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. A short introduction to the lambda calculus achim jung. As regards the pure lambda calculus no other primitives than the monadic lamb. In some sense, extensionality is already present in the emergent algebra moves. Request pdf some results on extensionality in lambda calculus in this paper we consider the problem due to a.
The most interesting aspect is a system of substitution primitives and an ac. We conclude that the lambda calculus is algebraic, in the sense that its canonical class of models is the class of lambda algebras. Appendix a studies the intermediatecoercion calculus used in section 4. Lambdacalculus and combinators, an introduction combinatory logic and. The lambda calculus, its syntax and semantics studies in. The name comes from the fact that it can be used for representing terms and reductions from untyped lambda calculus. Graphic lambda calculus consists of a class of graphs endowed with moves between them. In lambda calculus, there are 3 substitution rules for expression to be reduced. The \\ lambda\ calculus is, at heart, a simple notation for functions and application. The lambda calculus stanford encyclopedia of philosophy.
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. A lambda expression that cannot be reduced further, by either. March 18, 2004 abstract the lambda calculus can appear arcane on. In fact, this book successfully compiles almost all results on typefree lambda. A valid lambda calculus expression is called a lambda term. Originally developed in order to study some mathematical properties of e ectively computable functions, this formalism has provided a. While we have concentrated on models of the untyped lambda calculus, similar considerations apply to the algebraic modeling of any language with variables and binders. This explains why the lambda calculus has been called a calculus. Topics covered in these notes include the untyped lambda calculus, the. Functional languages are just the lambda calculus with a more palatable syntax. Lambda calculus as a formal system free and bound variables. 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. In lambda calculus, lambda expressions bound variables can be substituted with different name. Church 1936 invented the lambda calculus, written calculus.
Lambda calculus 5 scope, free and bound occurences lx. Combinatory logic and lambdacalculus, originally devised in the 1920s, have since developed into linguistic tools, especially useful in programming languages. Csc173 lambda calculus exercises 1 practice and practical. It was introduced by the mathematician alonzo church in the 1930s as part of his research into the. Lambda calculi with types, handbook of logic in comp. Lecture notes on the lambda calculus peter selinger department of mathematics and statistics university of ottawa abstract this is a set of lecture notes for the course mathematical foundations of computation, which i taught at the university of ottawa in fall 2001. I want to discuss here the introduction of extensionality in the graphic lambda calculus. In the simplytyped lambda calculus, term constructs can be prooftheoretically classi. In typed lambda calculus, functions can be applied only if they are capable of. Through lambda calculus, is pointed at from both \readings and \resources links and the course schedule. However, the etareduction from untyped lambda calculus needs a new move. Viewed pu rely as a naming device, however, it is a straighforward extension of ordinar y mathematical notation. The lambda calculus is a simple programming language a model of computation akin to turing machines and recursive functions, through which we can study the computability and complexity of functions and predicates. This book itself is purely theoretical and principally aimed for researchersstudents of its field.
The course is entitled lambda calculus and types and it appears in various forms. The previous exams at the \resources link might be helpful. It is a universal model of computation that can be used to simulate any turing machine. The following three rules give an inductive definition that can be applied to build all syntactically valid lambda terms. 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. They have since evolved into important tools for the development and study of programming languages. Finally, in 1960s, dana scott, then a professor at oxford university, and himself a former student of alonzo church, discovered a meaning for the lambda calculus. Lambda calculus is turing complete, that is, it is a universal model of computation that can be used to simulate any turing machine. The syntax of basic \\ lambda\ calculus is quite sparse, making it an elegant, focused notation for representing functions. We extend the equivalence relation with the ext rule. The authors previous book served as the main reference for introductory courses on lambdacalculus for over 20 years. Lambda calculus part of functional programming joseph cordina february 5, 2008 1 introduction function represented by its graph i.
Some results on extensionality in lambda calculus benedetto intrigilaa. In the pure lambda calculus, the only values are functions \x. The lambda calculus is algebraic dalhousie university. Lecture notes on the lambda calculus dalhousie university. Lambda calculus 6 beta reduction computation in the lambda calculus takes the form of betareduction. Extensional lambda calculus with surjective pairing 5 from the point of view of term rewriting. Pdf in this work we provide alternative formulations of the concepts of lambda theory and extensional theory without introducing the notion of. Lambda calculus as a programming language computability background. Programming in the lambdacalculus multiple arguments above, we wrote a function double that returns a function as an argument. A calculus is a method of moving symbols, not the undergrad course. Actually, lambda calculus is a generic label applying to several distinct equational theories. A tutorial introduction to the lambda calculus raul rojas fu berlin, ws9798 abstract this paper is a short and painless introduction to the calculus. It might be considered a visual language in the sense of erwig 1.
The main ideas are applying a function to an argument and forming functions by abstraction. These two settheoretical postulates are analogous to the postulates of beta and eta conversion in the lambda calculus, where the role of set abstraction is played by functional abstraction, and. Roughly speaking, the rule states that two terms, viewed as functions. Extensionality in graphic lambda calculus chorasimilarity. The idea is due to sch on nkel 1924 but is often called currying, after h. 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. It has relevant and representational problems at the end of the chapters, all with answers in the back. He is i think talking about models of the untyped lambda calculus without the following axiom of extensionality. One problem we found was that, in contrast to combinators, for hequality a n explicit definition of model did not exist. Modern processors are just overblown turing machines. In the parts conversion, reduction, theories, and models the view is respectively algebraic, computational, with more coinductive identifications, and. Lecture notes on the lambda calculus nanjing university. Values are lambda expressions that have properly finished being evaluated there is nothing more to do.
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