In mathematical logic and computer science, lambda calculus, also λ-calculus, is a formal system designed to investigate function definition, function application and recursion. Mathematical logic is a subfield of Logic and Mathematics with close connections to Computer science and Philosophical logic. Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their In formal logic, a formal system (also called a logical system, a logistic system, or simply a logic Formal systems in mathematics consist The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function Recursion, in Mathematics and Computer science, is a method of defining functions in which the function being defined is applied within its own definition It was introduced by Alonzo Church and Stephen Cole Kleene in the 1930s as part of an investigation into the foundations of mathematics, but has emerged as a useful tool in the investigation of problems in computability or recursion theory, and forms the basis of a paradigm of computer programming called functional programming. Alonzo Church ( June 14, 1903 – August 11, 1995) was an American Mathematician and logician Stephen Cole Kleene ( January 5, 1909, Hartford Connecticut, USA &ndash January 25, 1994, Madison Wisconsin The 1930s were described as an abrupt shift to more radical and conservative lifestyles as countries were struggling to find a solution to the Great Depression. Foundations of mathematics is a term sometimes used for certain fields of Mathematics, such as Mathematical logic, Axiomatic set theory, Proof theory Recursion theory, also called computability theory, is a branch of Mathematical logic that originated in the 1930s with the study of Computable functions In Computer science, functional programming is a Programming paradigm that treats Computation as the evaluation of mathematical functions and ^[1]

The lambda calculus can be thought of as an idealized, minimalistic programming language. It is capable of expressing any algorithm, and it is this fact that makes the model of functional programming an important one. In Mathematics, Computing, Linguistics and related subjects an algorithm is a sequence of finite instructions often used for Calculation In Computer science, functional programming is a Programming paradigm that treats Computation as the evaluation of mathematical functions and Functional programs are stateless and deal exclusively with functions that accept and return data (including other functions), but they produce no side effects in 'state' and thus make no alterations to incoming data. Modern functional languages, building on the lambda calculus, include Lisp, Scheme, ML, Haskell, and Erlang. Lisp (or LISP) is a family of Computer Programming languages with a long history and a distinctive fully parenthesized syntax Scheme is a Multi-paradigm programming language. It is one of the two main dialects of Lisp and supports a number of programming paradigms but is ML is a general-purpose Functional programming language developed by Robin Milner and others in the late 1970s at the University of Edinburgh, whose syntax Haskell is a standardized Purely functional Programming language with non-strict semantics, named after the Logician Haskell Curry Erlang is a general-purpose concurrent Programming language and Runtime system

The lambda calculus continues to play an important role in mathematical foundations, through the Curry-Howard correspondence. Foundations of mathematics is a term sometimes used for certain fields of Mathematics, such as Mathematical logic, Axiomatic set theory, Proof theory The Curry-Howard correspondence is the direct relationship between computer programs and mathematical proofs However, as a naïve foundation for mathematics, the untyped lambda calculus is unable to avoid set-theoretic paradoxes (see the Kleene-Rosser paradox). In Mathematics, the Kleene-Rosser paradox is a paradox that shows Church 's original Lambda calculus is inconsistent

This article deals with the "untyped lambda calculus" as originally conceived by Church. Most modern applications concern typed lambda calculi. A

Informal description

In lambda calculus, every expression is a unary function meaning a function with only one input (called an argument). A unary function is a function that takes one argument. In Computer science, a Unary operator is a subset of unary function When an expression is applied to another expression ('called' with the other expression as its argument), it returns a single value, called its result.

Since every expression is a unary function, and every argument and result are functions too, lambda calculus is quite interesting and unique within both computation and mathematics. Computation is a general term for any type of Information processing. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and

A function is anonymously defined by a lambda expression which expresses the function's action on its argument. For instance, the "add-two" function f such that  f(x) = x + 2  would be expressed in lambda calculus as  λ x. x + 2  (or equivalently as  λ y. y + 2;  the name of the formal parameter is immaterial) and the application of the function f(3) would be written as  (λ x. x + 2) 3.   Note that part of what makes this description "informal" is that the expression x + 2 (or even the number 2) is not part of lambda calculus; an explanation of how numbers and arithmetic can be represented in lambda calculus is below. Function application is left associative:  f x y = (f x) y. In Mathematics, associativity is a property that a Binary operation can have   Consider the function which takes a function as an argument and applies it to the number 3 as follows: λ f. f 3.   This latter function could be applied to our earlier "add-two" function as follows:  (λ f. f 3) (λ x. x + 2).   The three expressions:

• (λ f. f 3) (λ x. x + 2)
• (λ x. x + 2) 3
• 3 + 2

are equivalent.

