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1. Readings: Theory Of Computation
The lambda calculus Its Syntax and Semantics. NorthHolland (Amsterdam, 1981). Although it does not address Church s Thesis and effective Computability
Miser Project Readings
Theory of Computation
L ast updated 2003-02-10-22:35 -0800 (pst)
see also
Readings in Logic
Readings in Mathematics
Readings in Philosophy
Barendregt, Hendrik Pieter. The Lambda Calculus: Its Syntax and Semantics . North-Holland (Amsterdam, 1981). ISBN 0-444-85490-8. Studies in Logic and the Foundations of Mathematics, vol. 103.
This is one of my fundamental sources on the lambda calculus. I want to pay particular attention to combinatory logic (CL) and combinatory algebra (CA). My notes on this book focus on that.
Part I. Towards the Theory
1. Introduction
2. Conversion
3. Reduction
4. Theories 5. Models Part II. Conversion 6. Classical Lambda Calculus 7. The Theory of Combinators 8. Classical Lambda Calculus (continued) I -Calculus Part III.

2. MainFrame: The Lambda-calculus, Combinatory Logic, And Type Systems
Computability. The lambda calculus was first devised by Alonzo Church, first to provide a foundation for mathematics and then to show the existence of
The Lambda-calculus, Combinatory Logic, and Type Systems
Three interrelated topics at the heart of logic and computer science. The -Calculus A pure calculus of functional abstraction and function application, with applications throughout logic and computer science. Types The -calculus is good tool for exploring type systems, invaluable both in the foundations of mathematics and for practical programming languages. Pure Type Systems A further generalisation and systematic presentation of the class of type systems found in the -cube. Combinators Combinatory logic shows that bound variables can be eliminated without loss of expressiveness. It has applications both in the foundations of mathematics and in the implementation of functional programming languages. Programming Languages The connections between the lambda-calculus and programming languages are diverse and pervasive. Type systems are an important aspect of programming language design. The -cube A graphical presentation of the relationship between combinatory logic, lambda calculi and related logical systems. The -cube A graphical presentation of the relationship between various typed -calculi, illuminating the structure of Coquand's Calculus of Constructions.

3. Lambda The Ultimate Lc
The paper describes the history of the lambda calculus, and several of its uses. Along the way it discusses important notions like Computability and
Lambda the Ultimate The Programming Languages Weblog - join today! Home FAQ Feedback Departments ... Genealogical Diagrams
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A Security Kernel Based on the Lambda-Calculus (via our really great dicussion group) This report describes Scheme 48, a programming environment whose design is guided by established principles of operating system security. Scheme 48's security kernel is small, consisting of the call-by-value lambda-calculus with a few simple extensions to support abstract data types, object mutation, and access to hardware resources. Each agent (user or subsystem) has a separate evaluation environment that holds objects representing privileges granted to that agent. Because environments ultimately determine availability of object references, protection and sharing can be controlled largely by the way in which environments are constructed.
Posted to LC by Ehud Lamm on 5/21/04; 4:31:48 AM
Y derived I don't recall seeing this derivation of the applicative order Y combinator (from Richard Gabriel) mentioned here. It's short and sweet.

4. Models Of Comp
The fundamental ideas of (non)Computability and complexity will be presented. There will also be a section on the lambda calculus and its connection with
School of Computer Science Home Internal Modules SYLLABUS PAGE, 2007/08
Models of Computation
Level 2/I Dr V Sorge 10 credits in Sem2 Programmes Modules Links Outline ... Detailed Syllabus The School of Computer Science Module Description is a strict subset of this Syllabus Page. (The University module description has not yet been checked against the School's.)
Relevant Links
For more information (like notes, handouts) see the module web page at
The module will introduce various automata theoretic models of computation and discuss their practical and theoretical significance. Finite automata, grammars and stack automata and Turing machines will be introduced. The fundamental ideas of (non-)computability and complexity will be presented. There will also be a section on the Lambda Calculus and its connection with Functional Programming.
The aims of this module are to:
  • Introduce a variety of formal models of computation. Explain the significance of formal models from a practical and a foundational point of view. Make familiar with the fundamental results in the theory of computation.

