Closure (computer programming)

In computer science, a closure (also lexical closure, function closure or function value) is a function together with a referencing environment for the nonlocal names (free variables) of that function. Such a function is said to be "closed over" its free variables. The referencing environment binds the nonlocal names to the corresponding variables in scope at the time the closure is created, additionally extending their lifetime to at least as long as the lifetime of the closure itself.

The concept of closures was developed in the 1960s and was first fully implemented as a language feature in 1975 in the programming language Scheme to implement lexically-scoped first-class functions. Since then, many languages have been designed to support closures. The explicit use of closures is associated with functional programming and with languages such as ML and Lisp. Traditional imperative languages (such as Algol, C and Pascal) had no support for closures as these languages neither support nonlocal names (which can be introduced only in nested or anonymous functions) nor higher-order functions. Modern garbage-collected imperative languages (such as Smalltalk, the first object-oriented language featuring closures, C#, but notably not Java) and many interpreted and scripting languages (Such as Lua), do support higher-order functions and closures.

Closures are used to implement continuation-passing style, and in this manner, hide state. Constructs such as objects and control structures can thus be implemented with closures. In some languages, a closure may occur when a function is defined within another function, and the inner function refers to local variables of the outer function. At run-time, when the outer function executes, a closure is formed, consisting of the inner function’s code and references to any variables of the outer function required by the closure; such variables are called the upvalues of the closure.

The term closure is often mistakenly used to mean anonymous function. This is probably because most languages implementing anonymous functions allow them to form closures and programmers are usually introduced to both concepts at the same time. These are, however, distinct concepts. A closure retains a reference to the environment at the time it was created (for example, to the current value of a local variable in the enclosing scope) while a generic anonymous function need not do this.

Closures are closely related to function objects; the transformation from the former to the latter is known as defunctionalization or lambda lifting.

Etymology

Peter J. Landin defined the term closure in 1964 as having an environment part and a control part as used by his SECD machine for evaluating expressions. Joel Moses credits Landin with introducing the term closure to refer to a lambda expression whose open bindings (free variables) have been closed by (or bound in) the lexical environment, resulting in a closed expression, or closure. This usage was subsequently adopted by Sussman and Steele when they defined Scheme in 1975, and became widespread.

Closures and first-class functions

Further information: First-class function

Closures typically appear in languages in which functions are first-class values—in other words, such languages allow functions to be passed as arguments, returned from function calls, bound to variable names, etc., just like simpler types such as strings and integers. For example, consider the following Scheme function:

; Return a list of all books with at least THRESHOLD copies sold.
(define (best-selling-books threshold)
  (filter
    (lambda (book)
      (>= (book-sales book) threshold))
   book-list))

In this example, the lambda expression (lambda (book) (>= (book-sales book) threshold)) appears within the function best-selling-books. When the lambda expression is evaluated, Scheme creates a closure consisting of the code for the lambda expression and a reference to the threshold variable, which is a free variable inside the lambda expression.

The closure is then passed to the filter function, which calls it repeatedly to determine which books are to be added to the result list and which are to be discarded. Because the closure itself has a reference to threshold, it can use that variable each time filter calls it. The function filter itself might be defined in a completely separate file.

Here is the same example rewritten in JavaScript, another popular language with support for closures:

// Return a list of all books with at least 'threshold' copies sold.
function bestSellingBooks(threshold) {
  return bookList.filter(
      function (book) { return book.sales >= threshold; }
    );
}

The function keyword is used here instead of lambda, and an Array.filter method instead of a global filter function, but otherwise the structure and the effect of the code are the same.

A function may create a closure and return it, as in the following example:

// Return a function that approximates the derivative of f
// using an interval of dx, which should be appropriately small.
function derivative(f, dx) {
  return function (x) {
    return (f(x + dx) - f(x)) / dx;
  };
}

Because the closure in this case outlives the scope of the function that creates it, the variables f and dx live on after the function derivative returns. In languages without closures, the lifetime of a local variable coincides with the execution of the scope where that variable is declared. In languages with closures, variables must continue to exist as long as any existing closures have references to them. This is most commonly implemented using some form of garbage collection.

Closures and state representation

A closure can be used to associate a function with a set of "private" variables, which persist over several invocations of the function. The scope of the variable encompasses only the closed-over function, so it cannot be accessed from other program code.

