Lambda calculi with types handbook of logic in computer science

Download pdf an introduction to lambda calculi for computer. This item appears in the following collections open access publications 52423 freely accessible full text publications. Handbook of logic in computer science, ii, oxford university press 1992, pp. People who program and team sometimes are also good at mathematicians. The baeendregt of the lambda calculus in logic and computer science h barendregt bulletin of symbolic logic 3 2, mahady hasan rated it it was amazing apr 06, byerly journal of symbolic logic 47 1. Church, a set of postulates for the foundation of logic ann. This handbook is a sixvolume work covering all major areas of the application of logic to theoretical computer science.

Reynolds and later jeanyves girard formally developed this notion of polymorphism as an extension to lambda calculus called the polymorphic lambda calculus or system f. A formal survey of the various kinds of explicitly and implicitly typed lambda calculi i. Besides its historical role in computability theory it has had significant influence on. In this article, we address the problem of expansion with respect to rules of a calculus with explicit substitution. Church to logic is his invention of the lambda calculus. Logic is now widely recognized as one of the foundational disciplines of computing, and its applications reach almost every aspect of the subject, from software engineering and hardware to programming languages and ai. The ski cube is a way of presenting a number of related calculi and logics which are obtainable from pure combinatory logic by one or more of three kinds of extension the origin of the cube is pure combinatory logic c the three kinds of extension considered are called the axes of the cube the cube presents eight systems or kinds of system which are arranged at the vertices of the cube.

The undecidability of the third order dyadic unification problem, information and control 382, 170178 1978. Polymorphic types allow types allowing all usual mathematical entities to be constructed although without the. Maibaum, editors, handbook of logic in computer science, volume 2. Barendregts cube and programming with dependent types eric lu i. Initial algebra semantics for lambda calculi springerlink.

Nov 26, 2010 the inspiration for this form of abstraction mechanism comes from mathematical logic. Citeseerx document details isaac councill, lee giles, pradeep teregowda. A proper study of these foundations leads to a better understanding of some of the fundamental issues in computer science. The lambdacalculus, combinatory logic, and type systems. In the branches of mathematical logic known as proof theory and type theory, a pure type system pts, previously known as a generalized type system gts, is a form of typed lambda calculus that allows an arbitrary number of sorts and dependencies between any of these. The lambda calculus, volume 103 of studies in logic and the foundations of mathematics. Any parametrically polymorphic function is necessarily restricted in what it can do, working on the shape of the data instead of its value, leading to the concept of parametricity.

Handbook of logic in computer science, volume 2, oxford, new york. The first six chapters concern this calculus and cover the basic theory, reduction, models, computability, and the relationship between the lambdacalculus and combinatory. Mathematics department, swansea university, swansea sa2 8pp, uk. Under the formulaeastypes correspondence, the simplytyped lambda calculus has as its matching logic propositional calculus, adding dependent types adds universal and existential quantification at higher finite types to its logic. In handbook of logic in computer science, vol 2, oup, 1992, pp 117309. The semantics is realized as the unique morphism from the initial algebra to an appropriate semantic algebra.

The lics symposium is an annual international forum on theoretical and practical topics in computer science that relate to logic, broadly construed. An introduction to lambda calculi for computer scientists. Lambda calculi with types, handbook of logic in computer science vol. Academic publications 177407 academic output radboud university. Im referring to the postscript version available on the web. Go to page top go back to contents go back to site navigation. The calculus a pure calculus of functional abstraction and function application, with applications throughout logic and computer science. This handbook with exercises reveals the mathematical beauty of formalisms hitherto mostly used for software and hardware design and verification. Barendregt, lambda calculi with types, handbook of logic in computer science, oxford univ. Barendregts cube and programming with dependent types. The lambda calculus forms a prototype universal programming language, which in its untyped version is related to lisp, and was treated in the first authors classic the lambda calculus 1984.

