Full abstraction in the lazy lambda calculus pdf

The rst basic operation of the calculus is application. This model is shown to be fully abstract with respect to applicative simulation. D d we set up the domain logic corresponding to the same type expression. In this paper we deene the lazy lambda calculus with constants, which extends abramskys pure lazy lambda calculus. We propose an extension of lambda calculus for which the berarducci trees equality coincides with observational equivalence, when we observe rootstable or rootactive behavior of terms. Therefore, we can say that it is equivalent to turing machines. The \\ lambda\ calculus is, at heart, a simple notation for functions and application. This also shows that the natural embedding of abramskys lazy lambda calculus into the callbyneed lambda calculus with letrec is an isomorphism between the respective termmodels. Similarity implies equivalence in a class of nondeterministic callbyneed lambda calculi. Pdf full abstraction in the lazy lambda calculus luke ong. The lazy lambda calculus in a concurrency scenario core. We present a calculus that captures the operational semantics of callby need. Games and full abstraction for the lazy lambda calculus. The result is then extended to the full calculus by means of a taylor expansion formula.

Games and full abstraction for the lazy lambdacalculus core. Their combined citations are counted only for the first article. The calculus with resources is a nondeterministic refinement of lazy. In this paper, we study a functionality theory in the style of coppo et al. For example, the standard binders allow only one argument of a binary function to be bound. In order to obtain this result we introduce a new category of games, the monotonic games, whose main characteristic consists in. We prove that congruence and coincidence with the corresponding context relation hold for both bisimilarity and similarity, the latter known to be impossible in sequential languages.

A model of the lazy calculus, a typefree functional language based on evaluation to weak head normal form, is given in g, yielding an extensional model in e. The pure lambda calculus does not have a concept of named constants since all atomic lambda terms are variables, but one can emulate having named constants by setting aside a variable as the name of the constant, using abstraction to bind that variable in the main body, and apply that abstraction to the intended definition. Programs of the pure lazy lambda calculus 2, 15 are closed. Lambda calculus 2 outline syntax of the lambda calculus abstraction over variables operational semantics beta reduction substitution programming in the lambda calculus representation tricks.

It arose in connection with milners encoding of lazy lambda calculus into. Full abstraction in the lazy lambda calculus pdf download. We shall see that by contrast with this situation, the lazy. However, there is a fundamental mismatch between the standard theory of sensible lambda calculus and the practice of lazy evaluation which is a distinctive feature of functional programming. This calculus forms a model for modern lazy functional programming languages.

Lambda calculus cs 152 spring 2020 harvard university thursday, february 20, 2020. A beginners look at lambda calculus perpetual enigma. We study how applicative bisimilarity behaves when instantiated on a callbyvalue probabilistic \\lambda \calculus, endowed with plotkins parallel disjunction operator. Interaction categories and the foundations of typed concurrent programming. Full abstraction in the lazy lambda calculus, when was the last time you read a book or an abstract magazine article. The full version of the typed lambda calculus fits into montagues intensional logic with its type theory. I \lazy languages, such as haskell, typically use cbn. The expression f a or fa denotes the data f considered as algorithm applied to the data aconsidered as input. A lambda expression that cannot be reduced further, by either. The lazy lambda calculus in a concurrency scenario citeseerx. Picalculus, dialogue games and full abstraction pcf. The strategies therein are historyfree and satisfy a monotonicity condition. Games and full abstraction for the lazy lambdacalculus 1995.

Deadlock sensitive types for lambda calculus with resources. 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. Intuition a function is a rule for determining a value from an. The lazy lambda calculus in a concurrency scenario. Lambda calculus 1 lesson2 lambda calculus basics 11002 chapter 5. See 6, 7 for a more indepth discussions about these restrictions. The main ideas are applying a function to an argument and forming functions by abstraction. Viewed pu rely as a naming device, however, it is a straighforward extension of ordinar y mathematical notation.

This cited by count includes citations to the following articles in scholar. It was introduced by the mathematician alonzo church in the 1930s as part of his. I full nondeterministic betareduction programming in the lambdacalculus multiple arguments above, we wrote a function double that returns a function as an argument. Non conservativity of embeddings manfred schmidtschauss 1and elena machkasova2 and david sabel 1 dept. He has made contributions to the areas of domain theory, the lazy lambda calculus, strictness analysis, concurrency theory, interaction categories.