5. Domains And Lambda Calculi (book Announcement)
A basic link between Scott continuity and Computability (the This is a simply typed lambdacalculus extended with fixpoints and arithmetic operators.
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Domains and Lambda Calculi (book announcement)
Book announcement: DOMAINS AND LAMBDA CALCULI by Roberto AMADIO and Pierre-Louis CURIEN Cambridge Tracts in Theoretical Computer Science 46, ISBN 0-521-62277-8, Cambridge University Press We are pleased to announce that the book is now available. Order informations can be found at and for North-America at

6. Re^2: Pissed Off About Functional Programming
Church is the one who defined what he called effective calculabilty (which we call Computability today) and linked lambda calculus to the murecursive

7. Manzonetto Giulio - Ph.D Student In Computer Science - Home Page
My main interest in computer science is in Computability theory, lambda calculus, topology, term rewriting systems, abstract semantics and functional
Manzonetto Giulio
x .gmanzone x
Dottore Magistrale
Ph.D. student in Computer Science
Room 5A09, tel. (+33)(0)1 44 27 69 30 Fo lo manto al caco macaco
e'l truffo sgarruffo a lo spino del baco
(Marius, poeta efficace)
  • 01 nov: Alea iacta est: today, I've sent my Ph.D. thesis to the referees. 16 oct: The only four people with who I wouln't like to exchange myself are my referees. 08 oct: The first days of november I must send my Ph.D. Thesis to the referees. I've never been stressed as in this period. 10 sep: I'm in Lausanne to attend the conference CSL 2007. I will held a talk on ''lambda theories of effective lambda models'' the 12th september, and another one titled ''not enough points is enough'' the 13th september. 1 sep: Happy New (psycological) Year! The summer is passed quickly, and I came back in Paris for another exciting year of research. 18 jul: During the academic year 2007/2008 I will be teacher assistant at Paris 7 for the following courses: Intelligence Artificielle (M1), Algorithmique (L3BI), Analyse syntaxique et Compilation (L3) and Suivi projets longs (M1). Summing-up: 101 hours of fun. 16 jul: The following people have accepted to be the referees of my thesis: H.P. Barendregt

8. Functional Programming With Haskell
The lambdacalculus grew out of an attempt by Alonzo Church and Stephen Kleene in the early 1930s to formalize the notion of Computability.
Functional Programming Using Haskell
Wade Estabrooks Michael Goit Mark Steeves
Table of Contents
1. Introduction to Haskell
2. Evaluation of the Language 2.1 Readability 2.2 Writability ... 3. Our Program Appendix: Source Code Appendix A - source code for the Dictionary module. Appendix B - source code for the Words module. Appendix C - source code for the Fixwords module.
1. Introduction to Haskell
Haskell is a general purpose, purely functional programming language incorporating many recent innovations in programming language design. Haskell provides higher-order functions, non-strict semantics, static polymorphic typing, user-defined algebraic datatypes, pattern-matching, list comprehensions, a module system, a monadic I/O system, and a rich set of primitive datatypes, including lists, arrays, arbitrary and fixed precision integers, and floating-point numbers. Haskell is both the culmination and solidification of many years of research on lazy functional languages. True, this definition is a bit long, but it almost completely shows the power that Haskell has as both a functional language, and as a programming language in general. Because Haskell is a purely functional language is has certain characteristics, and also because Haskell is modern language, many historical mistakes made with language design have been avoided. Recent advances in Typed-Lambda Calculus have formed a basis for Haskell, and the addition of very strong typing to a functional language makes Haskell (unlike LISP) "safe" to use. The typing rules also allow a program Haskell to be validated.

9. Lambda Calculus - Wikipedia, The Free Encyclopedia
Then he assumes that this predicate is computable, and can hence be expressed in lambda calculus. Building on earlier work by Kleene and constructing a
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Lambda calculus
From Wikipedia, the free encyclopedia
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You can improve this article by introducing more precise citations. It has been suggested that Church encoding be merged into this article or section. ( Discuss In mathematical logic and computer science lambda calculus , also λ-calculus , is a formal system designed to investigate function definition, function application and recursion . It was introduced by Alonzo Church and Stephen Cole Kleene in the as part of a larger effort to base the foundation of mathematics upon functions rather than sets (in the hopes of avoiding obstacles like Russell's Paradox ). The