In stateful languages, closures can thus be used to implement paradigms for state representation and information hiding, since the closure's upvalues (its closed-over variables) are of indefinite extent, so a value established in one invocation remains available in the next. Closures used in this way no longer have referential transparency, and are thus no longer pure functions; nevertheless, they are commonly used in "near-functional" languages such as Scheme.

Uses of closures

Closures have many uses:

• Because closures delay evaluation—i.e., they do not "do" anything until they are called—they can be used to define control structures. For example, all Smalltalk's standard control structures, including branches (if/then/else) and loops (while and for), are defined using objects whose methods accept closures. Users can easily define their own control structures also.
• In languages that allow assignment, multiple functions can be produced that close over the same environment, enabling them to communicate privately by altering that environment. In Scheme:

(define foo #f)
(define bar #f)

(let ((secret-message "none"))
  (set! foo (lambda (msg) (set! secret-message msg)))
  (set! bar (lambda () secret-message)))

(display (bar)) ; prints "none"
(newline)
(foo "meet me by the docks at midnight")
(display (bar)) ; prints "meet me by the docks at midnight"

• Closures can be used to implement object systems.

Note: Some speakers call any data structure that binds a lexical environment a closure, but the term usually refers specifically to functions.

Differences in semantics

Lexical environment

As different languages do not always have a common definition of the lexical environment, their definitions of closure may vary also. The commonly held minimalist definition of the lexical environment defines it as a set of all bindings of variables in the scope, and that is also what closures in any language have to capture. However the meaning of a variable binding also differs. In imperative languages, variables bind to relative locations in memory that can store values. Although the relative location of a binding does not change at runtime, the value in the bound location can. In such languages, since closure captures the binding, any operation on the variable, whether done from the closure or not, are performed on the same relative memory location. Here is an example illustrating the concept in ECMAScript, which is one such language:

var f, g;
function foo() {
  var x = 0;
  f = function() { return ++x; };
  g = function() { return --x; };
  x = 1;
  alert('inside foo, call to f(): ' + f()); // "2"
}
foo();
alert('call to g(): ' + g()); // "1"
alert('call to f(): ' + f()); // "2"

Note how function foo and the closures referred to by variables f and g all use the same relative memory location signified by local variable x.

On the other hand, many functional languages, such as ML, bind variables directly to values. In this case, since there is no way to change the value of the variable once it is bound, there is no need to share the state between closures—they just use the same values.

Yet another subset, lazy functional languages such as Haskell, bind variables to results of future computations rather than values. Consider this example in Haskell:

foo :: Num -> Num -> (Num -> Num)
foo x y = let r = x / y
          in (\z -> z + r)

f :: Num -> Num
f = foo 1 0

main = print (f 123)

The binding of r captured by the closure defined within function foo is to the computation (x / y) - which in this case results in division by zero. However, since it is the computation that is captured, and not the value, the error only manifests itself when the closure is invoked, and actually attempts to use the captured binding.

Closure leaving

Yet more differences manifest themselves in the behavior of other lexically-scoped constructs, such as return, break and continue statements. Such constructs can, in general, be considered in terms of invoking an escape continuation established by an enclosing control statement (in case of break and continue, such interpretation requires looping constructs to be considered in terms of recursive function calls). In some languages, such as ECMAScript, return refers to the continuation established by the closure lexically innermost with respect to the statement—thus, a return within a closure transfers control to the code that called it. In Smalltalk, however, the superficially similar ^ operator invokes the escape continuation established for the method invocation, ignoring the escape continuations of any intervening nested closures. The escape continuation of a particular closure can only be invoked in Smalltalk implicitly by reaching the end of the closure's code. The following examples in ECMAScript and Smalltalk highlight the difference:

"Smalltalk"
foo
  | xs |
  xs := #(1 2 3 4).
  xs do: [:x | ^x].
  ^0
bar
  Transcript show: (self foo printString) "prints 1"

// ECMAScript
function foo() {
  var xs = [1, 2, 3, 4];
  xs.forEach(function (x) { return x; });
  return 0;
}
alert(foo()); // prints 0

The above code snippets will behave differently because the Smalltalk ^ operator and the JavaScript return operator are not analogous. In the ECMAScript example, return x will leave the inner closure to begin a new iteration of the forEach loop, whereas in the Smalltalk example, ^x will abort the loop and return from the method foo.