A function of two variables is expressed in lambda calculus as a function of one argument which returns a function of one argument (see currying). In Computer science, currying, invented by Moses Schönfinkel and Gottlob Frege, and independently by Haskell Curry, is the technique of For instance, the function  f(x, y) = x - y  would be written as  λ x. λ y. x - y. A common convention is to abbreviate curried functions as, in this example,  λ x y. x - y. While it is not part of the formal definition of the language,

λ x₁ x₂ … x_n. expression

is used as an abbreviation for

λ x₁. λ x₂. … λ x_n. expression

Not every lambda expression can be reduced to a definite value like the ones above; consider for instance

(λ x. x x) (λ x. x x)

or

(λ x. x x x) (λ x. x x x)

and try to visualize what happens when you start to apply the first function to its argument.  (λ x. x x)  is also known as the ω combinator;  ((λ x. Combinatory logic is a notation introduced by Moses Schönfinkel and Haskell Curry to eliminate the need for Variables in Mathematical logic x x) (λ x. x x))  is known as Ω,  ((λ x. x x x) (λ x. x x x))  as Ω₂, etc.

Lambda calculus expressions may contain free variables, i. e. variables not bound by any λ. For example, the variable  y  is free in the expression  (λ x. y) , representing a function which always produces the result y. Occasionally, this necessitates the renaming of formal arguments. For example, in the formula below, the letter y is used first as a formal parameter, then as a free variable:

(λ x y. y x) (λ x. y).

To reduce the expression, we rename the first identifier z so that the reduction does not mix up the names:

(λ x z. z x) (λ x. y)

the reduction is then

λ z. z (λ x. y).

If one only formalizes the notion of function application and replaces the use of lambda expressions by the use of combinators, one obtains combinatory logic. Combinatory logic is a notation introduced by Moses Schönfinkel and Haskell Curry to eliminate the need for Variables in Mathematical logic

Formal definition

Definition

Lambda expressions are composed of

variables v₁, v₂, . . . v_n

the abstraction symbols λ and .

parentheses ( )

The set of lambda expressions, Λ, can be defined recursively:

1. If x is a variable, then x Λ
2. If x is a variable and M Λ, then ( λ x . Recursion, in Mathematics and Computer science, is a method of defining functions in which the function being defined is applied within its own definition M ) Λ
3. If M, N Λ, then ( M N ) Λ

Instances of 2 are known as abstractions and instances of 3, applications. ^[2]

Notation

To keep the notation of lambda expressions uncluttered, the following conventions are usually applied.

Outermost parentheses are dropped: M N instead of (M N).

Applications are assumed to be left associative: M N P means (M N) P.

The body of an abstraction extends as far right as possible: λ x. M N means (λ x. M N) and not (λ x. M) N

A sequence of abstractions are contracted: λ x λ y λ z. N is abbreviated as λ x y z . N^[3]

Free and bound variables

The abstraction operator, λ, is said to bind its variable wherever it occurs in the body of the abstraction. Variables that fall within the scope of a lambda are said to be bound. All other variables are called free. For example in the following expression y is a bound variable and x is free:

λ y . xxy

Also note that a variable binds to its "nearest" lambda. In the following expression the single occurrence of x is bound by the second lambda:

λ x . y (λ x . z x)

The free variables of a lambda expression, M, is specified with FV(M) and are defined inductively as follows:

1. FV( x ) = x, where x is a variable
2. FV ( λ x . M ) = FV ( M ) - {x}
3. FV ( M N ) = FV ( M ) FV ( N )^[4]

An expression which contains no free variables is said to be closed. Closed lambda expressions are also known as combinators and are equivalent to terms in combinatory logic. Combinatory logic is a notation introduced by Moses Schönfinkel and Haskell Curry to eliminate the need for Variables in Mathematical logic

Reduction

Substitution

Substitution, written E[V := E′], corresponds to the replacement of a variable V by expression E′ every place it is free within E. The precise definition must be careful in order to avoid accidental variable capture (See also Hygienic macro). Hygienic macros are macros whose expansion is guaranteed not to cause collisions with existing symbol definitions For example, it is not correct for (λ x. y)[y := x] to result in (λ x. x), because the substituted x was supposed to be free but ended up being bound. The correct substitution in this case is (λ z. x), up-to α-equivalence.

