Lambda calculus

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In computer science, the lambda 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 the lambda calculus in 1936 to give a negative answer to the Entscheidungsproblem. The calculus can be used to cleanly define what is a computable function. 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.

The lambda calculus can be called the smallest universal programming language. The lambda calculus consists of a single transformation rule (variable substitution) and a single function definition scheme. The lambda calculus is universal in the sense that any computable function can be expressed and evaluated using this formalism. It is thus equivalent to Turing machines. However, the 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.

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 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. 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 argument and applies it to the argument 3: λ x. x 3.  This latter function could be applied to our earlier "add-two" 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. A common convention is to abbreviate curried functions as, for instance,  λ x y. x - y.

The three expressions

x 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.  (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

x 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 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 so-called alpha-conversion rule and the beta-reduction rule.

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. 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 xxx  is the same as  λ y. (λ xxy.

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. 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.

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 iff 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 xf x == λ xg 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 xf xy == f y,  we have  λ xf x  ==  f;  i.e., eta-conversion is found to be valid.

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 power 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. 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 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. ng h. h (g f)) (λ u. x) (λ u. u

or alternatively

PRED = λ n. ng 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.

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:

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 and only if its argument is the Church numeral 0:

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.

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 either returns the constant 1, or returns 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:

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: NN 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

Most programming languages are equivalent to the lambda calculus extended with some additional programming language constructs. The classical work where this viewpoint was put forward was Peter Landin's "A Correspondence between ALGOL 60 and Church's Lambda-notation", published in CACM in 1965. The key point is that the lambda calculus expresses the kind of procedural abstraction and application useful for any programming language. Prominently, functional programming languages are basically the lambda calculus with some constants and datatypes added. Lisp uses a variant of lambda notation for defining functions, but only its purely functional subset is really equivalent to lambda calculus. Strictly speaking, this holds only for modern dialects of Lisp, such as Common Lisp and Scheme. More archaic Lisps, such as Emacs Lisp, still use dynamic binding, and so are not based on the lambda calculus. Rather, they are based on the syntax of the lambda calculus, together with a misunderstanding of the notion of binding and substitution in the lambda calculus. In Lisp terminology, this is known as the Funarg problem. The lambda calculus is thus a fundamental theory in theoretical computer science and in programming language theory.

Theory of the lambda calculus says that lambda calculus computations can always be carried out sequentially not that they must be carried out sequentially. The lambda calculus is suitable for expressing some kinds of parallelism, e.g., the parallel evaluation of the arguments of a procedure. However the lambda calculus does not in general implement concurrency, e.g., a shared financial account that is updated concurrently. On the other hand concurrent computation as in the Actor model and Process calculi can perform the parallelism of the lambda calculus. The difference between parallelism in the lambda calculus and concurrency in Actors is reflected in that the Actor model has unbounded nondeterminism whereas the nondeterministic lambda calculus has bounded nondeterminism.

References

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