10. Pietro Di Gianantonio Home Page
University of Udine Real number Computability, semantics of concurrency, lambda-calculus.
Pietro Di Gianantonio
Dipartimento di Matematica e Informatica Università di Udine
Real number computability, semantics of programming languages, lambda-calculus, game semantics, formal proofs.
Teaching (Didattica)
Orario di Ricevimento:
  • mercoledì ore 13:30 - 15:30, o su appuntamento.
2nd floor, room 1, Stecca Nord,
phone: fax: mail:
pietro at dimi dot uniud dot it
Dipartimento di Matematica e Informatica Università di Udine Via delle Scienze, 206 33100 Udine - Italy

11. Lambda Calculus
1 S. Abramsky and C.H. L. Ong. Full abstraction in the lazy lambda calculus. Information and Computation, 105159-267, 1993. 2 H. Barendregt.
Lambda Calculus
C.-H. L. Ong Sixteen-hour lecture course. Final-year computer science undergraduate / MSc
Nature and aim of the course
Lambda calculus is a theory of functions that is central to (theoretical) computer science. It is well known that all recursive functions are representable as lambda terms: the representation is so compelling that definability in the calculus may as well be regarded as a definition of computability. This forms part of the standard foundations of computer science and mathematics. Less familiar are two separate developments one in programming, the other in proof theory in which lambda calculus has played a key role:
  • Lambda calculus is the commonly accepted basis of functional programming languages; and it is folklore that the calculus is the prototypical functional language in purified form.
  • The idea that there is a close relation between proof theory and a theory of functions is an old one. It underlies the Kolmogorov-Brouwer-Heyting interpretation of intuitionistic logic, and the Curry-Howard isomorphism between natural deduction and typed lambda calculus.
We develop the syntax and semantics of lambda calculus along these two themes. The aim of this course is to provide the foundation for an important aspect of the semantics of programming languages with a view to helping enthusiastic research students appreciate (perhaps even begin to address) some of the open problems in the field. The second theme in particular will be followed up by two new courses

12. Lambda
An interesting aspect is that + and even 2 can themselves be defined in terms of lambda calculus it is a complete description of discrete computation.

13. CCCs And The λ-calculus
One of the most wellknown is the lambda calculus, invented by Church and Kleene in the 1930s as a model of computation. Any function computable by the
John Baez
September 28, 2006
Categorical semantics was born in Lawvere's celebrated 1963 thesis on algebraic theories: Algebraic theories are a simple formalism for reasoning about operations that satisfy equations. For example, since the concept of a "group" involves only some operations (multiplication, inverses...) satisfying equations, this concept can be formalized using an algebraic theory called Th(Grp). The role of semantics enters when we consider "models" of an algebraic theory. Loosely speaking, a model is just one of the things the theory seeks to describe. For example, a "model" of Th(Grp) is just a group. Technically, an algebraic theory T is a category with finite products, and a model is a functor that preserves finite products: from T to the category of sets. The basic idea is simple: if for example T = Th(Grp), then Z maps the abstract concept of "group" to a specific set, the abstract concept of "multiplication" to a specific multiplication on the chosen set, and so on, thus picking out a specific group. Dual to the concept of semantics is the concept of syntax , which deals with symbol manipulation. Just as semantics deals with models, syntax deals with "proofs". For example, starting from Th(Grp) we can prove consequences of the group axioms merely by juggling equations. In the case of algebraic theories, the syntax often goes by the name of

14. BOOK-SPRINGER: The Parametric Lambda Calculus: A Metamodel For Computation
It is well known that lambdacalculus is Turing complete, in both its call-by-name and call-by-value variants, i.e. it has the power of the computable
A Metamodel for Computation
by Simona Ronchi Della Rocca and Luca Paolini
Series : Texts in Theoretical Computer Science . An EATCS Series
Springer-Verlag Berlin,Hidelberg,New York,Hong Kong,London,Milan,Paris,Tokyo
2004, XIII, 252 p., Hardcover ISBN: 3-540-20032-0
The lambda-calculus was invented by Church in the 1930s with the purpose of supplying a logical foundation for logic and mathematics.
Its use by Kleene as a coding for computable functions makes it the first programming language, in an abstract sense, exactly as the Turing machine can be considered the first computer machine. The lambda-calculus has quite a simple syntax (with just three formation rules for terms) and a simple operational semantics (with just one operation, substitution), and so it is a very basic setting for studying computation properties.
The first contact between lambda-calculus and real programming languages was in the years 1956-1960, when McCarthy developed the LISP programming language