Common Lisp provides a construct that can express either of the above actions: Smalltalk ^x behaves as (return-from foo x), while JavaScript return x behaves as (return-from nil x). Hence, Smalltalk makes it possible for a captured escape continuation to outlive the extent in which it can be successfully invoked. Consider:

foo
    ^[ :x | ^x ]
bar
    | f |
    f := self foo.
    f value: 123 "error!"

When the closure returned by the method foo is invoked, it attempts to return a value from the invocation of foo that created the closure. Since that call has already returned and the Smalltalk method invocation model does not follow the spaghetti stack discipline to allow multiple returns, this operation results in an error.

Some languages, such as Ruby, allow the programmer to choose the way return is captured. An example in Ruby:

# ruby
def foo
  f = Proc.new { return "return from foo from inside proc" }
  f.call # control leaves foo here
  return "return from foo"
end

def bar
  f = lambda { return "return from lambda" }
  f.call # control does not leave bar here
  return "return from bar"
end

puts foo # prints "return from foo from inside proc"
puts bar # prints "return from bar"

Both Proc.new and lambda in this example are ways to create a closure, but semantics of the closures thus created are different with respect to the return statement.

In Scheme, definition and scope of the return control statement is explicit (and only arbitrarily named 'return' for the sake of the example). The following is a direct translation of the Ruby sample.

(define call/cc call-with-current-continuation)

(define (foo)
  (call/cc
   (lambda (return)
     (define (f) (return "return from foo from inside proc"))
     (f) ; control leaves foo here
     (return "return from foo"))))

(define (bar)
  (call/cc
   (lambda (return)
     (define (f) (call/cc (lambda (return) (return "return from lambda"))))
     (f) ; control does not leave bar here
     (return "return from bar"))))

(display (foo)) ; prints "return from foo from inside proc"
(newline)
(display (bar)) ; prints "return from bar"

Implementation and theory

Closures are typically implemented with a special data structure that contains a pointer to the function code, plus a representation of the function's lexical environment (e.g., the set of available variables and their values) at the time when the closure was created.

A language implementation cannot easily support full closures if its run-time memory model allocates all local variables on a linear stack. In such languages, a function's local variables are deallocated when the function returns. However, a closure requires that the free variables it references survive the enclosing function's execution. Therefore, those variables must be allocated so that they persist until no longer needed. This explains why, typically, languages that natively support closures also use garbage collection. The alternative is for the language to accept that certain use cases will lead to undefined behaviour, as in the proposal for lambda expressions in C++. The Funarg problem (or "functional argument" problem) describes the difficulty of implementing functions as first class objects in a stack-based programming language such as C or C++. Similarly in D version 1, it is assumed that the programmer knows what to do with delegates and local variables, as their references will be invalid after return from its definition scope (local variables are on the stack) - this still permits many useful functional patterns, but for complex cases needs explicit heap allocation for variables. D version 2 solved this by detecting which variables must be stored on the heap, and performs automatic allocation. Because D uses garbage collection, in both versions, there is no need to track usage of variables as they are passed.

In strict functional languages with immutable data (e.g. Erlang), it is very easy to implement automatic memory management (garbage collection), as there are no possible cycles in variables references. For example in Erlang, all arguments and variables are allocated on the heap, but references to them are additionally stored on the stack. After a function returns, references are still valid. Heap cleaning is done by incremental garbage collector.

In ML, local variables are allocated on a linear stack . When a closure is created, it copies the values of those variables that are needed by the closure into the closure's data structure.

Scheme, which has an ALGOL-like lexical scope system with dynamic variables and garbage collection, lacks a stack programming model and does not suffer from the limitations of stack-based languages. Closures are expressed naturally in Scheme. The lambda form encloses the code and the free variables of its environment, persists within the program as long as it can possibly be accessed, and can be used as freely as any other Scheme expression.

Closures are closely related to Actors in the Actor model of concurrent computation where the values in the function's lexical environment are called acquaintances. An important issue for closures in concurrent programming languages is whether the variables in a closure can be updated and, if so, how these updates can be synchronized. Actors provide one solution.

Closure-like constructs in other languages

In C, libraries that support callbacks sometimes allow a callback to be registered using two values: a function pointer and a separate void* pointer to arbitrary data of the user's choice. Each time the library executes the callback function, it passes in the data pointer. This allows the callback to maintain state and to refer to information captured at the time it was registered. The idiom is similar to closures in functionality, but not in syntax.

