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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.
- Motivation
- Basics of Lambda Calculus
- Arithmetic
- Data Types
Lambda calculus is a fascinating topic for the following reasons. 1. It is simple. It just consists of variables, functions and function applications. 2. Despite of being simple it is possible to express any computable function in the calculus.This is the reason why Alonzo Church (1936) invented the lambda calculus. He wanted to explore the limits ...
Combinators
Lambda terms where all variables are bound are called combinators.
Type Annotations
Untyped lambda calculus does not know of any types. Therefore the name. However the programmer thinks in types. We have already talked about booleans and pairs. These are types. We use types to express our intentions. Since we want to do programmingin lambda calculus, we want to be able to express our intentions in the source code.
Terminating and Non Terminating Computations
Up to now all lambda terms we have used have terminatedin the sense that we reached a state, where no more reducible expressions are in the term. Is this always the case? Unfortunately not. In the following we show how to construct potentially endless loops in lambda calculus.
Church Numerals
We want to encode the natural numbers 0, 1, 2, ...in lambda calculus. Since lambda calculus only has functions, we have to figure out a way to encode numbers as functions. What can we do with a number n? We can do something n times. And what can lambda calculus do? Correct answer: Function application. Therefore we encode a natural number as a function which takes a function argument and a start value and iterates the function ntimes on the start value. This is called the Church encoding of n...
Simple Arithmetics
Church numerals are iterations. We can use that to do simple arithmetics. To add the numbers n and m which are represented as church numerals we simply apply the successor function n times with start value m. Multiplication of the numbers n and m is defined as n times the iterated addition of m on the number zero. The exponentiation n ^ m is defined as n * n * .... * n * one, i.e. it is an mtimes iterated multiplication. There is no problem to define exponentiation in lambda calculus
Simple Predicates
A predicate is a function returing a boolean value. Predicates are deciders. We want to be able to decide, if a number is zero, is an even number or is an odd number. The encoding of the predicate isZero as an iteration is surprisingly simple. Evidently the start value of the iteration is true, because the number zero is zero. The iteration function just ignores the result of the previous iterations and returns false. The evenness and oddness predicates can be represented as iterations as wel...
Data types are an important means to structure computations. We think of data types like Boolean, Natural, List, Tree, etc. In the previous chapters represented the types Boolean and Naturalin lambda calculus. Since lambda calculus has only functions, we have to represent objects of a certain type as functions. We represented the type Boolean as a ...
Lambda calculus (also \(\lambda\)-calculus), introduced by Alonzo Church in the 1930s, is a model of computation based on functions. All functions in lambda calculus are anonymous, providing the inspiration for lambda expressions in modern programming languages.
The Lambda calculus is an abstract mathematical theory of computation, involving \(\lambda\) functions. The lambda calculus can be thought of as the theoretical foundation of functional programming. It is a Turing complete language; that is to say, any machine which can compute the lambda calculus can compute everything a Turing machine can ...
Lambda Calculus. Lambda calculus (λ-calculus), originally created by Alonzo Church, is the world’s smallest programming language. Despite not having numbers, strings, booleans, or any non-function datatype, lambda calculus can be used to represent any Turing Machine! Lambda calculus is composed of 3 elements: variables, functions, and ...
The lambda-calculus, first published by the logician Alonzo Church in 1932, is a core calculus with only three syntactic constructs: variables, abstraction, and application. It captures the key concept of functional abstraction, which appears in pretty much every programming language, in the form of either functions, procedures, or methods.
one-line universal program: Here’s a lambda calculus self-interpreter: ( λ f. ( λ x. f ( x x)) ( λ x. f ( x x))) ( λ e m. m ( λ x. x) ( λ m n. e m ( e n)) ( λ m v. e ( m v))) . In contrast, universal Turing machines are so tedious that classes often skip the details and just explain why they exist.