15. No Title
Miraculously, the lambda calculus model is both simpler than the Turing machine model and more practical for real computation. The programming language Lisp
Note: If you came here from Simon Cozens' article looking for an application of closures, you may be disappointed, because this isn't really an application at all, although it is very interesting. If you want a practical application, you should visit my article about memoization I'm also writing a whole book about practical applications of closures! The outline and samples may interest you I also teach a class called Stolen Secrets of the Wizards of the Ivory Tower that is about practical applications for closures in Perl. I've given this class at the Open Source Conference in 2000 and 2001 and at some other places also. You might like to look at the samples. Thanks for visting my site. Enjoy.
Lambda Calculus
When computer scientists want to study what is computable, they need a model of computation that is simpler than real computers are. The usual model they use involves a Turing Machine , which has the following parts:
  • One state register which can hold a single number, called the state ; the state register has a maximum size specified in advance.
  • 16. Lambda Calculus - A Definition From
    lambda calculus, considered to be the mathematical basis for programming Alonzo Church and Stephen Kleene in the 1930s to express all computable functions.,,sid9_gci341298,00.html
    lambda calculus
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    lambda calculus
    Lambda calculus, considered to be the mathematical basis for programming language, is a calculus developed by Alonzo Church and Stephen Kleene in the 1930s to express all computable�functions.� In an effort to formalize the concept of computability (also known as constructibility and effective calculability), Church and Kleene developed a powerful language with a simple� syntax and few grammar restrictions. The language deals with the application of a function to its arguments (a function is a set of rules) and expresses any entity as either a variable, the application of one function to another, or as a "lambda abstraction" (a function in which the Greek letter�lambda is defined as the abstraction operator). Lambda calculus, and the closely related theories of combinators and type systems, are important foundations in the study of mathematics, logic, and computer programming language.

    17. BOOK ANNOUNCEMENT: The Parametric Lambda Calculus
    BOOK ANNOUNCEMENT THE PARAMETRIC lambda calculus A Metamodel for Computation by Simona Ronchi Della Rocca and Luca Paolini Series Texts in Theoretical
    var addthis_pub = 'comforteagle'; science.mathematics.frogs Top All Lists Date Thread
    BOOK ANNOUNCEMENT: The Parametric Lambda Calculus
    Subject BOOK ANNOUNCEMENT: The Parametric Lambda Calculus - BOOK ANNOUNCEMENT THE PARAMETRIC LAMBDA CALCULUS - A Metamodel for Computation by Simona Ronchi Della Rocca and Luca Paolini Series : Texts in Theoretical Computer Science. An EATCS Series 2004, XIII, 252 p., Hardcover ISBN: 3-540-20032-0 We are pleased to announce that the book is available. Order informations can be found at,11855,5-40356-72-14202886-0,00.html You can found the table of contents at,11996,0-0-45-117802-0,00.pdf More with this subject... Current Thread Previous by Date: Counterexamples Alessio Guglielmi Previous by Thread: Counterexamples Alessio Guglielmi Indexes: Date Thread Top All Lists Recently Viewed: linux.nfsv4/200... debian.apt.deve... security.risks/... qplus.devel/200... ... Perl 5.10 21/12/07 23:57 from