Several object-oriented techniques and language features simulate some features of closures. For example:

Eiffel

Eiffel includes inline agents defining closures. An inline agent is an object representing a routine, defined by giving the code of the routine in-line. For example, in

ok_button.click_event.subscribe (
	agent (x, y: INTEGER) do
		map.country_at_coordinates (x, y).display
	end
)

the argument to subscribe is an agent, representing a procedure with two arguments; the procedure finds the country at the corresponding coordinates and displays it. The whole agent is "subscribed" to the event type click_event for a certain button, so that whenever an instance of the event type occurs on that button - because a user has clicked the button - the procedure will be executed with the mouse coordinates being passed as arguments for x and y.

The main limitation of Eiffel agents, which distinguishes them from true closures, is that they cannot reference local variables from enclosing scope, but this can easily be worked around by providing additional closed operands to the agent. Only Current (a reference to current object, analogous to this in Java), its features, and arguments of the agent itself can be accessed from within the agent body.

Erlang

In Erlang, closures are supported simply using the keyword fun (Erlang's name for anonymous function) with references to outer variables. Because Erlang is a functional language with immutable value passing semantics, it is both easy to construct closures, execute them, or manage memory. Implementation is done by hidden module-level functions with N+M arguments (N: number of closed outer variables; M: number of own arguments), which is also very simple (see Lambda lifting).

construct_filter(L) ->
  Filter = fun (X) -> lists:member(X, L) end,  % by using L in this fun,
  Filter.                                      % we construct closure

complex_filter(SmallListOfSearchedElements, BigListToBeSearched) ->
  Filter = construct_filter(SmallListOfSearchedElements),
  Result = lists:filter(Filter, BigListToBeSearched),
  Result.

C++ (operator overloading)

C++ allows defining function objects by overloading operator(). These objects behave somewhat like functions in a functional programming language. They may be created at runtime and may contain state, but they do not implicitly capture local variables as closures do. Two proposals to introduce C++ language support for closures (both proposals call them lambda functions) are being considered by the C++ Standards Committee. The main difference between these proposals is that one stores a copy of all the local variables in a closure by default, and another stores references to original variables. Both provide functionality to override the default behavior. If some form of these proposals is accepted, one would be able to write

void foo(string myname) {
    typedef vector<string> names;
    int y;
    names n;
    // ...
    names::iterator i = std::find_if(n.begin(), n.end(), [&](const string& s) { 
            return s != myname && s.size() > y; 
        });
    // 'i' is now either 'n.end()' or points to the first string in 'n'
    // which is not equal to 'myname' and whose length is greater than 'y'
}

At least two C++ compilers, Visual C++ 2010 and GCC 4.5, already support this notation.

C#

C# anonymous methods and lambda expressions support closure to local variables:

var data = new[] {1, 2, 3, 4};
var multiplier = 2;
var result = data.Select(x => x * multiplier);

D

Closures are implemented by delegates in D.

auto test1() {
    int a = 7;
    return delegate() { return a + 3; }; // anonymous delegate construction
}

auto test2() {
    int a = 20;
    int foo() { return a + 5; } // inner function
    return &foo;  // other way to constructing delegate
}

void bar() {
    auto dg = test1();
    dg();    // =10   // ok, test1.a is in a closure and still exists

    dg = test2();
    dg();    // =25   // ok, test2.a is in a closure and still exists
}

D version 1, has limited closure support. For example, the above code will not work correctly, because the variable a is on the stack, and after returning from test(), it is no longer valid to use it (most probably calling foo via dg(), will return a 'random' integer). This can be solved by explicitly allocating the variable a on heap, or using structs or class to store all needed closed variables and construct a delegate from a method implementing the same code. Closures can be passed to other functions, as long as they are only used while the referenced values are still valid (for example calling another function with a closure as a callback parameter), and are useful for writing generic data processing code, so this limitation often not an issue in practice.

This limitation was fixed in D version 2 - the variable 'a' will be automatically allocated on the heap because it is used in the inner function, and a delegate of that function is allowed to escapes the current scope (via assignment to dg or return). Any other local variables (or arguments) that are not referenced by delegates or that are only referenced by delegates that don't escape the current scope, remain on the stack, which is simpler and faster than heap allocation. The same is true for inner's class methods that references a function's variables.

