The definition generalizes easily to boolean functions, or functions of other data types. Recursion lecture 9 tuesday, february 23, 2010 1 lambda calculus encodings the pure lambda calculus contains only functions as values. The course is entitled lambda calculus and types and it appears in various forms. Lambda expressions a lambda calculus expression is defined as e x variable. The simply typed lambdacalculus stlc is a tiny core calculus embodying the key concept of functional abstraction, which shows up in pretty much every realworld programming language in some form functions, procedures, methods, etc. Viewed pu rely as a naming device, however, it is a straighforward extension of ordinar y mathematical notation. The definition generalizes easily to boolean functions, or functions of other data. The lambda calculus can be thought of as the theoretical foundation of functional programming. Lambda calculus is a language with clear operational and denotational semantics capable of expressing algorithms. It was introduced by the mathematician alonzo church in the 1930s as part of his research into the. We can add constants and builtin primitives to enrich. It was introduced by the mathematician alonzo church in. Lambda calculus 10 computation in the lambda calculus takes the form. Lambda calculus lambda calculus stanford university.
Functional programming is rooted in lambda calculus, which constitutes the worlds smallest programming language. In this paper we show that the stone representation the orem for boolean algebras can be generalized to combi natory algebras. With a little syntax sugar, lambda calculus becomes a practical programming language. False is represented as function taking two arguments and returning the second.
The lambda calculus is an abstract mathematical theory of computation, involving. Lecture notes on the lambda calculus nanjing university. A tutorial introduction to the lambda calculus raul rojas fu berlin, ws9798 abstract this paper is a short and painless introduction to the calculus. But how can i now get from there to a formulation for the. This data type can be encoded in the lambda calculus. Formulate boolean logic in lambda calculus mathematics. The method is named for alonzo church, who first encoded data in the lambda calculus this way. It is a universal model of computation that can be used to simulate any turing machine. The idea is due to sch on nkel 1924 but is often called currying, after h. Values are lambda expressions that have properly finished being evaluated there is nothing more to do. Lecture notes on the lambda calculus department of mathematics. There are just a few keywords, and most of the utility of the language is inherent in its minimal, unornamented structure, unlike, say, \public static void main java.
This means that you can exclude pretty much all the usual. March 18, 2004 abstract the lambda calculus can appear arcane on. Functional languages are just the lambda calculus with a more palatable syntax. Programming in the lambda calculus church booleans common lisp church booleans. Recently, i was challenged to write a clojures macro called if which behaves like the reif but doesnt use anything that expands to it. In lambda calculus true is represented as a function taking two arguments and returning the first. If the rst is one boolean it evaluates to the second.
Lecture notes on the lambda calculus peter selinger department of mathematics and statistics dalhousie university, halifax, canada abstract this is a set of lecture notes that developed out of courses on the lambda calculus that i taught at the university of. Originally developed in order to study some mathematical properties of e ectively computable functions, this formalism has provided a. A underscore is used at the beginning of each method to highlight these are not part of the lambda calculus or church encoding. In lambda calculus, lambda expressions bound variables can be substituted with different name. Already, our factorial example above is shorter than equivalent code in many highlevel languages. With the above 2 helper methods, the unit tests become extremely easy. Lecture notes on the lambda calculus peter selinger department of mathematics and statistics dalhousie university, halifax, canada abstract this is a set of lecture notes that developed out of courses on the lambda calculus that i taught at the university of ottawa in 2001 and at dalhousie university in 2007. Even though the untyped lambda calculus already is very minimalistic, we can further reduce it to just three s,k,i or even two s,k symbols, or combinators. Query on booleans in lambda calculus stack overflow. True let tru lambda tt f f tt f l s lambda tt f f f f test lambda b then else funcall b. Encoding booleans the \boolean adt i there are two booleans and one conditional expression. Lambda calculus 9 programming in the lambda calculus multiple parameters through currying booleans pairs church numerals and arithmetic lists recursion call by name and call by value versions 11002 lesson 2.
The general technique is to represent the values of a given type by normal forms, that is. It is not exactly easy to write large or in teresting programs in the pure lambda calculus. The lambda calculus department of computer science, columbia. Lambda calculus is a notation for describing mathematical functions. The lambda calculus notes by don blaheta october 12, 2000 \a little bondage is always a good thing.
