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