Delphi

Delphi includes support for closures implemented as anonymous methods since the 2009 release. They can be declared inline within another method, and capture references to local state. For example:

type
  TCounter = reference to function: integer;

function MakeCounter: TCounter;
var
  i: integer;
begin
  i := 0;
  result := function: integer
            begin
              inc(i);
              result := i;
            end;
end;

Calling MakeCounter will return a new anonymous method. Invoking it will return 1 the first time, 2 the second, and so on.

Perl

Anonymous functions in Perl become closures when they reference lexical variables defined in enclosing scopes.

sub number_of_bagels {
    my $bagels = shift;
    return sub { my $sub_arg = shift; return $sub_arg + $bagels; };
}

my $func = number_of_bagels(5);
print "I now have " . $func->(2) . " bagels!\n";  #I now have 7 bagels!
print "I now have " . $func->(10) . " bagels!\n"; #I now have 15 bagels!

PHP

Since PHP 5.3+, closures can be implemented as anonymous functions:

$greet = function($name) {
    printf("Hello %s\r\n", $name);
};

$greet('World');
$greet('PHP');

Or they can be implemented truer to the closure definition by using the use keyword which makes the calling environment available:

$foo = function() use (&$bar) {
    echo($bar);
};

$bar = "Hello, World!";
$foo();

Python

Python has two types of closures: lambda expressions and named closures (functions declared inside of functions).

Lambda expressions look like lambda x: x+3. A two-argument equivalent is lambda x,y: x*y and a no-argument equivalent is lambda: foo(). These are limited by the fact that the result of the lambda expression must be one expression rather than a statement. Thus, it is not possible to put a "for", "while", or "if" block in a lambda expression. (However, "for" loops can be simulated using a list comprehension, e.g. lambda x: [foo(y) for y in x], and "if" blocks can be simulated using conditional expressions, e.g. lambda x: foo(x) if x == 2 else bar(x).) The two can be combined using "if" modifiers on list comprehensions, e.g. lambda x: [foo(y) for y in x if y > 5].

The other type of closure is named closures, as shown in the following Python 3.1 example:

#tested on python 3.1

def outer():
    y = 0
    def inner():
        nonlocal y
        y += 1
        return y
    return inner

f = outer()
print(f(), f(), f()) #prints 1 2 3

The nested function declaration def inner() actually declares a closure and assigns it the name of "inner". These nested functions have the full power of closures, i.e. the lexical environment at the time the "def" expression is evaluated is captured and can be referenced or assigned to even after the enclosing function goes out of scope, as shown.

In Python, a def statement is like a special assignment statement for functions—just like the assignment x = y binds x to the value y, def foo(): body binds the name foo to the code body. The best way to conceive of how a nested function declaration works is to consider it to be equivalent to the creation of a closure followed by its assignment to a variable. For example, the following is what the above function might look like in a hypothetical extended Python language where lambda expressions are allowed to contain a full statement block:

# NOTE: The following is NOT valid Python code!
# It is pseudo-code using a Python-like syntax.

def outer():
    y = 0
    inner = lambda: (
        nonlocal y
        y += 1
        return y
    )
    return inner

f = outer()
print(f(), f(), f()) #prints 1 2 3

Java

Java allows defining "anonymous classes" inside a method; an anonymous class may refer to names in lexically enclosing classes, or read-only variables (marked as final) in the lexically enclosing method.

class CalculationWindow extends JFrame {
	private volatile int result;
	...
	public void calculateInSeparateThread(final URI uri) {
		// The expression "new Runnable() { ... }" is an anonymous class.
		new Thread(
			new Runnable() {
				void run() {
					// It can read final local variables:
					calculate(uri);
					// It can access private fields of the enclosing class:
					result = result + 10;
				}
			}
		).start();
	}
}

Some features of full closures can be emulated by using a final reference to a mutable container, for example, a single-element array. The inner class will not be able to change the value of the container reference itself, but it will be able to change the contents of the container.