    18. Research/Lambda Calculus And Type Theory - Foundations
    This system is called now the (typefree) lambda calculus. Representing computable functions as lambda terms gives rise to so called functional programming.
    Foundations Group
    of the ICIS
    • People Seminars
      Lambda calculus and Type Theory
      The lambda calculus was originally conceived by Church in 1932 as part of a general theory of functions and logic, intended as a foundation for mathematics. Although the system turned out to be inconsistent, the subsystem dealing with functions only became a succesful model for the computable functions. This system is called now the (type-free) lambda calculus. Representing computable functions as lambda terms gives rise to so called functional programming. People::
      • Henk Barendregt Wil Dekkers Herman Geuvers Jan Willem Klop Iris Loeb
      • Typed Lambda Calculus and Applications.
        • The aim is to produce a research monograph on typed lambda calculus with its mentioned applications. This book will serve as a sequel to Barendregt's monograph on type-free lambda calculus (North-Holland, 1984, also translated into Russian and Chinese), a classical work that is considered as the standard reference to lambda calculus. The editors and main authors of the book are Henk Barendregt and Wil Dekkers of the University of Nijmegen and Rick Statman of Carnegie Mellon University, Pittsburgh, USA. Several co-authors (all of them leading experts in the field) will contribute to this work. The project is embedded in the larger project `Lambda-calculus and Applications', a 7-year research effort at the Computing Science Institute in Nijmegen, supported by a special grant of the University Council.

    19. Classical Vs Quantum Computation (Week 1) | The N-Category Café
    the lambda calculus and its role in classical computation,; how quantum computation differs from classical computation,; the quantum lambda calculus and its
    @import url("/category/styles-site.css");
    A group blog on math, physics and philosophy
    Skip to the Main Content
    Enough, already! Skip to the content. Note: These pages make extensive use of the latest XHTML and CSS Standards only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser. Main
    October 3, 2006
    Classical vs Quantum Computation (Week 1)
    Posted by John Baez
    PDF files here:
    • John Baez and Derek Wise, Classical versus quantum computation Week 1 (Sept. 28) - Computation, the lambda calculus and cartesian closed categories: an overview.
      here In these lectures I hope to talk about:
      • the lambda calculus and its role in classical computation, how quantum computation differs from classical computation, the quantum lambda calculus and its role in quantum computation, cartesian closed categories, algebraic theories, PROPs and operads

    20. Lambda Calculus Tutorial
    The lambda calculus was developed in 1936 by Lorenzo Church, and is a mathematical system for defining computable functions (i.e., a model of computation).
    Lambda Calculus Tutorial
    Programming Languages Group 16 Cody Robbins [
    Jonathan Dance [
    Jeffrey Lynch [
    Matthew Cherian [
    The lambda calculus was developed in 1936 by Lorenzo Church, and is a mathematical system for defining computable functions (i.e., a model of computation). Church’s lambda calculus is equivalent in power to the Turing machine, although Church and Turing both developed their respective models of computation independently. We attempt to explain to the fundamental principles of the lambda calculus in a clear, concise, and easy to understand fashion. We provide examples and self-tests to facilitate in the conceptualization of the material.
    Table of Contents
  • Introduction Syntax of the Lambda Calculus Evaluation Strategies Reductions ...
  • Download the entire tutorial as a gzipped tar Created April 15, 2002

    21. Incremental Reduction In The Lambda Calculus
    SIGACT ACM Special Interest Group on Algorithms and Computation Theory for the lambda calculus, Information and Computation, v.75 n.3, p.191231, Dec.

    22. Lambda-Calculus And Computer Science Theory 1975
    Corrado Böhm (Ed.) lambdacalculus and Computer Science Theory, 272-286 BibTeX Marisa Venturini Zilli A model with nondeterministic computation.
    Lambda-Calculus and Computer Science Theory 1975: Rome, Italy
    (Ed.): Lambda-Calculus and Computer Science Theory, Proceedings of the Symposium Held in Rome, March 25-27, 1975. Lecture Notes in Computer Science 37 Springer 1975, ISBN 3-540-07416-3 BibTeX DBLP

    23. Research Laboratory For Logic And Computation, GC CUNY
    CT A function f N N is algorithmically computable iff it is (general) recursive. Cartesian closed categories and lambda calculus II.
    Research Laboratory for Logic and Computation

    FALL 2003
    Tuesday, 2pm - 4pm, room 4421
    December 2 talk
    Yegor Bryukhov. Type Theory for a practicing mathematician.

    Abstract: continue
    December 2 talk
    Yegor Bryukhov. Type Theory for a practicing mathematician.