The precise rules are defined inductively as follows:

x[x := N]        ≡ x

y[x := N]        ≡ y, if x ≠ y

(M₁ M₂)[x := N]  ≡ (M₁[x := N]) (M₂[x := N])

(λ y. M)[x := N] ≡ λ y. (M[x := N]), if x ≠ y and y∉fv(N)

α-conversion

Alpha conversion allows bound variable names to be changed. For example, an alpha conversion of  λx. x  would be  λy. y . Frequently in uses of lambda calculus, terms that differ only by alpha conversion are considered to be equivalent.

The precise rules for alpha conversion are not completely trivial. First, when alpha-converting an abstraction, the only variable occurrences that are renamed are those that are bound to the same abstraction. For example, an alpha conversion of  λx. λx. x  could result in  λy. λx. x , but it could not result in  λy. λx. y . The latter has a different meaning from the original.

Second, alpha conversion is not possible if it would result in a variable getting captured by a different abstraction. For example, if we replace x with y in λx. λy. x, we get λy. λy. y, which is not at all the same.

β-reduction

Beta reduction expresses the idea of function application. The beta reduction of  ((λ V. E) E′)  is simply  E[V := E′] .

η-conversion

Eta conversion expresses the idea of extensionality, which in this context is that two functions are the same if and only if they give the same result for all arguments. In Logic, extensionality refers to principles that judge objects to be equal if they have the same external properties ↔ Eta-conversion converts between  λ x. f x  and  f  whenever x does not appear free in f.

This conversion is not always appropriate when lambda expressions are interpreted as programs. Evaluation of  λ x. f x  can terminate even when evaluation of f does not.

Arithmetic in lambda calculus

There are several possible ways to define the natural numbers in lambda calculus, but by far the most common are the Church numerals, which can be defined as follows:

0 := λ f x. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an In Mathematics, Church encoding is a means of embedding data and operators into the Lambda calculus, the most familiar form being the Church numerals, a x

1 := λ f x. f x

2 := λ f x. f (f x)

3 := λ f x. f (f (f x))

and so on. A Church numeral is a higher-order function—it takes a single-argument function f, and returns another single-argument function. In Mathematics and Computer science, higher-order functions or '''functionals''' are functions which do at least one of the following The Church numeral n is a function that takes a function f as argument and returns the n-th composition of f, i. e. the function f composed with itself n times. This is denoted f⁽ⁿ⁾ and is in fact the n-th power of f (considered as an operator); f⁽⁰⁾ is defined to be the identity function. Such repeated compositions (of a single function f) obey the laws of exponents, which is why these numerals can be used for arithmetic. Note that 1 returns f itself, i. e. it is essentially the identity function, and 0 returns the identity function. (Also note that in Church's original lambda calculus, the formal parameter of a lambda expression was required to occur at least once in the function body, which made the above definition of 0 impossible. )

We can define a successor function, which takes a number n and returns n + 1 by adding an additional application of f:

SUCC := λ n f x. f (n f x)

Because the m-th composition of f composed with the n-th composition of f gives the m+n-th composition of f, addition can be defined as follows:

PLUS := λ m n f x. n f (m f x)

PLUS can be thought of as a function taking two natural numbers as arguments and returning a natural number; it can be verified that

PLUS 2 3    and    5

are equivalent lambda expressions. Since adding m to a number, n can be accomplished by adding 1 m times, an equivalent definition is:

PLUS := λ n m. m SUCC n^[5]

Similarly, multiplication can be defined as

MULT := λ m n f . m (n f)^[6]

Alternatively

MULT := λ m n. m (PLUS n) 0,

since multiplying m and n is the same as repeating the "add n" function m times and then applying it to zero. The predecessor function defined by  PRED n = n - 1  for a positive integer n and  PRED 0 = 0  is considerably more difficult. The formula

PRED := λ n f x. n (λ g h. h (g f)) (λ u. x) (λ u. u) 

can be validated by showing inductively that if T denotes (λ g h. h (g f)), then T⁽ⁿ⁾(λ u. x) = (λ h. h(f^(n-1)(x)) ) for n > 0. Two other definitions of PRED are given below, one using conditionals and the other using pairs. With the predecessor function, subtraction is straightforward. Defining

SUB := λ m n. n PRED m,

SUB m n yields m - n when m > n and 0 otherwise.

