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Lambda calculus

The lambda calculus is a formal system designed to investigate function definition, function application and recursion. It was introduced by Alonzo Church and Stephen Kleene in the 1930s; Church used the lambda calculus in 1936 to solve the Entscheidungsproblem. (He actually proved the non-existence of the sought algorithm that would have been considered a "solution" to the Entscheidungsproblem.) The calculus can be used to cleanly define what a "computable function" is. The question of whether two lambda calculus expressions are equivalent cannot be solved by a general algorithm, and this was the first question, even before the halting problem, for which undecidability could be proved. Lambda calculus has greatly influenced functional programming languages, especially LISP.

This article deals with the "untyped lambda calculus" as originally conceived by Church. Since then, some typed lambda calculi have been developed, which are not (yet) covered here.

Table of contents

Informal description

In lambda calculus, every expression stands for a function with a single argument; the argument of the function is in turn a function with a single argument, and the value of the function is another function with a single argument. Functions are anonymously defined by a lambda expression which expresses the function's action on its argument. For instance, the "add-two" function 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 number f(3) would be written as (λ x. x + 2) 3. Function application is left associative: f x y = (f x) y. Consider the function which takes a function as argument and applies it to the argument 3: λ x. x 3. This latter function could be applied to our earlier "add-2" function as follows: (λ x. x 3) (λ x. x+2). It is clear that the three expressions

x. x 3) (λ x. x+2)   and    (λ x. x + 2) 3    and    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. The three expressions
x. λ y. x - y) 7 2    and    (λ y. 7 - y) 2    and    7 - 2
are equivalent. It is this equivalence of lambda expressions which in general can not be decided by an algorithm.

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

x. x x) (λ x. x x)
or
x. x x x) (λ x. x x x)
and try to visualize what happens as you start to apply the first function to its argument.

While the lambda calculus itself does not contain symbols for integers or addition, these can be defined as abbreviations within the calculus and arithmetic can be expressed as we will see below.

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

x. λ y. y x) (λ x. y)     to     λ z. zx. y)

If one only formalizes the notion of function application and does not allow lambda expressions, one obtains combinatory logic.

Formal definition

Formally, we start with a countably infinite set of identifiers, say {a, b, c, ..., x, y, z, x1, x2, ...}. The set of all lambda expressions can then be described by the following context-free grammar in BNF:

  1. <expr> → <identifier>
  2. <expr> → (λ <identifier> . <expr>)
  3. <expr> → (<expr> <expr>)

The first two rules generate functions, while the third describes the application of a function to an argument. Usually the brackets for lambda abstraction (rule 2) and function application (rule 3) are omitted if there is no ambiguity under the assumptions that (1) function application is left-associative, and (2) a lambda binds to the entire expression following it. For example, the expression ((λ x. (x x)) (λ y. y)) can be simply written as (λ x. x x) λ y.y.

Lambda expressions such as λ x. (x y) do not define a function because the occurrence of the variable y is free, i.e., it is not bound by any λ in the expression. The binding of occurrences of variables is (with induction upon the structure of the lambda expression) defined by the following rules:

  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 those of V. In this case the occurrences of V in E are said to be bound by the λ before V.
  3. In an expresssion of the form (E E' ) the free occurrences are the free occurrences in E and E' .

Over the set of lambda expressions an equivalence relation (here denoted as ==) is defined that captures the intuition that two expressions denote the same function. This equivalence relation is defined by the so-called alpha-conversion rule and the beta-reduction rule.

The alpha-conversion rule is intended to express the idea that the names of the bound variables are unimportant; for example that λx.x and λy.y are the same function. However, the rule is not as simple as it first appears. There are a number of restrictions on when one bound variable may be replaced with another.

The alpha-conversion rule states that if V and W are variables, E is a lambda expression and E[V/W] means the expression E with every free occurrence of V in E replaced with W then

λ V. E == λ W. E[V/W]
if W does not appear freely in E and W is not bound by a λ in E whenever it replaces a V. This rule tells us for example that λ x. (λ x. x) x is the same as λ y. (λ x. x) y.

The beta-reduction rule expresses the idea of function application. It states that

((λ V. E) E') == E[V/E']
if all free occurrences in E' remain free in E[V/E'].

The relation == is then defined as the smallest equivalence relation that satisfies these two rules.

