Lambda calculus

In mathematical logic and computer science, lambda calculus, also λ-calculus, is a formal system designed to investigate function definition, function application and recursion. It was introduced by Alonzo Church and Stephen Cole Kleene in the 1930s as part of a larger effort to base the foundation of mathematics upon functions rather than sets (in the hopes of avoiding obstacles like Russell's Paradox). The Kleene-Rosser paradox shows that the lambda calculus is unable to avoid set-theoretic paradoxes, but the lambda calculus 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.

The lambda calculus can be thought of as an idealized, minimalistic programming language. It is a close cousin of the Turing machine, another minimalist abstraction capable of expressing any algorithm. The difference between the two is that the lambda calculus takes a functional view of algorithms, while the original Turing machine takes an imperative view. That is, a Turing machine maintains 'state' - a 'notebook' of symbols that can change from one instruction to the next. The imperative paradigm can be seen in programming languages like C or BASIC. By contrast, the lambda calculus is stateless, it deals exclusively with functions which accept and return data (including other functions), but produce no side effects in 'state' and do not make alterations to incoming data (immutability.) The functional paradigm can be seen in modern languages like Lisp, Scheme and Haskell.

Church used the lambda calculus to give a negative answer to the Entscheidungsproblem and to the halting problem. The lambda calculus - and the paradigm of functional programming - is still influential, especially among the artificial intelligence community.

This article deals with the "untyped lambda calculus" as originally conceived by Church. Since then, some typed lambda calculi have been developed.

History

Originally, Church had tried to construct a complete formal system for the foundations of mathematics; when the system turned out to be susceptible to the analog of Russell's paradox, he separated out the lambda calculus and used it to study computability, culminating in his negative answer to the Entscheidungsproblem.

Informal description

In lambda calculus, every expression is a unary function, that is, a function with only one input (called an argument). When it is applied to another expression (that is, called with the expression as its argument), it returns a single value, called its result. Further, since again every expression is a unary function, every argument and every result is also a function (indeed, a unary function just like this one), making lambda calculus quite interesting & unique within both computation even mathematics. 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 argument 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.  Consider the function which takes a function as an argument and applies it to the number 3: 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). 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. 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 instance in order to reduce

(λ x y. y x) (λ x. y)

one first needs to rename the immaterial occurrences of y with a name not already occurring in the term, say z

(λ x z. z x) (λ x. y)

so that the reduction does not mix up the names and correctly yields

λ z. z (λ x. y).

In the first line of this example, the first y defines the formal parameter, the second uses the formal parameter, and the third, written in this font, is a free variable.

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.

Formal definition

Formally, a lambda expression is defined inductively as one of the following:

1. V, a variable, where V is any identifier. (The precise set of identifiers is arbitrary, but must be infinite.)
2. (λ V. E), an abstraction, where V is any identifier and E is any lambda expression. An abstraction corresponds to an anonymous function.
3. E E′, an application, where E and E′ are any lambda expressions. An application corresponds to calling a function (E) with an argument (E′).

To unclutter the notation, parentheses may be omitted if they are redundant. When there are parentheses missing, function application is left associative, and a lambda binds as much as possible after it. For example, the expression  ((λ x. (x x)) (λ y. y))  can be simply written as  (λ x. x x) λ y. y.

Free and bound variables

Each variable in a lambda expression is free or bound. For example, the x in  (x y)  is free, while the x in  λ x. (x y)  is bound. A bound variable has a specific lambda it is associated with, while a free variable does not. Precisely, the free variables of a lambda expression are defined inductively as follows:

1. In an expression of the form  V,  where V is a variable, this V is the single free occurrence.
2. In an expression of the form  λ V. E,  the free occurrences are the free occurrences in E except for V. In this case the occurrences of V in E are said to be bound by the λ before V.
3. In an expression of the form  (E E′),  the free occurrences are the free occurrences in E and E′.

An expression which contains no free variables is said to be closed.

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.

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). 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).

The precise rules are defined inductively as follows:

1. V[V := E] == E
2. W[V := E] == W, if W and V are different
3. (E₁ E₂)[V := E] == (E₁[V := E] E₂[V := E])
4. (λ V. E′)[V := E] == (λ V. E′)
5. (λ W. E′)[V := E] == (λ W. E′[V := E]), if V and W are different and W is not free in E.
6. (λ W. E′)[V := E] == (λ W′. E′[W := W′])[V := E], if V and W are different and if W′ is not free in E.

