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Lambda calculusFrom CryptoDox, The Online Encyclopedia on Cryptography and Information SecurityIn 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. HistoryOriginally, 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 descriptionIn lambda calculus, every expression stands for a function with a single input, called its 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 application of the function 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
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
or
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 is shown 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
to
In the unsimplified expression in this example, the first "y" defines the formal parameter, the second uses the formal parameter, and the third is a free variable. If one only formalizes the notion of function application and does not allow lambda expressions, one obtains combinatory logic. Formal definitionFormally, a lambda expression is defined inductively as one of the following:
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 variablesEach 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:
α-conversionAlpha 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. SubstitutionSubstitution, 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:
β-reductionBeta reduction expresses the idea of function application. The beta reduction of ((λ V. E) E′) is simply E[V := E′] . η-conversionEta 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 calculusThere 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:
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:
Addition is defined as follows:
PLUS can be thought of as a function taking two natural numbers as arguments and returning a natural number; it can be verified that
are equivalent lambda expressions. Multiplication can then be defined as
the idea being that multiplying m and n is the same as m times adding n to zero. Alternatively
The predecessor PRED n = n - 1 of a positive integer n is more difficult:
or alternatively
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 predicatesBy convention, the following two definitions (known as Church booleans) are used for the boolean values TRUE and FALSE:
Then, with these two λ-terms, we can define some logic operators (these are just possible formulations; other expressions are equally correct):
We are now able to compute some logic functions, as for example:
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:
The availability of predicates and the above definition of TRUE and FALSE make it convenient to write "if-then-else" statements in lambda calculus. PairsA pair (2-tuple) datatype can be defined in terms of TRUE and 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. RecursionRecursion 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
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:
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:
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:
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 calculusA 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 equivalenceThere 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 languagesAs 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 * x 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 parallelismThe 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 alsoReferences
Some parts of this article are based on material from FOLDOC, used with permission. External links
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