According to a Java 7 proposal, closures will allow the above code to be executed as:

class CalculationWindow extends JFrame {
	private volatile int result;
	...
	public void calculateInSeparateThread(final URI uri) {
                // the code #(){ /* code */ } is a closure
                new Thread(#(){
                    calculate( uri );
                    result = result + 10;
                }).start();
	}
}

Java also supports another form of classes, which are called inner (or nested) classes. These are defined in the body of an enclosing class and have full access to each and every instance variable of the enclosing class, thus resembling standard function closures. Due to their binding to these instance variables, a nested inner class may only be instantiated with an explicit binding to an instance of the enclosing class using a special syntax.

public class EnclosingClass {
	/* Define the inner class */
	public class InnerClass {
		public int incrementAndReturnCounter() {
			return counter++;
		}
	}

	private int counter;

	{
		counter = 0;
	}

	public int getCounter() {
		return counter;
	}

	public static void main(String[] args) {
		EnclosingClass enclosingClassInstance = new EnclosingClass();
		/* Instantiate the inner class, with binding to the instance */
		EnclosingClass.InnerClass innerClassInstance =
			enclosingClassInstance.new InnerClass();

		for(int i = enclosingClassInstance.getCounter(); (i =
		innerClassInstance.incrementAndReturnCounter()) < 10;) {
			System.out.println(i);
		}
	}
}

Upon execution, this will print the integers from 0 to 9. Beware to not confuse this type of class with the so called static inner class, which is declared in the same way with an accompanied usage of the "static" modifier; those have not the desired effect but are instead just classes with no special binding defined in an enclosing class.

There have been a number of proposals for adding more fully featured closures to Java.

Lua

The Lua programming language provides fully featured closures . In combination with functions, which are first-class values in Lua, one may write generators (functions that create functions) as shown in the following example:

    function newCounter ()
      local i = 0
      return function ()   -- anonymous function
               i = i + 1
               return i
             end
    end
    
    c1 = newCounter()
    print(c1())  --> 1
    print(c1())  --> 2

C, C++, Objective-C 2.0 (blocks)

Apple introduced Blocks, a form of closure, into C, C++, Objective-C 2.0 and in Mac OS X 10.6 "Snow Leopard" and iOS 4.0. Closure variables are marked with __block and pointers to block and block literals are marked with ^.

typedef int (^IntBlock)();

IntBlock downCounter(int start) {
	 __block int i = start;
	 return [[ ^int() {
		 return i--;
	 } copy] autorelease];
}

IntBlock f = downCounter(5);
NSLog(@"%d", f());
NSLog(@"%d", f());
NSLog(@"%d", f());

JavaScript

JavaScript features closures also.

function countDownFrom(count) {
  return function() {
    return count--;
  };
}

var countDownFromOne = countDownFrom(1);
var countDownFromTwo = countDownFrom(2);

// example uses console
// which may not be available depending on browser/environment;
// substitute accordingly (e.g.: with alert).

console.log(countDownFromOne()); //>1
console.log(countDownFromTwo()); //>2
console.log(countDownFromOne()); //>0
console.log(countDownFromTwo()); //>1
console.log(countDownFromTwo()); //>0

Go

In Go a function literal represents an anonymous function, so a function literal is a closure.

// A strategy chooses an action for any given score.
type strategy func(score) action

// stayAtK returns a strategy that rolls until thisTurn is at least k, then stays.
func stayAtK(k int) strategy {
  return func(s score) action {
    if s.thisTurn >= k {
      return stay
    }
    return roll
  }
}

See also

• Anonymous function
• Blocks (C language extension)
• Command pattern
• Continuation
• Currying
• Funarg problem
• Lambda calculus
• Lazy evaluation
• Partial application
• Spaghetti stack
• Syntactic closure
• Value-level programming

External links

• The Original "Lambda Papers": A classic series of papers by Guy Steele and Gerald Sussman discussing, among other things, the versatility of closures in the context of Scheme (where they appear as lambda expressions).
• Neal Gafter (28 January 2007). "A Definition of Closures".
• Gilad Bracha, Neal Gafter, James Gosling, Peter von der Ahé. "Closures for the Java Programming Language (v0.5)".{{cite web}}: CS1 maint: multiple names: authors list (link)
• Closures: An article about closures in dynamically-typed imperative languages, by Martin Fowler.
• Collection closure methods: An example of a technical domain where using closures is convenient, by Martin Fowler.

Javascript

• What are closures: A post on closures in Javascript.
• Javascript Closures for Dummies: An article teaching closures in Javascript by examples.

Java and .NET

• The beauty of closures: An article about using closures in Java and .NET

Delphi

• Nick Hodges, "Delphi 2009 Reviewers Guide", October 2008, CodeGear Developer Network, CodeGear.
• Craig Stuntz, "Understanding Anonymous Methods", October 2008
• Dr. Bob, "Delphi 2009 Anonymous Methods"

Ruby

• Robert Sosinski, "Understanding Ruby Blocks, Procs and Lambdas", December 2008