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. Eta-conversion converts between  λ x. f x  and  f  whenever x does not appear free in f.

This conversion is not always equivalent 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. 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. 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 can be accomplished by adding 1 m times, an equivalent definition is:

PLUS := λ m. m SUCC

Similarly, multiplication can be defined as

MULT := λ m n f. m (n f)

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. 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.

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. 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.

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. 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 the unsolvability could be proven. Of course, in order to do so, the notion of algorithm has to be cleanly defined; Church used a definition via recursive functions, which is now known to be equivalent to all other reasonable definitions of the notion.

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. If e is applied to its own Gödel number, a contradiction results.

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.

Implementing the lambda calculus on a computer involves treating "functions" as first-class objects, which raises implementation issues for stack-based programming languages. This is known as the Funarg problem.

The most prominent counterparts to lambda calculus in programming are functional programming languages, which essentially implement the calculus augmented with some constants and datatypes. Lisp uses a variant of lambda notation for defining functions, but only its purely functional subset ("Pure Lisp") is really equivalent to lambda calculus.

Functional languages are not the only ones to support functions as first-class objects. Numerous imperative languages, e.g. Pascal, have long supported passing subprograms as arguments to other subprograms. In C and the C-like subset of C++ the equivalent result is obtained by passing pointers to the code of functions (subprograms). 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"). 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.

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 can also treat any other function created with the standard def statement as first-class objects.

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.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. This function variable can then be passed to other methods (or function delegates)

   //Declare a delegate signature
   delegate double MathDelegate(double i);
   //Create an delegate instance
   MathDelegate 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 func)
   {
       return func(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 instance
   MathDelegate f = i => i * i;
   Execute(f);
   // or more simply put
   Execute(i => i * i);

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. (Indeed, the lambda calculus is referentially transparent.) While this means the lambda calculus can model the various nondeterministic evaluation strategies, it does not offer any richer notion of parallelism, nor can it express any concurrency issues. The Actor model and Process calculi such as CSP, the CCS, the π calculus and the ambient calculus have been designed for such purposes.

Although the nondeterministic lambda calculus is capable of expressing all partial recursive functions, it is not capable of expressing all computations. For example it is not capable of expressing unbounded nondeterminism (i.e. every nondeterministic lambda expression that is guaranteed to terminate; terminates in a finite number of expressions). However, there are concurrent programs guaranteed to halt that terminate in an infinite number of states [Clinger 1981, Hewitt 2006].

See also

• Anonymous recursion
• Combinatory logic
• Curry-Howard isomorphism
• Evaluation strategy
• Knights of the Lambda Calculus
• Lambda cube
• Rewriting
• SKI combinator calculus
• System F
• Calculus of constructions
• Typed lambda calculus
• Unlambda
• Lambda-mu calculus
• Cartesian closed category

References

References

• Abelson, Harold & Gerald Jay Sussman. Structure and Interpretation of Computer Programs. The MIT Press. ISBN 0-262-51087-1.
• Barendregt, Hendrik Pieter, The Lambda Calculus, Its Syntax and Semantics. Studies in Logic and the Foundations of Mathematics, Volume 103, North-Holland (1984), ISBN 0-444-87508-5 is the comprehensive reference on the (untyped) lambda calculus; see also the paper 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. 345–363. This paper contains the proof that the equivalence of lambda expressions is in general not decidable.
• Clinger, William, Foundations of Actor Semantics. MIT Mathematics Doctoral Dissertation, June 1981.
• Punit,Gupta, Amit & Ashutosh Agte, Untyped lambda-calculus, alpha-, beta- and eta- reductions and recursion
• Henz, Martin, The Lambda Calculus. Formally correct development of the Lambda calculus.
• Hewitt, Carl, What is Commitment? Physical, Organizational, and Social COIN@AAMAS. April 27, 2006.
• Kleene, Stephen, A theory of positive integers in formal logic, American Journal of Mathematics, 57 (1935), pp. 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. 8, no. 2 (1965), pages 89-101. 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.

External links

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