Lambda Calculus
Introduction
Lambda 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. It was first introduced by mathematician Alonzo Church in the 1930s as part of his research into the foundations of mathematics.
History
Lambda calculus was developed by Alonzo Church and his students at Princeton University in the 1930s. Church sought a clear, simple basis for logic and computation, and his work resulted in the creation of lambda calculus. It was initially developed to achieve a formal system for defining functions and their application. However, it was soon recognized as a model of computation, equivalent in power to the Turing machines being developed by Alan Turing around the same time.
Syntax and Semantics
Lambda calculus consists of constructing lambda terms and using them to create more complex expressions. The syntax of lambda calculus is quite simple, consisting of variables, abstraction, and application.
Variables
Variables in lambda calculus are usually represented by lowercase letters. They are placeholders for values and can be replaced (or substituted) by other lambda terms.
Abstraction
Abstraction in lambda calculus is the process of defining a function. It is represented by the Greek letter lambda (λ), followed by one or more variables, a dot, and the function body.
Application
Application in lambda calculus is the process of applying a function to an argument. It is represented by juxtaposition, with the function and its argument written side by side.
Reduction Strategies
Lambda calculus features several reduction strategies, which are rules used to simplify or evaluate lambda expressions. The three primary reduction strategies are normal order, call by name, and call by value.
Normal Order
Normal order reduction is a strategy where the leftmost, outermost redex (reducible expression) is always reduced first. This strategy guarantees that if a lambda expression has a normal form, normal order reduction will find it.
Call by Name
Call by name is a reduction strategy where only the outermost redexes are reduced, and a redex is only reduced when its right-hand side has been reduced to a value.
Call by Value
Call by value is a reduction strategy where only the leftmost, innermost redex is reduced first. Unlike normal order and call by name, call by value does not guarantee that a normal form will be found if one exists.
Lambda Calculus and Programming Languages
Lambda calculus has had a significant influence on the design and development of programming languages. It serves as the theoretical foundation for functional programming languages such as LISP, Scheme, and Haskell, where functions are first-class citizens and can be manipulated like any other data.