Logic and predicates

By convention, the following two definitions (known as Church booleans) are used for the boolean values TRUE and FALSE:

TRUE := λ x y. x

FALSE := λ x y. y

(Note that FALSE is equivalent to the Church numeral zero defined above)

Then, with these two λ-terms, we can define some logic operators (these are just possible formulations; other expressions are equally correct):

AND := λ p q. p q p

OR := λ p q. p p q

NOT := λ p a b. p b a

IFTHENELSE := λ p. p

We are now able to compute some logic functions, for example:

AND TRUE FALSE

≡ (λ p q. p q p) TRUE FALSE →_β TRUE FALSE TRUE

≡ (λ x y. x) FALSE TRUE →_β FALSE

and we see that AND TRUE FALSE is equivalent to FALSE.

A predicate is a function which returns a boolean value. The most fundamental predicate is ISZERO which returns TRUE if its argument is the Church numeral 0, and FALSE if its argument is any other Church numeral:

ISZERO := λ n. n (λ x. FALSE) TRUE

The following predicate tests whether the first argument is less-than-or-equal-to the second:

LEQ := λ m n. ISZERO (SUB m n),

and since m = n iff LEQ m n and LEQ n m, it is straightforward to build a predicate for numerical equality.

The availability of predicates and the above definition of TRUE and FALSE make it convenient to write "if-then-else" expressions in lambda calculus. For example, the predecessor function can be defined as' '

PRED := λ n. n (λ g k. ISZERO (g 1) k (PLUS (g k) 1) ) (λ v. 0) 0

which can be verified by showing inductively that n (λ g k. ISZERO (g 1) k (PLUS (g k) 1) ) (λ v. 0) is the "add n - 1" function for n > 0.

Pairs

A pair (2-tuple) can be defined in terms of TRUE and FALSE, by using the Church encoding for pairs. In Mathematics, Church encoding is a means of embedding data and operators into the Lambda calculus, the most familiar form being the Church numerals, a For example, PAIR encapsulates the pair (x,y), FIRST returns the first element of the pair, and SECOND returns the second.

PAIR := λ x y f. f x y

FIRST := λ p. p TRUE

SECOND := λ p. p FALSE

NIL := λ x. TRUE

NULL := λp. p (λx y. FALSE)

A linked list can be defined as either NIL for the empty list, or the PAIR of an element and a smaller list. The predicate NULL tests for the value NIL.

As an example of the use of pairs, the shift-and-increment function that maps (m, n) to (n, n+1) can be defined as

Φ := λ x. PAIR (SECOND x) (SUCC (SECOND x))

which allows us to give perhaps the most transparent version of the predecessor function:

PRED := λ n. FIRST (n Φ (PAIR 0 0))

Recursion

Recursion is the definition of a function using the function itself; on the face of it, lambda calculus does not allow this. Recursion, in Mathematics and Computer science, is a method of defining functions in which the function being defined is applied within its own definition However, this impression is misleading. Consider for instance the factorial function f(n) recursively defined by

f(n) = 1, if n = 0; and n·f(n-1), if n>0. Definition The factorial function is formally defined by n!=\prod_{k=1}^n k

In lambda calculus, one cannot define a function which includes itself. To get around this, one may start by defining a function, here called g, which takes a function f as an argument and returns another function that takes n as an argument:

g := λ f n. (1, if n = 0; and n·f(n-1), if n>0).

The function that g returns is either the constant 1, or n times the application of the function f to n-1. Using the ISZERO predicate, and boolean and algebraic definitions described above, the function g can be defined in lambda calculus.

However, g by itself is still not recursive; in order to use g to create the recursive factorial function, the function passed to g as f must have specific properties. Namely, the function passed as f must expand to the function g called with one argument -- and that argument must be the function that was passed as f again!

In other words, f must expand to g(f). This call to g will then expand to the above factorial function and calculate down to another level of recursion. In that expansion the function f will appear again, and will again expand to g(f) and continue the recursion. This kind of function, where f = g(f), is called a fixed-point of g, and it turns out that it can be implemented in the lambda calculus using what is known as the paradoxical operator or fixed-point operator and is represented as Y -- the Y combinator:

Y = λ g. (λ x. g (x x)) (λ x. g (x x))

In the lambda calculus, Y g is a fixed-point of g, as it expands to g (Y g). Now, to complete our recursive call to the factorial function, we would simply call  g (Y g) n,  where n is the number we are calculating the factorial of.

