A short introduction to the lambda calculus achim jung. This means that you can exclude pretty much all the usual. 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. In lambda calculus, there are 3 substitution rules for expression to be reduced. 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. The lambda calculus is an abstract mathematical theory of computation, involving. Originally developed in order to study some mathematical properties of e ectively computable functions, this formalism has provided a. If the rst is one boolean it evaluates to the second. 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. 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. Idioms in lambda calculus now we will study some of the common programming idioms in lambda calculus that serve to demonstrate its expressive power.
Often handy is the function iszero from natural numbers to booleans, which. Lambda calculus is the theoretical foundation for functional programming. Functional languages are just the lambda calculus with a more palatable syntax. Perhaps the simplest interesting kind of value is a boolean. Cs 329 notes on untyped lambda calculus cse, iit bombay. Functional programming is rooted in lambda calculus, which constitutes the worlds smallest programming language. 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. 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. From the looks of it m and n are supposed to be booleans and a and b some other values. Despite not having numbers, strings, booleans, or any nonfunction datatype, lambda calculus can be used to represent any turing machine. An introduction to functional programming through lambda. Boolean values can be defined in the lambda calculus, although they are often built into programming languages based on lambda calculus. Substitution e 1xe 2 captureavoiding, see lecture 2 conversion x.
The function you showed above takes four arguments. Lambda calculus as a formal system free and bound variables. Boolean values encoded as lambda expressions are called church booleans in honor of church who invented lambda calculus. 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. We can however encode objects, such as booleans, and integers. 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. The definition generalizes easily to boolean functions, or functions of other data. March 18, 2004 abstract the lambda calculus can appear arcane on. Investigate implementing the untyped lambda calculus in haskell. Encoding booleans the \boolean adt i there are two booleans and one conditional expression. Lecture notes on the lambda calculus department of mathematics.
Dantam csci561, colorado school of mines fall 2019. In mathematics, church encoding is a means of representing data and operators in the lambda calculus. 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. Query on booleans in lambda calculus stack overflow. Any set of three expressions meeting this speci cation is a proper encoding of booleans. The definition generalizes easily to boolean functions, or functions of other data types. Lecture notes on the lambda calculus nanjing university. 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.
Lambda calculus pdcs 2 alpharenaming, beta reduction, eta conversion, applicative and normal evaluation orders, churchrosser theorem, combinators, booleans carlos varela rennselaer polytechnic institute september 6, 2019. This wellrespected text offers an accessible introduction to functional programming concepts and techniques for students of mathematics and computer science. The course is entitled lambda calculus and types and it appears in various forms. Dantam csci561, colorado school of mines fall 2019 dantam mines csci561the lambda calculus pre lecturefall 2019162. Values are lambda expressions that have properly finished being evaluated there is nothing more to do. Modern processors are just overblown turing machines. In this paper we show that the stone representation the orem for boolean algebras can be generalized to combi natory algebras. In lambda calculus, lambda expressions bound variables can be substituted with different name. The simplest nontrivial data type are the booleans, a type with two elements. We will use it as a foundation for sequential computation. 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.
But they can be encoded in pure lambda forms as abstractions. Spare a thought for students struggling to make turing machines do simple tasks. An introduction to functional programming through lambda calculus greg michaelson. Varela 2 mathematical functions take the mathematical function. It was introduced by the mathematician alonzo church in the 1930s as part of his research into the. Lambda calculus is a language with clear operational and denotational semantics capable of expressing algorithms. Recursion lecture 9 tuesday, february 23, 2010 1 lambda calculus encodings the pure lambda calculus contains only functions as values. Formulate boolean logic in lambda calculus mathematics. The lambda calculus can be thought of as the theoretical foundation of functional programming. 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. It was introduced by the mathematician alonzo church in. Recursion lecture 8 thursday, february 21, 20 1 lambda calculus encodings the pure lambda calculus contains only functions as values. The lambda calculus notes by don blaheta october 12, 2000 \a little bondage is always a good thing. There may, indeed, be other applications of the system than its use as a logic.
Lambda calculus 10 computation in the lambda calculus takes the form. The idea is due to sch on nkel 1924 but is often called currying, after h. If it is the other boolean it evaluates to the third. Heres how to multiply two numbers in lambda calculus. Lambda abstraction the only other thing in the lambda calculus is lambda abstraction. 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. Lambda calculus is a notation for describing mathematical functions. Viewed pu rely as a naming device, however, it is a straighforward extension of ordinar y mathematical notation. The method is named for alonzo church, who first encoded data in the lambda calculus this way. We can add constants and builtin primitives to enrich. There are many reasonable encodings into lambda calculus.
Terms that are usually considered primitive in other notations. True let tru lambda tt f f tt f l s lambda tt f f f f test lambda b then else funcall b. Lambda calculus university of maryland, college park. 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. With the above 2 helper methods, the unit tests become extremely easy. It is a universal model of computation that can be used to simulate any turing machine.
With a little syntax sugar, lambda calculus becomes a practical programming language. The church numerals are a representation of the natural numbers using lambda notation. A tutorial introduction to the lambda calculus raul rojas fu berlin, ws9798 abstract this paper is a short and painless introduction to the calculus. 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. But how can i now get from there to a formulation for the. In lambda calculus true is represented as a function taking two arguments and returning the first. False is represented as function taking two arguments and returning the second.
Lambda calculus lambda calculus stanford university. Along the way, a variety of topics are discussed including arithmetic, linear list and binary tree processing, and alternative evaluation strategies. A underscore is used at the beginning of each method to highlight these are not part of the lambda calculus or church encoding. The lambda calculus is a language of pure functions. Lambda expressions a lambda calculus expression is defined as e x variable. 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. This data type can be encoded in the lambda calculus. A generic version of church boolean and its boolean logic will be introduced later in the church pair part. In the pure lambda calculus, the only values are functions \x.
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