    Abstract: In this talk we will follow R.Constable's paper "Naive Computational Type Theory" which in turn "follows" the book of Paul Halmos "Naive Set Theory". This paper gives some new perspectives in Type Theory, including a new meaning of openness of Type Theory. We'll start from the fundamentals: "what is type", "propositions as types", and then go to type-theoretic analogues of a set, subset, pair, union, intersection, function, relation, etc. We will consider two meanings of this popular statement "Type Theory is open-ended", one is old and the other one is new. They are related but the new one is much deeper. It shows that the Type Theory is VERY different from the Set Theory. Time permitting we'll discuss dependent intersection (a relatively new result by Alexei Kopylov) and records. November 25 talk Walter Dean (GC and Rutgers). From Church's Thesis to Extended Church's Thesis.

    24. Wiktionary:Tea Room/Archive 2007 - Wiktionary
    lambda calculus is a subfield of Computability theory studied first by Alonzo Church. The strong linking to him is because most places which discuss
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    Wiktionary:Tea room/Archive 2007
    From Wiktionary
    Wiktionary:Tea room Jump to: navigation search This is an archive page that has been kept for historical purposes. The conversations on this page are no longer live.
    • January
      morphing of word meaning
      I am wanting to find the word that is used to describe the process by which the meaning of a word changes over time. For example, the word "gay" used to mean "happy, lighthearted, etc." and now is predominantly used to indicate an alternate lifestyle. What word can be used to describe the morphing of the meaning of a word over time? —This comment was unsigned.
      I call that the linguistic evolution (of a term) but there probably is a shorter way to say it. Connel MacKenzie 20:08, 2 January 2007 (UTC)
      It's not shorter, but I would probably talk about semantic change in that case. Widsith 10:02, 3 January 2007 (UTC)

    25. Education, Master Class 1988/1999, MRI Nijmegen
    The lambda calculus is a mathematical theory of computable functions. lambda calculus gives representations of algorithms and of constructive proofs.
    Education, Master Class, Master Class 1998/1999, Detailed Course Content
    Detailed Content of the Courses
    Course content
    1st semester:

    Model Theory
    W. Veldman
    Lambda Calculus
    H. Barendregt, E. Barendsen
    Recursion Theory and Proof Theory
    H. Schellinx
    Logic Panorama
    2nd semester:
    Type Theory and Applications
    H. Barendregt, E. Barendsen
    Incompleteness Theorems
    J. van Oosten Sheaves and Logics I. Moerdijk Mathematical Logic seminar Courses Name of the course: Model Theory Lecturer: W. Veldman Prerequisites: Some familiarity with mathematical reasoning. Literature: C.C. Chang, H.J. Keisler, Model Theory, North Holland Publ. Co. 1977 W. Hodges, Model Theory, Cambridge UP, 1993 Contents: Model theory studies the variety of mathematical structures that satisfy given formal theory. It may also be described as a study of mathematical structures from the logician's point of view. Model theory at its best is a delightful blend of abstract and concrete reasoning. Among the topics to be treated in this course are Fraisse's characterisation of the notion 'elementary equivalence' (structures A,B are called elementarily equivalent if they satisfy the same first-order-sentences), the compactness theorem and its many consequences, ultraproducts, some non-standard-analysis, Tarski's decision method for the field of real numbers by quantifier elimination and Robinson's notion of model completeness. If time permits, some attention will be given to constructive and recursive model theory.

    26. Arithmetic In Lambda Calculus - Wolfram Demonstration Project
    lambda calculus was developed by Alonzo Church and Stephen Kleene in 1930 and It is a system capable of universal computation, that is, any computable
    Arithmetic in Lambda Calculus
    loadFlash(644, 387, 'ArithmeticInLambdaCalculus'); Lambda calculus was developed by Alonzo Church and Stephen Kleene in 1930 and consists of a single transformation rule (variable substitution) and a single function definition scheme. It is a system capable of universal computation, that is, any computable function that can be computed in any of the standard programming languages can also be done in lambda calculus, though it might be very hard to actually carry out. Only the basic arithmetic operations successor, testing for zero, addition, multiplication, and exponentiation are considered here. The second numeral is not used for successor or testing for zero.
    The central concept in λ calculus is the "expression". A "name" (or "variable") is an identifier that can be any letter. An expression is defined recursively as follows: In order to apply a function to an argument by substituting the argument for a variable in the body of the function and for giving a name to the function determined by a rule, it is necessary to define the following terms: 1) The identity function: (( 2) Self-application: . Applying this to any expression expr results in (expr expr), which may or may not make sense.