Given n = 5, for example, this expands to:

(λ n. (1, if n = 0; and n·((Y g)(n-1)), if n>0)) 5

1, if 5 = 0; and 5·(g(Y g)(5-1)), if 5>0

5·(g(Y g) 4)

5·(λ n. (1, if n = 0; and n·((Y g)(n-1)), if n>0) 4)

5·(1, if 4 = 0; and 4·(g(Y g)(4-1)), if 4>0)

5·(4·(g(Y g) 3))

5·(4·(λ n. (1, if n = 0; and n·((Y g)(n-1)), if n>0) 3))

5·(4·(1, if 3 = 0; and 3·(g(Y g)(3-1)), if 3>0))

5·(4·(3·(g(Y g) 2)))

. . .

And so on, evaluating the structure of the algorithm recursively. Every recursively defined function can be seen as a fixed point of some other suitable function, and therefore, using Y, every recursively defined function can be expressed as a lambda expression. In particular, we can now cleanly define the subtraction, multiplication and comparison predicate of natural numbers recursively.

Computable functions and lambda calculus

A function F: N → N of natural numbers is a computable function if and only if there exists a lambda expression f such that for every pair of x, y in N,  F(x) = y  if and only if  f x == y,  where x and y are the Church numerals corresponding to x and y, respectively. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an Computable functions are the basic objects of study in computability theory. ↔ This is one of the many ways to define computability; see the Church-Turing thesis for a discussion of other approaches and their equivalence.

Undecidability of equivalence

There is no algorithm which takes as input two lambda expressions and outputs TRUE or FALSE depending on whether or not the two expressions are equivalent. This was historically the first problem for which undecidability could be proven. As is common for a proof of undecidability, the proof shows that no computable function can decide the equivalence. Computable functions are the basic objects of study in computability theory. Church's thesis is then invoked to show that no algorithm can do so.

Church's proof first reduces the problem to determining whether a given lambda expression has a normal form. A normal form is an equivalent expression which cannot be reduced any further. Then he assumes that this predicate is computable, and can hence be expressed in lambda calculus. Building on earlier work by Kleene and constructing a Gödel numbering for lambda expressions, he constructs a lambda expression e which closely follows the proof of Gödel's first incompleteness theorem. In Mathematical logic, a Gödel numbering is a function that assigns to each symbol and Well-formed formula of some Formal language a unique In Mathematical logic, Gödel's incompleteness theorems, proved by Kurt Gödel in 1931 are two Theorems stating inherent limitations of all but the most If e is applied to its own Gödel number, a contradiction results. In Mathematical logic, a Gödel numbering is a function that assigns to each symbol and Well-formed formula of some Formal language a unique

Lambda calculus and programming languages

As pointed out by Peter Landin's 1965 classic A Correspondence between ALGOL 60 and Church's Lambda-notation, most programming languages are rooted in the lambda calculus, which provides the basic mechanisms for procedural abstraction and procedure (subprogram) application. Peter Landin is a British Computer scientist. He was one of the first to realize that the Lambda calculus could be used to model a programming language A programming language is an Artificial language that can be used to write programs which control the behavior of a machine particularly a Computer.

Implementing the lambda calculus on a computer involves treating "functions" as first-class objects, which raises implementation issues for stack-based programming languages. In Computing, a first-class object (also value, entity, and citizen) in the context of a particular Programming language, is an entity This is known as the Funarg problem. Funarg is an abbreviation for "functional argument" in Computer science, the funarg problem relates to a difficulty in implementing functions

The most prominent counterparts to lambda calculus in programming are functional programming languages, which essentially implement the calculus augmented with some constants and datatypes. In Computer science, functional programming is a Programming paradigm that treats Computation as the evaluation of mathematical functions and A data type in Programming languages is an attribute of a datum which tells the computer (and the programmer something about the kind of datum it is Lisp uses a variant of lambda notation for defining functions, but only its purely functional subset ("Pure Lisp") is really equivalent to lambda calculus. Lisp (or LISP) is a family of Computer Programming languages with a long history and a distinctive fully parenthesized syntax Lispkit Lisp is a lexically scoped, Purely functional subset of Lisp (" Pure Lisp " developed as a testbed for Functional programming