The handbook of logic in computer science is a sixvolume, internationally authored work which offers a comprehensive treatment of the application of logic to theoretical computer science. Ordinary and polymorphic typed lambda calculi are constructed as initial algebras for suitable endofunctors. A typed lambda calculus with callbyname and callbyvalue iteration, manuscript. The first six chapters concern this calculus and cover the basic theory, reduction, models, computability, and the relationship between the lambda calculus and combinatory logic. Barendregt, lambda calculi with types, handbook of logic in computer science, volume ii, oxford university press, 1993. We extend the framework of pure type systems with subtyping. Under the formulaeas types correspondence, the simplytyped lambda calculus has as its matching logic propositional calculus, adding dependent types adds universal and existential quantification at higher finite types to its logic. Pdf an introduction to lambada calculi for computer. Lambda calculi with types handbook of logic in computer science. Lambda calculi with types, booktitle handbook of logic in computer science, year.

A tutorial on proof systems and typed lambdacalculi. Mathematics, logic, categories and sets, programming languages and applied logic. Bohm, editors, fifth international conference on automata, languages and programming, lecture notes in computer science 62 1978, pp. Types the calculus is good tool for exploring type systems, invaluable both in the foundations of mathematics and for practical programming languages. Type systems, chapter 18 of the handbook of automated reasoning vol 2, eds. The inspiration for this form of abstraction mechanism comes from mathematical logic. Buy an introduction to lambda calculi for computer scientists by chris hankin isbn. Pure type systems with subtyping proceedings of the 4th.

Introduction to generalized type systems, journal of functional programming, 1 1991, pp 125154. So when they create programming languages or features they recognize patterns and abstractions from mathematics. Each volume is comprised of an average of five 100page monographs and presents an indepth overview of one of the fields major topics. Spend a moment from your computer, open the lambda calculus with types book, and rebuild your soul a bit. Combinatory logic and lambdaconversion were originally devised in the 1920s for investigating the foundations of mathematics using the basic concept of operation instead of set. Papers from the 1974 summer research institute of the australian mathematical society, monash university, australia, lecture notes in mathematics, vol. Barwise, handbook of mathematical logic, 90, northholland, berlinnew york 1977, pp. Lambda calculi with types, handbook of logic in computer science, volume 2, background. Get special knowledge after reading this book cambridge university press. Higher inductive types in programming, journal of universal computer.

Newman, which decides whether a lambda calculus term is typable without actually computing its principal type. An arithmetical proof of the strong normalization for the lambdacalculus with recursive equations on types. The impact of the lambda calculus in logic and computer science. In this paper, we develop a formal model for probabilistic reasoning about lambda terms with intersection types, which is a combination of lambda calculus and probabilistic logic. Prasad, editors, logic and its applications 7th indian. We present the genesis of this theory and its two major areas of application. The framework can be seen as a generalisation of barendregts lambda cube, in the sense that all corners of the cube can be. Typed lambda calculi are closely related to mathematical logic and proof theory via the curryhoward isomorphism and they can be considered as the internal language of classes of categories, e. The formalism has since been extended with types and used in functional programming haskell, clean and proof assistants coq, isabelle, hol, used in. Home browse by title books handbook of logic in computer science vol. Combinatory logic and lambda conversion were originally devised in the 1920s for investigating the foundations of mathematics using the basic concept of operation instead of set. Synopsis the lambdacalculus lies at the very foundations of computer science.

Barendregt, lambda calculi with types, in handbook of logic in computer science, volume ii, s. They have now developed into linguistic tools, useful in several branches of logic and computer science, especially in the study of programming languages. The handbook of logic in computer science is a multivolume work covering all the major areas of application of logic to theoretical computer science. Joseph added it aug 02, who am i to criticize one of the greatest of all time. In handbook of logic in computer science, volume 2, oxford, new york. Pdf the impact of the lambda calculus in logic and. Lambda calculi internet encyclopedia of philosophy. Handbook for logic in computer science, abramsky, s. Everyday low prices and free delivery on eligible orders. The emergence in the past ten years of a coherent field of research often named. Linked bibliography for the sep article type theory by. The purpose of this paper is to give an exposition of material dealing with constructive logic, typed. Maibaum, editors, handbook of logic in computer science, volume ii.