Recall that in lambda calculus, we have function expressions and function applications. Simply typed lambda calculus, implemented in ocaml. It is a universal model of computation that can be used to simulate any turing machine. Various results on the lambda calculus involving bigstep evaluation. If youre a beginner you can start with some basics. An introduction to functional programming through lambda calculus greg michaelson department of computing and electrical engineering heriotwatt university riccarton campus edinburgh eh14 4as 2 preface overview this book aims to provide a gentle introduction to 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 and 20. What is the relationship between lambda calculus and. Lambda calculus is the smallest programming language. First, through performing statistical sampling experiments with randomly generated terms, we can study interesting properties of lambda calculus terms. Well, lambda calculus is basically a simple notation for functions and applications in mathematics and computer science.
Ppl lecture lambda calculus alpha,beta,eta reduction. Evaluating core ocaml in the substitution model 10. Citeseerx document details isaac councill, lee giles, pradeep teregowda. The application operator binds tighter than abstraction operator, so lx.
Simple typed lambda calculus typechecking function in ocaml. Lambda calculus lambda calculus stanford university. It includes the automatic translation of a subset of ocaml expressions into lambda expressions. I believe ocaml is a good starting point into functional programming for a c programmer, and that scheme is a good starting point to dive into the lambda calculus. How do you define a simple lambda calculuslike dsl in ocaml using gadts. If you don tcant use opam, download and compile the source tarball github mirror. Specifically, i cant figure out how to properly define the type checker to translate from an untyped ast to a typed ast nor can i figure out the correct type for the context and environment. Reduction consists of replacing a part pof eby another expression p0 according to the given rewrite rules. That is, a lambda expression can have a single restid that is not surrounded by parentheses. Using, understanding, and unraveling the ocaml language inria. Simple lambda calculus dsl using gadts in ocaml stack. As we saw in my previous post, the only building blocks available are functions, variables, and expressions. Originally developed in order to study some mathematical properties of e ectively computable functions, this formalism has provided a strong theoretical foundation. Back in the 1930s, when the theories of computing were finding their footing, alonso church came up with a thing he called the lambda calculus as a way of modeling what is meant by a computation.
Simply typed lambda calculus, implemented in ocaml onionhoneystlc ocaml. It includes the automatic translation of a subset of ocaml expressions into lambdaexpressions. The lambda calculus is an abstract mathematical theory of computation, involving. If that seems obvious, its because youre used to it from ocaml. Computation in the lambda calculus takes place by substitution. The whole point of programming in a functional language is to define functions.
The ocaml dialect of ml is one of the few attempts to combine the two paradigms. The lambda calculus, treated in this book mainly in its untyped version, consists of a collection of expressions, called lambda terms, together with ways how to rewrite and identify these. You must implement the last six of the listed functions in lambda. A lambda calculus expression is defined as e x variable. Weve already seen ocamls fun lambda expression, and we have seen how to name values, and since a function is a value like any other, that means we already know how to define functions. A tutorial introduction to the lambda calculus raul rojas fu berlin, ws9798 abstract this paper is a short and painless introduction to the calculus. Selective calculus is a conservative extension of lambda calculus which, by labeling abstractions and applications, enables some. Syntactically, expressions of the lambdacalculus written with letter a are of three. In the lambda calculus, each lambda abstraction binds a single variable, and the lambda abstractions body may be another lambda abstraction. Now we need to define the substitution operation for the lambda calculus. The formalism became popular and has provided a strong theoretical foundation for the family of functional programming languages. Intuitions suppose we want to describe a function that.
Do you write caml or caml, and ocaml, ocaml or ocaml. This book itself is purely theoretical and principally aimed for researchersstudents of its field. Evaluating the lambda calculus in the environment model 10. We explore the problem of generating lambda calculus terms of a given size uniformly at random. Back in school, you must remember studying differential and integral calculus. Although the lambda calculus arose as a branch of mathematical logic to provide a foundation for mathematics, it has led to considerable rami. The lambda calculus is not associated with calculus which ought to be called analysis instead.
It is a mathematical system for studying the interaction of functional abstraction and functional application. It supports many advanced features such as recursion, userdefined operators and multiple evaluation strategies, all based on the pure calculus. Simple typed lambda calculus typechecking function in ocaml typecher. With a little syntax sugar, lambda calculus becomes a practical programming language. It was introduced by the mathematician alonzo church in the 1930s as part of his. Lambda abstraction the only other thing in the lambda calculus is lambda abstraction. Your next task is to implement a few functions related to lambda calculus, which you learned about in cmsc 330.
The y combinator no, not that one ayaka nonaka medium. I am not sure who the user is supposed to be in your question, but if you want to implement lambda calculus with ocaml and encode lambda with fun directly, you are going to inherit the evaluation order of ocaml the one i linked. Church developed the lambda calculus in the 1930s as a theory of functions that provides rules for manipulating functions in a purely syntactic manner. Languages defines how and when terms are evaluated, which includes how functions are applied. The lambda calculus, its syntax and semantics studies in. In lambda calculus, lambda expressions bound variables can be substituted with different name. First real ocaml project to discover what is lambda calculus.
The lambda calculus is a formal mathematical system for expressing the notion of computation. If you dontcant use opam, download and compile the source tarball github mirror. Currying a function of two inputs turns that function into a function with one input by passing one of the inputs into it. A small web application for learning lambdacalculus. Already, our factorial example above is shorter than equivalent code in many high. The lambda calculus is often called the assembly language of functional programming, and variations and extensions on it form the basis of many functional compiler intermediate forms for languages like haskell, ocaml, standardml, etc. Weve got quite far into the tutorial, yet we havent really considered functional programming. In the parts conversion, reduction, theories, and models the view is respectively algebraic, computational, with more coinductive identifications, and. An educational web application for experimenting with lambda calculus and with the reduction strategies which are used in functional languages ocaml, scheme, haskell. Spare a thought for students struggling to make turing machines do simple tasks. An educational web application for experimenting with lambdacalculus and with the reduction strategies which are used in functional languages ocaml, scheme, haskell. Another way of thinking about more than one input is currying. The resulting function accepts any number of arguments, and the arguments are put into a list bound to restid. Functional programming what is functional programming.
The lambda calculus can be thought of as the theoretical foundation of functional programming. It has a significant impact in the field of programming language theory. The lambda calculus the lambdacalculus i if our previous language of arithmetic expressions was the simplest nontrivial programming language, then the. It captures some of the essential, common features of a wide variety of programming languages.
All of the features given so far rich data types, pattern matching, type inference, nested functions you could imagine could exist in a kind of super c language. Ppl lecture lambda calculus alpha,beta,eta reduction with example hindi in. A beginners look at lambda calculus perpetual enigma. Lambda calculus is a notation for describing mathematical functions and programs. In mathematics and computer science in general, a fixed point of a function is a value that is. Lambda is certainly fun, but what i mean here is that the. Want to be notified of new releases in onionhoneystlc ocaml. This formalism was developed by alonzo church as a tool for studying the mathematical properties of effectively computable functions.
Currying recall that in lambda calculus, a function can have more than one input, each preceded by a. Contribute to na0214linear lambda calculus development by creating an account on github. Most functional programming languages are based upon the lambda calculus. In the classical untyped lambda calculus, every function has a fixed point. A small web application for learning lambda calculus. Fundamental to all functional languages is the most atomic notion of composition, function abstraction of a single variable. Heres how to multiply two numbers in lambda calculus.
1581 543 1105 1471 867 339 580 299 377 423 664 474 704 919 814 429 579 1187 865 934 1226 299 596 1230 898 524 726 355 3 655 938 970 598 620 548 1004 122