Functional languages are not the only ones to support functions as first-class objects. In Computing, a first-class object (also value, entity, and citizen) in the context of a particular Programming language, is an entity Numerous imperative languages, e. In Computer science, imperative programming is a Programming paradigm that describes computation in terms of statements that change a program state g. Pascal, have long supported passing subprograms as arguments to other subprograms. Pascal is an influential imperative and procedural Programming language, designed in 1968/9 and published in 1970 by Niklaus Wirth as a small In C and the C-like subset of C++ the equivalent result is obtained by passing pointers to the code of functions (subprograms). tags please moot on the talk page first! --> In Computing, C is a general-purpose cross-platform block structured C++ (" C Plus Plus " ˌsiːˌplʌsˈplʌs is a general-purpose Programming language. Such mechanisms are limited to subprograms written explicitly in the code, and do not directly support higher-level functions. Some imperative object-oriented languages have notations that represent functions of any order; such mechanisms are available in C++, Smalltalk and more recently in Eiffel ("agents") and C# ("delegates"). An object-oriented Programming language (also called an OO language) is one that allows or encourages to some degree Object-oriented programming C++ (" C Plus Plus " ˌsiːˌplʌsˈplʌs is a general-purpose Programming language. Smalltalk is an object-oriented, dynamically typed, reflective programming language. Eiffel is an ISO -standardized Object-oriented Programming language designed to enable programmers to efficiently develop extensible reusable reliable C# (pronounced C Sharp is a Multi-paradigm As an example, the Eiffel "inline agent" expression

agent (x: REAL): REAL do Result := x * x end

denotes an object corresponding to the lambda expression λ x . x*x (with call by value). It can be treated like any other expression, e. g. assigned to a variable or passed around to routines. If the value of square is the above agent expression, then the result of applying square to a value a (β-reduction) is expressed as square. item ([a]), where the argument is passed as a tuple. In Mathematics, a tuple is a Sequence (also known as an "ordered list" of values called the components of the tuple

A Python example of this uses the lambda form of functions:

func = lambda x: x ** 2

This creates a new anonymous function and names it func which can be passed to other functions, stored in variables, etc. Python is a general-purpose High-level programming language. Its design philosophy emphasizes programmer productivity and code readability Python can also treat any other function created with the standard def statement as first-class objects. In Computing, a first-class object (also value, entity, and citizen) in the context of a particular Programming language, is an entity

The same holds for Smalltalk expression

[ :x | x * x ]

This is first-class object (block closure), which can be stored in variables, passed as arguments, etc.

A similar C++ example (using the Boost. The Boost C++ Libraries are a collection of Peer-reviewed Open source libraries that extend the functionality of C++. Lambda library):

std::for_each(c. begin(), c. end(), std::cout << _1 * _1 << std::endl);

Here the standard library function for_each iterates over all members of container 'c', and prints the square of each element. The _1 notation is Boost. Lambda's convention (originally derived from Boost. Bind) for representing the first placeholder element (the first argument), represented as x elsewhere.

A simple C# delegate taking a variable and returning the square. A delegate is a form of type-safe Function pointer used by the. This function variable can then be passed to other methods (or function delegates)

//Declare a delegate signaturedelegate double MathDelegate(double i);//Create a delegate instanceMathDelegate f = delegate(double i) { return Math. Pow(i, 2); };

/* Passing 'f' function variable to another method, executing,   and returning the result of the function */double Execute(MathDelegate f){    return f(100);}

In the . NET Framework 3. 5, C# has lambda expressions in a form similar to python or lisp. The expression resolves to a delegate like in the previous example but the above can be simplified to below.

//Create an delegate instanceMathDelegate f = i => i * i;Execute(f);// or more simply putExecute(i => i * i);

Reduction strategies

For more details on this topic, see Evaluation strategy. In Computer science, an evaluation strategy is a set of (usually deterministic rules for determining the evaluation of expressions in a Programming language

Whether a term is normalising or not, and how much work needs to be done in normalising it if it is, depends to a large extent on the reduction strategy used. The distinction between reduction strategies relates to the distinction in functional programming languages between eager evaluation and lazy evaluation. Eager evaluation or strict evaluation is the Evaluation strategy in most traditional Programming languages In eager evaluation an expression In Computer programming, lazy evaluation (or delayed evaluation) is the technique of delaying a computation until such time as the result of the computation is

The following uses the term 'redex', short for 'reducible expression'. For example, (λ x. M) N is a beta-redex; λ x. M x is an eta-redex if x is not free in M. The expression to which a redex reduces is called its reduct; using the previous example, the reducts of these expressions are respectively M[x:=N] and M.

Full beta reductions

Any redex can be reduced at any time. This means essentially the lack of any particular reduction strategy — with regard to reducibility, "all bets are off".

