Lambda calculus

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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; Church used lambda calculus in 1936 to give a negative answer to the Entscheidungsproblem. Lambda calculus can be used to define what a computable function is. The question of whether two lambda calculus expressions are equivalent cannot be solved by a general algorithm. This was the first question, even before the halting problem, for which undecidability could be proved. Lambda calculus has greatly influenced functional programming languages, such as Lisp, ML and Haskell.

Lambda calculus can be called the smallest universal programming language. It consists of a single transformation rule (variable substitution) and a single function definition scheme. Lambda calculus is universal in the sense that any computable function can be expressed and evaluated using this formalism. It is thus equivalent to the Turing machine formalism. However, lambda calculus emphasizes the use of transformation rules, and does not care about the actual machine implementing them. It is an approach more related to software than to hardware.

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

Contents

[edit] 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.

[edit] Informal description

In lambda calculus, every expression stands for a function with a single argument (argument::input of a function); 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. 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 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 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)    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. A common convention is to abbreviate curried functions as, for instance,  λ x y. x - y. 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 as 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.

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

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

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

[edit] 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 parentheses 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 functions 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 expression 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 α-conversion rule and the β-reduction rule. Sometimes an alternate equivalence relation, obtained by adopting a third rule called η-conversion, is used.

[edit] α-conversion

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 (see Lambda lifting for a practical implementation). There are a number of restrictions on when one bound variable may be replaced with another. For example, if we replace x with y in λx.λy.x, we get λy.λy.y, which is not at all the same.

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.

As an example we note that

for (int i = 0; i < max; i++) { proc(i); }

is equivalent to

for (int j = 0; j < max; j++) { proc(j); }

[edit] β-reduction

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. An expression of the form ((λ V. E) E′) is called a beta redex. A lambda expression which does not allow any beta reduction, i.e., has no beta redexes as subexpressions 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 bound variables. Furthermore, there is an algorithm for computing normal forms: keep replacing the first (left-most) bound variable with its corresponding actual 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 bound variables if and only if they are equivalent.

[edit] η-conversion

There is a third rule, eta-conversion, which may be added to these two to form a new equivalence relation. 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 can be seen to be equivalent to extensionality as follows:

If f and g are extensionally equivalent, i.e. if  f a==g a  for all lambda expressions a, then in particular by taking a to be a variable x not appearing free in f nor g we have  f x == g x  and hence  λ x. f x == λ x. g x,  and so by eta-conversion  f == g.  So if we take eta-conversion to be valid, we find extensionality is valid.

Conversely if extensionality is taken to be valid, then since by beta-reduction for all y we have  (λ x. f x) y == f y,  we have  λ x. f x  ==  f;  i.e., eta-conversion is found to be valid.

[edit] 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. Intuitively, the number n in lambda calculus is a function that takes a function f as argument and returns the n-th composition of f. That is to say, a Church numeral is a higher-order function -- it takes a single-argument function f, and returns another single-argument function.

(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.) Given this definition of the Church numerals, we can define a successor function, which takes a number n and returns n + 1:

SUCC := λ n f x. f ((n f) x)

Addition is 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 is fun to verify that

PLUS 2 3    and    5

are equivalent lambda expressions. Multiplication can then be defined as

MULT := λ m n. m (PLUS n) 0,

the idea being that multiplying m and n is the same as m times adding n to zero. Alternatively

MULT := λ m n f. m (n f)

The predecessor  PRED n = n - 1  of a positive integer n is more difficult:

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

or alternatively

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

[edit] 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 FALSE
OR := λ p q. p TRUE q
NOT := λ p. p FALSE TRUE
IFTHENELSE := λ p x y. p x y

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

AND TRUE FALSE
≡ (λ p q. p q FALSE) TRUE FALSE →β TRUE FALSE FALSE
≡ (λ x y. x) FALSE FALSE →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 availability of predicates and the above definition of TRUE and FALSE make it convenient to write "if-then-else" statements in lambda calculus.

[edit] Pairs

A pair (2-tuple) datatype can be defined in terms of TRUE and FALSE.

CONS := λf.λs. λb. b f s
CAR := λp. p TRUE
CDR := λp. p FALSE
NIL := λx.TRUE
NULL := λp. p (λx y.FALSE)

A linked list datatype can be defined as either NIL for the empty list, or the CONS of an element and a smaller list.

[edit] 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.

[edit] 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.

[edit] 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.

[edit] 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 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 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 * x

This creates a new anonymous function named 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.

[edit] 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. Process calculi such as CSP, the CCS, the π calculus and the ambient calculus have been designed for such purposes.

[edit] See also

[edit] References

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

[edit] External links

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