A more operational definition of the equivalence relation can be given by applying the rules only from left to right. A lambda expression which does not allow any beta reduction, i.e., has no subexpression of the form ((λ V. E) E' ), is called a normal form. Not every lambda expression is equivalent to a normal form, but if it is, then the normal form is unique up to naming of the formal arguments. Furthermore, there is an algorithm for computing normal forms: keep replacing the first (left-most) formal argument with its corresponding concrete argument, until no further reduction is possible. This algorithm halts if and only if the lambda expression has a normal form. The Church-Rosser theorem then states that two expressions result in the same normal form up to renaming of the formal arguments if and only if they are equivalent.

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 integers, which can be defined as follows:

0 = λ f. λ x. x
1 = λ f. λ x. f x
2 = λ f. λ x. f (f x)
3 = λ f. λ x. f (f (f x))
and so on. Intuitively, the number n in lambda calculus is a function that takes a function f as argument and returns the n-th power of f. (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.) Addition is defined as follows:
PLUS = λ m. λ n. λ f. λ x. m f (n f x)
PLUS can be thought of as a function taking two natural numbers as arguments and returning a natural number; it is fun to verify that
PLUS 2 3    and    5
are equivalent lambda expressions. Multiplication can then be defined as
MULT = λ m. λ n. mk. PLUS k n) 0,
the idea being that multiplying m and n is the same as m times adding n to zero. Alternatively
MULT = λ m. λ n. λ f. m (n f)
The predecessesor PRED n = n - 1 of a positive integer n is more difficult:
PRED = λ n. λ f. λ x. ng. λ h. h (g f)) (λ u. x) (λ u. u)
or alternatively
PRED = λ n. ng. λ k. (g 1) (λ u. PLUS (g k) 1) k) (λ l. 0) 0
Note the trick (g 1) (λ u. PLUS (g k) 1) k which evaluates to k if g(1) is zero and to g(k) + 1 otherwise.

Logic and Predicates

By convention, the following two definitions are used for the boolean values TRUE and FALSE:

TRUE = λ u. λ v. u
FALSE = λ u. λ v. v
A predicate is a function which returns a boolean value. The most fundamental predicate is ISZERO which returns true if and only if its argument is zero:
ISZERO = λ n. nx. FALSE) TRUE
The availability of predicates and the above definition of TRUE and FALSE make it convenient to write "if-then-else" statements in lambda calculus.

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.

One may view the right-hand side of this definition as a function g which takes a function f as an argument and returns another function g(f). Using the ISZERO predicate, the function g can be defined in lambda calculus. The factorial function is then a fixed-point of g:

f = g(f).
In fact, every recursively defined function can be seen as a fixed point of some suitable other function. This allows the definition of recursive functions in lambda calculus, because every function in lambda calculus has a fixed point, and the fixed point can be easily described: the fixed point of a function g is given by
x. g (x x)) (λ x. g (x x))
and if we define the Y combinator as
Y = λ g. (λ x. g (x x)) (λ x. g (x x))
then Y g is a fixed point of g, meaning that the two expressions
Y g   and   g (Y g)
are equivalent. 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.

Computability and lambda calculus

A function F : NN of natural numbers is defined to be computable if there exists a lambda expression f such that for every pair of x, y in N, F(x) = y if and only if the expressions f x and y are equivalent. 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 output "YES" or "NO" 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 utilizing Gödel's procedure of Gödel numbers 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.

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.

Lambda calculus and programming languages

Most functional programming languages are equivalent to lambda calculus extended with constants and datatypes. LISP uses a variant of lambda notation for defining functions but only its purely functional subset is really equivalent to lambda calculus.


References:
  • Stephen Kleene, 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.
  • Alonzo Church, 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.
  • Jim Larson, An Introduction to Lambda Calculus and Scheme (http://www.jetcafe.org/~jim/lambda). A gentle introduction for programmers.
  • Martin Henz, The Lambda Calculus (http://www.comp.nus.edu.sg/~cs6202/). Formally correct development of the Lambda calculus.
  • Henk Barendregt, The lambda calculus, its syntax and semantics (http://www.elsevier.nl/inca/publications/store/5/0/1/7/2/7/), North-Holland (1984), is the comprehensive reference on the (untyped) lambda calculus.
  • Amit Gupta and Ashutosh Agte, Untyped lambda-calculus, alpha-, beta- and eta- reductions and recursion (http://www.cis.ksu.edu/~stefan/Teaching/CIS705/Reports/GuptaAgte-2.pdf)

Some parts of this article are based on material from FOLDOC, used with permission.



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