Applicative order

The rightmost, innermost redex is always reduced first. Intuitively this means a function's arguments are always reduced before the function itself. Applicative order always attempts to apply functions to normal forms, even when this is not possible.

Most programming languages (including Lisp, ML and imperative languages like C and Java) are described as "strict", meaning that functions applied to non-normalising arguments are non-normalising. A lisp ( OE wlisp, stammering is a Speech impediment, historically also known as sigmatism. tags please moot on the talk page first! --> In Computing, C is a general-purpose cross-platform block structured This is done essentially using applicative order, call by value reduction (see below), but usually called "eager evaluation".

Normal order

The leftmost, outermost redex is always reduced first. That is, whenever possible the arguments are substituted into the body of an abstraction before the arguments are reduced.

Call by name

As normal order, but no reductions are performed inside abstractions. For example λ x. (λ x. x)x is in normal form according to this strategy, although it contains the redex (λ x. x)x.

Call by value

Only the outermost redexes are reduced: a redex is reduced only when its right hand side has reduced to a value (variable or lambda abstraction).

Call by need

As normal order, but function applications that would duplicate terms instead name the argument, which is then reduced only "when it is needed". Called in practical contexts "lazy evaluation". In implementations this "name" takes the form of a pointer, with the redex represented by a thunk. The word thunk has at least three related meanings in computer science

Applicative order is not a normalising strategy. The usual counterexample is as follows: define Ω = ωω where ω = λ x. xx. This entire expression contains only one redex, namely the whole expression; its reduct is again Ω. Since this is the only available reduction, Ω has no normal form (under any evaluation strategy). Using applicative order, the expression KIΩ = (λ x y . x)(λ x. x)Ω is reduced by first reducing Ω to normal form (since it is the rightmost redex), but since Ω has no normal form, applicative order fails to find a normal form for KIΩ.

In contrast, normal order is so called because it always finds a normalising reduction if one exists. In the above example, KIΩ reduces under normal order to I, a normal form. A drawback is that redexes in the arguments may be copied, resulting in duplicated computation (for example, (λ x. xx)((λ x. x)y) reduces to ((λx. x)y)((λx. x)y) using this strategy; now there are two redexes, so full evaluation needs two more steps, but if the argument had been reduced first, there would now be none).

The positive tradeoff of using applicative order is that it does not cause unnecessary computation if all arguments are used, because it never substitutes arguments containing redexes and hence never needs to copy them (which would duplicate work). In the above example, in applicative order (λ x. xx)((λ x. x)y) reduces first to (λ x. xx)y and then to the normal order yy, taking two steps instead of three.

Most purely functional programming languages (notably Miranda and its descendents, including Haskell), and the proof languages of theorem provers, use lazy evaluation, which is essentially the same as call by need. Automated theorem proving ( ATP) or automated deduction, currently the most well-developed subfield of Automated reasoning (AR is the In Computer programming, lazy evaluation (or delayed evaluation) is the technique of delaying a computation until such time as the result of the computation is This is like normal order reduction, but call by need manages to avoid the duplication of work inherent in normal order reduction using sharing. In the example given above, (λ x. xx)((λ x. x)y) reduces to ((λx. x)y)((λx. x)y), which has two redexes, but in call by need they are represented using the same object rather than copied, so when one is reduced the other is too.

A note about complexity

While the idea of beta reduction seems simple enough, it is not an atomic step, in that it must have a non-trivial cost when estimating computational complexity. Computational complexity theory, as a branch of the Theory of computation in Computer science, investigates the problems related to the amounts of resources ^[7] To be precise, one must somehow find the location of all of the occurrences of the bound variable V in the expression E, implying a time cost, or one must keep track of these locations in some way, implying a space cost. A naïve search for the locations of V in E is O(n) in the length n of E. In mathematics big O notation (so called because it uses the symbol O) describes the limiting behavior of a function for very small or very large arguments This has led to the study of systems which use explicit substitution. In Computer science, Explicit substitution is an umbrella term used to describe several calculi based on the Lambda calculus that pay special attention to the formalization Sinot's director strings ^[8] offer a way of tracking the locations of free variables in expressions. In Mathematics, in the area of Lambda calculus and Computation, directors or director strings are a mechanism for keeping track of the Free