In part ii, we recast the full abstraction problem in the lazy. The syntax of basic \\ lambda\ calculus is quite sparse, making it an elegant, focused notation for representing functions. Lambda calculus explained everything explained today. Lambda calculus is a language with clear operational and denotational semantics capable of expressing algorithms. Pdf game semantics for the pure lazy lambdacalculus. I if we add additional data types and operations such as integers and addition, we have an applied calculus. This is, so far as we know, the first purely semantic construction of a fully abstract model for a reflexivelytyped sequential language. Programs of the pure lazy lambda calculus 2, 15 are closed terms, and. The yrst fully abstract game model of the pure lazy lambda calculus was constructed by the second author in 6.

A fresh look at the lambdacalculus drops schloss dagstuhl. I full nondeterministic betareduction programming in the lambda calculus multiple arguments above, we wrote a function double that returns a function as an argument. Full abstraction in the lazy lambda calculus sciencedirect. Samson abramsky simple english wikipedia, the free.

Lambda calculus is commonly thought to be the basis for functional programming. Lesson2 lambda calculus basics university of chicago. Moreover, the lambda abstraction with the standard tools is full of restrictions. Lambda calculus is universal in the sense that any computable function can be expressed and evaluated using this formalism. It was first introduced by mathematician alonzo church in the 1930s as part of his. It is a universal model of computation that can be used to simulate any turing machine.

From the universality theorem, it is a small step to show that the model is fully abstract for the lazy lambda calculus. The text is unique in that it presents a new calculus parametric lambda calculus which can be instantiated to obtain already known lambdacalculi. W e shall see that by contrast with this situation, the lazy calculus we shall. Lambda calculus and types department of computer science. Introduction to the lambda calculus iowa state university. An introduction historically, lambda calculus and combinatory logic were invented and developed separately, starting in the 1930s. Proving congruence of bisimulation in functional programming languages. Typed lambda calculi are weaker than the untyped lambda calculus, which is the primary subject of this article, in the sense that typed lambda calculi can express less than the untyped calculus can, but on the other hand typed lambda calculi allow more things to be proved. In this paper we present a fully abstract game model for the pure lazy. March 18, 2004 abstract the lambda calculus can appear arcane on. The lambda calculus stanford encyclopedia of philosophy. Nov 10, 2015 we study how applicative bisimilarity behaves when instantiated on a callbyvalue probabilistic \\lambda \calculus, endowed with plotkins parallel disjunction operator.

By manual analysis of the various combinations we can. On applicative similarity, sequentiality, and full abstraction. The book contains a completely new presentation of classical results in the field of lambda calculus, together with new results. A short introduction to the lambda calculus achim jung. From the viewpoint of concurrent programming, this was a natural step to take since one of the main features of this programming setting is the consumption of resources which. Are your daily reading habits directed against tweets, facebook updates, or directions to your instant oatmeal pack. Full abstraction in the lazy lambda calculus full description. Oct 25, 2014 lambda calculus is universal in the sense that any computable function can be expressed and evaluated using this formalism. Simulation in the callbyneed lambdacalculus with letrec. The course is entitled lambda calculus and types and it appears in various forms. This paper proposes modification of a number of key notions in the sensible. Pdf full abstraction in the lazy lambda calculus luke.

Lambda by alonzo church and his illustrious students at princeton combinatory by haskell curry and colleagues, some of them during his sojourns in holland and gottingen. This is motivated by a mismatch between the standard i. Pdf the benefit of being nonlazy in probabilistic \lambda. Home conferences fpca proceedings fpca 95 picalculus, dialogue games and full abstraction pcf. The lazy lambda calculus samson abramsky department of computing imperial college of science and technology march 6, 2006 1 introduction the commonly accepted basis for functional programming is the. Full abstraction for lambda calculus with resources and. Lambda abstraction the only other thing in the lambda calculus is lambda abstraction. Application and abstraction the rst basic operation of the calculus is application. If youre one of the many people who do not have a habit of reading regularly, you might miss out. We prove that congruence and coincidence with the corresponding context relation hold for both bisimilarity and similarity, the latter known to be impossible in. However, the difference here is that lambda calculus emphasizes the use of transformation rules, and it does not care about the actual machine implementing them. 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.

Pdf games and full abstraction for the lazy lambdacalculus. We consider the probabilistic applicative bisimilarity pab a coinductive relation comparing the applicative behaviour of probabilistic untyped. Samson abramsky frs, frse born 12 march 1953 is a computer scientist who holds the christopher strachey professorship at the department of computer science formerly the computing laboratory, university of oxford. We start from a characterization of formula presented in sangiorgi d. The parts we will use the most will be the type theory, the lambda calculus rule 7, and the rule. Lecture notes on the lambda calculus dalhousie university.