used as computational units. What is interesting about the McCulloch-Pitts model of a neural network is that it can be used as the components of computer-like systems. That is, where neural networks are commonly used to learn something, a McCulloch-Pitts neuron is constructed to do a particular job.

It gives you an idea, although probably not the whole idea, of how the brain might just be a computer. The basic idea of a McCulloch-Pitts model is to use components which have some of the characteristics of real neurons. What the neuron does depends on the sum of inputs. The excitatory inputs tend to make the cell fire and the inhibitory inputs make is not fire — i. When the cell fires a signal propagates down the axon which acts like the output wire and then is applied to the inputs of other neurons via the axon terminals.

You can think of it as a sort of battle to control the neuron with the excitatory inputs pushing it to fire and the inhibitory inputs stopping it. There are lots of complications to this basic model — in particular a neuron fires a train of pulses rather than just changing its state - but the idea of inhibitory fighting excitatory seems to be the key behaviour. What we need is a model of the neuron that doesn't capture all of its properties and behaviour but just enough to capture the way it performs compuation.

So what we have to do is throw away aspects of the way the cell behaves in an effort to simplify without destroying its action. This turns out to be a bit too complicated to analyse so we make one more big simplification — the inhibitory input overrides the excitatory. That is, if an inhibitory input is on the cell cannot fire i. With this small change things are made a lot easier and we can now define precisely what a McCulloch-Pitts neuron is:.

The rule for its operation is that at time t it looks at its excitatory inputs and counts up the number of ones present. Notice that this rule is more complicated than you might think at first and it does give rise to behaviours that are quite complicated. The key factor that you might just overlook is that we are using sequential logic that involves time.

A clock is used to synchronise all of the cells in a network and they change state at each clock pulse. This is necessary to stop infinite period oscillation in some types of circuit. Minsky uses a symbol that shows a cell as a circle with its threshold written in.

It is very easy to invent equivalents of the standard Boolean gates. You should be able to see that the truth tables for each of these cells is equivalent to the corresponding Boolean gate. The OR gate fires if either or both of its inputs are one because its threshold is one. The only tricky one is the NOT gate and you can see here that a single inhibitory input shown with a circle at the end combined with a threshold of zero gives the desired result. When the inhibitor is off the threshold is equalled, i.

If the inhibitor is on then the cell is obviously off — a NOT gate as promised. Given we can make up the basic gates of Boolean logic it is clear that that McCulloch-Pitts networks are up to building anything that can be built with standard logic gates, but are they capable of anything more? The answer is yes and the reason is partly that they have time built into their specification and partly because they are threshold gates.

For example, a delay gate can be built using a single excitatory input with a threshold of one. Stacking such gates up provides a delay of more than one time unit. Delaying a signal is the same as remembering it for n time periods, so cells have memory! If you would like a clearer example of the memory inherent in McCulloch-Pitts cells have a look at the following simple latch. The top cell passes the input to the output if it has a one from the other cell.

You can see that the feedback loop in the lower cell keeps it switched on once it is on irrespective of the input on the Start line. Once on the only way to switch it off is to put a one on the Stop line which, being an inhibitor, turns everything off.

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