Concurrency and parallelism

The Church-Rosser property of the lambda calculus means that evaluation (β-reduction) can be carried out in any order, even concurrently. This means that various nondeterministic evaluation strategies are relevant. In Computer science, an evaluation strategy is a set of (usually deterministic rules for determining the evaluation of expressions in a Programming language However, the lambda calculus does not offer any explicit constructs for parallelism. Parallel computing is a form of computation in which many instructions are carried out simultaneously operating on the principle that large problems can often Various process calculi have been proposed as minimal languages for concurrency and distributed computation. In Computer science, the process calculi (or process algebras) are a diverse family of related approaches to formally modelling Concurrent systems Process

See also

References

Further Reading

• Abelson, Harold & Gerald Jay Sussman. Structure and Interpretation of Computer Programs. Structure and Interpretation of Computer Programs ( SICP) is a textbook published in 1985 about general The MIT Press. The MIT Press is a University press affiliated with the Massachusetts Institute of Technology (MIT in Cambridge Massachusetts ( USA) ISBN 0-262-51087-1.
• Hendrik Pieter Barendregt Introduction to Lambda Calculus.
• Barendregt, Hendrik Pieter, The Type Free Lambda Calculus pp1091-1132 of Handbook of Mathematical Logic, North-Holland (1977) ISBN 0-7204-2285-X
• Church, Alonzo, An unsolvable problem of elementary number theory, American Journal of Mathematics, 58 (1936), pp. Hendrik Pieter (Henk Barendregt (born 1947 is a Dutch Logician, known for his work in Lambda calculus and Type theory. North Holland ( Dutch: Noord-Holland,, West Frisian: Noôrd-Holland) is a province situated on the North Sea in the Also 1977 (album by Ash. Year 1977 ( MCMLXXVII) was a Common year starting on Saturday (link displays American Journal of Mathematics is a Bimonthly mathematics journal published by the Johns Hopkins University Press, founded in 1878 by James Joseph Sylvester 345–363. This paper contains the proof that the equivalence of lambda expressions is in general not decidable.
• Kleene, Stephen, A theory of positive integers in formal logic, American Journal of Mathematics, 57 (1935), pp. American Journal of Mathematics is a Bimonthly mathematics journal published by the Johns Hopkins University Press, founded in 1878 by James Joseph Sylvester Year 1935 ( MCMXXXV) was a Common year starting on Tuesday (link will display full calendar of the Gregorian calendar. 153–173 and 219–244. Contains the lambda calculus definitions of several familiar functions.
• Landin, Peter, A Correspondence Between ALGOL 60 and Church's Lambda-Notation, Communications of the ACM, vol. Peter Landin is a British Computer scientist. He was one of the first to realize that the Lambda calculus could be used to model a programming language Communications of the ACM ( CACM) is the flagship monthly Journal of the Association for Computing Machinery (ACM 8, no. 2 (1965), pages 89-101. Year 1965 ( MCMLXV) was a Common year starting on Friday (link will display full calendar of the 1965 Gregorian calendar. Available from the ACM site. A classic paper highlighting the importance of lambda-calculus as a basis for programming languages.
• Larson, Jim, An Introduction to Lambda Calculus and Scheme. A gentle introduction for programmers.
• Schalk, A. and Simmons, H. (2005) An introduction to λ-calculi and arithmetic with a decent selection of exercises. Notes for a course in the Mathematical Logic MSc at Manchester University.

Some parts of this article are based on material from FOLDOC, used with permission. The Free On-line Dictionary of Computing ( FOLDOC) is an online searchable encyclopedic Dictionary of Computing subjects

External links

• Henk Barendregt, Erik Barendsen Introduction to Lambda Calculus-(PDF)
• Achim Jung, A Short Introduction to the Lambda Calculus-(PDF)
• David C. Keenan, To Dissect a Mockingbird: A Graphical Notation for the Lambda Calculus with Animated Reduction
• Raúl Rojas, A Tutorial Introduction to the Lambda Calculus-(PDF)
• Peter Selinger, Lecture Notes on the Lambda Calculus-(PDF)
• L. Allison, Some executable λ-calculus examples
• Chris Barker, Some executable (Javascript) simple examples, and text.
• Georg P. Loczewski, The Lambda Calculus and A++
• Bret Victor, Alligator Eggs: A Puzzle Game Based on Lambda Calculus
• Lambda Calculus on Safalra’s Website
• Lambda Calculus on PlanetMath
• LCI Lambda Interpreter a simple yet powerful pure calculus interpreter
• Lambda Calculus links on Lambda-the-Ultimate
• Mike Thyer, Lambda Animator, a graphical Java applet demonstrating alternative reduction strategies. PlanetMath is a free, collaborative online Mathematics Encyclopedia.