Dantam csci561, colorado school of mines fall 2019 dantam mines csci561the lambda calculus pre lecturefall 2019162. Note that this is cfg is ambiguous, but thats not a problem for defining the terms in the language we are not using it for parsing i. In the pure lambda calculus, the only values are functions \x. Also it forms a compact language to denote mathematical proofs logic provides a formal language in which mathematical statements can be formulated and provides deductive power to derive these.
Heres how to multiply two numbers in lambda calculus. Modern processors are just overblown turing machines. The simply typed lambda calculus stlc is a tiny core calculus embodying the key concept of functional abstraction, which shows up in pretty much every realworld programming language in some form functions, procedures, methods, etc. Lambda abstraction the only other thing in the lambda calculus is lambda abstraction. Lecture notes on the lambda calculus peter selinger department of mathematics and statistics dalhousie university,halifax, canada abstract this is a set of lecture notes that developed out of courses on the lambda calculus that i taught at the university of. The church numerals are a representation of the natural numbers using lambda notation.
There may, indeed, be other applications of the system than its use as a logic. In mathematics, church encoding is a means of representing data and operators in the lambda calculus. Recursion lecture 8 thursday, february 21, 20 1 lambda calculus encodings the pure lambda calculus contains only functions as values. The simplest nontrivial data type are the booleans, a type with two elements. Substitution e 1xe 2 captureavoiding, see lecture 2 conversion x. From the looks of it m and n are supposed to be booleans and a and b some other values. We can however encode objects, such as booleans, and integers. This wellrespected text offers an accessible introduction to functional programming concepts and techniques for students of mathematics and computer science. A generic version of church boolean and its boolean logic will be introduced later in the church pair part. In lambda calculus, there are 3 substitution rules for expression to be reduced. An introduction to functional programming through lambda. Idioms in lambda calculus now we will study some of the common programming idioms in lambda calculus that serve to demonstrate its expressive power. Lecture notes on the lambda calculus peter selinger department of mathematics and statistics dalhousie university,halifax, canada abstract this is a set of lecture notes that developed out of courses on the lambda calculus that i taught at the university of ottawa in 2001 and at dalhousie university in 2007 and 20. The function you showed above takes four arguments.
It is not exactly easy to write large or interesting programs in the pure lambda calculus. Lambda calculus is the theoretical foundation for functional programming. There are many reasonable encodings into lambda calculus. Any set of three expressions meeting this speci cation is a proper encoding of booleans. We would like to define terms that act like the boolean constants true and false and the boolean operators if, and, or, not, so that all these terms behave in the expected way, obeying the boolean abstraction. Cs 329 notes on untyped lambda calculus cse, iit bombay. Perhaps the simplest interesting kind of value is a boolean.
But they can be encoded in pure lambda forms as abstractions. The method is named for alonzo church, who first encoded data in the lambda calculus this way terms that are usually considered primitive in other notations such as integers, booleans, pairs, lists, and. We will use it as a foundation for sequential computation. In this post i will introduce some of the basic concepts of the lambda calculus and use them to define basic terms and operators of the boolean logic.
The lambda calculus is a language of pure functions. Boolean values encoded as lambda expressions are called church booleans in honor of church who invented lambda calculus. Terms that are usually considered primitive in other notations. Along the way, a variety of topics are discussed including arithmetic, linear list and binary tree processing, and alternative evaluation strategies. Boolean values can be defined in the lambda calculus, although they are often built into programming languages based on lambda calculus. Spare a thought for students struggling to make turing machines do simple tasks.
If it is the other boolean it evaluates to the third. A short introduction to the lambda calculus achim jung. Lambda calculus university of maryland, college park. Dantam csci561, colorado school of mines fall 2019. Despite not having numbers, strings, booleans, or any nonfunction datatype, lambda calculus can be used to represent any turing machine. Varela 2 mathematical functions take the mathematical function. All terms that can be expressed in the lambda calculus can also be expressed using just the combinators s,k,i and parentheses, and the i. An introduction to functional programming through lambda calculus greg michaelson. Lambda calculus as a formal system free and bound variables.