Lambda calculus (nonfiction)

Lambda calculus (also written as λ-calculus) is a formal system in mathematical logic for expressing computation based on function abstraction and application using variable binding and substitution.

It is a universal model of computation that can be used to simulate any Turing machine, introduced by mathematician Alonzo Church circa 1935-1936 as part of his research of the foundations of mathematics.

Lambda calculus consists of constructing lambda terms and performing reduction operations on them.

In the simplest form of lambda calculus, terms are built using only the following rules:

• x = Name : A character or string representing a parameter or mathematical/logical value
• λx.M = Abstraction : Function definition (M is a lambda term). The variable x becomes bound in the expression.
• M N = Application : Applying a function to an argument. M and N are lambda terms.

producing expressions such as: (λx.λy.(λz.(λx.z x) (λy.z y)) (x y)).

Parentheses can be dropped if the expression is unambiguous.

For some applications, terms for logical and mathematical constants and operations may be included.

Lambda calculus is Turing complete, that is, it is a universal model of computation that can be used to simulate any Turing machine. Its namesake, the Greek letter lambda (λ), is used in lambda expressions and lambda terms to denote binding a variable in a function.

Lambda calculus may be untyped or typed. In typed lambda calculus, functions can be applied only if they are capable of accepting the given input's "type" of data. Typed lambda calculi are weaker than the untyped lambda calculus that is the primary subject of this article, in the sense that typed lambda calculi can express less than the untyped calculus can, but on the other hand typed lambda calculi allow more things to be proved; in the simply typed lambda calculus it is for example a theorem that every evaluation strategy terminates for every simply typed lambda-term, whereas evaluation of untyped lambda-terms need not terminate. One reason there are many different typed lambda calculi has been the desire to do more (of what the untyped calculus can do) without giving up on being able to prove strong theorems about the calculus.

Lambda calculus has applications in many different areas in mathematics, philosophy, linguistics, and computer science. Lambda calculus has played an important role in the development of the theory of programming languages. Functional programming languages implement the lambda calculus.

Lambda calculus also is a current research topic in Category theory.

The lambda calculus was introduced by mathematician Alonzo Church in the 1930s as part of an investigation into the foundations of mathematics. The original system was shown to be logically inconsistent in 1935 when Stephen Kleene and J. B. Rosser developed the Kleene–Rosser paradox.

Subsequently, in 1936 Church isolated and published just the portion relevant to computation, what is now called the untyped lambda calculus. In 1940, he also introduced a computationally weaker, but logically consistent system, known as the simply typed lambda calculus.

Until the 1960s when its relation to programming languages was clarified, the λ-calculus was only a formalism. Thanks to Richard Montague and other linguists' applications in the semantics of natural language, the λ-calculus has begun to enjoy a respectable place in linguistics and computer science.

Fiction cross-reference

• Crimes against mathematical constants

Nonfiction cross-reference

• Alonzo Church (nonfiction)
• Computation (nonfiction)
• Computer science (nonfiction)
• Mathematics (nonfiction)
• Turing machine (nonfiction)

External links

• Lambda calculus @ Wikipedia
• From Type Theory to Haskell in 10 Minutes @ Stackbuilders.com by by Matt Campbell

Related topics

• To Mock a Mockingbird by the mathematician and logician Raymond Smullyan. It contains many nontrivial recreational puzzles of the sort for which Smullyan is well known. It is also a gentle and humorous introduction to combinatory logic and the associated metamathematics, built on an elaborate ornithological metaphor.