    27. Fall 2001, CSE 520: Lectures
    1937 Turing proves that every function computable by a Turing machine can be represented in the lambda calculus, and viceversa. In the same years,
    Fall 2001, CSE 520: Lecture 1
    The Lambda Calculus
    Some history
    • 1932: Alonzo Church develops the (type-free) Lambda Calculus as part of a general theory of functions and logic, intended as a foundation for mathematics. The whole system was proved inconsistent by Kleene and Rosser (1935), but the Lambda Calculus (taken alone) was proved consistent by Chuch and Rosser (as a consequence of the Church-Rosser property of the Lambda Calculus).
    • 1937: Turing proves that every function computable by a Turing machine can be represented in the Lambda Calculus, and viceversa.
    • In the same years, Church formulates the statement which is known as Church's Thesis Every computable function can be defined in the Lambda Calculus Namely every function which can ever be computed algoritmically can be expressed in the Lambda Calculus. This statement cannot be proved formally, of course, because the notion of "algoritmic computability" is not defined formally. But, as a matter of fact, this "conjecture" has never been "disproved": not even the most sophisticated modern machines and programming languages can define more functions than those defined by the lambda calculus (or any of the other equivalent formalisms: Recursive functions, Turing machines, etc.).
    • Based on the Lambda Calculus, a class of programming languages have been developed: the so-called Functional Languages (Lisp, Scheme, ML, Haskell, ...)

    28. Abstracts Of The Lectures At The School In Logic And Computation
    Mariangola DezaniCiancaglini University of Turin; The lambda calculus in Different from a computation and orders it according to the order it produced,

    29. Definitions Of Computable
    Church s Hypothesis on Computability. Turing Machines; lambda calculus; Post Formal Systems; Partial Recursive Functions; Unrestricted Grammars

    30. Introduction To Lambda Calculus
    Introduction to lambda calculus. lambda calculus. The calculus is universal in the sense that any computable function can be expressed and evaluated
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    31. Summer School And Workshop On Proof Theory, Computation And Complexity
    Like for last year’s events on `Proof Theory and Computation´ (Dresden) and `Proof, After introducing the simply typed lambda calculus, it is planned to
    Summer School and Workshop on
    Proof Theory, Computation and Complexity
    June 23-July 4, 2003 Call for Participation
    (Dresden) and For attending courses, we ask for a fee of 100 EUR (to be paid in cash at the school). Registration is requested before May 25, 2003; please send an email to PTEvent@Janeway.Inf.TU-Dresden.DE , making sure you include a very brief bio (5-10 lines) stating your experience, interests, home page, etc. We select applicants in case of excessive demand. A limited number of grants covering all expenses is available. Applications for grants must include an estimate for travel costs and they should be sent together with the registration. We provide assistance in finding an accommodation in Dresden. Week 1, June 23-27: courses on
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    32. European Masters Program In Computational Logic - Modules At TU Wien
    Logic programs with constraints are introduced and basic computation mechanisms given. Keywords higher order logics; lambda calculus; lambda prolog;
    Modules in Computational Logic at TU Wien
    Module ECTS Hours Responsible Person Foundation Modules Foundations Alexander Leitsch Logic and Constraint Programming Thomas Eiter Advanced Logics Matthias Baaz Integrated Logic Systems Georg Gottlob Advanced Modules Principles of Computation Thomas Eiter Logical Foundations Alexander Leitsch Mathematical Methods Alexander Leitsch Computational Logic for Information Technology Georg Gottlob Inference in Classical and Nonclassical Logic Matthias Baaz Student Project Student Project Georg Gottlob
    Alexander Leitsch
    Thomas Eiter MSc Thesis MSc Thesis Georg Gottlob
    Alexander Leitsch
    Thomas Eiter
    Matthias Baaz
    Foundation Modules
    Keywords: propositional logic; first order logic; deduction; proof theory; abduction and induction; knowledge representation and reasoning; complexity theory; computer algebra. The module offers a comprehensive introduction to Computational Logic covering the main subareas as well as main methods and techniques. After recalling basic notions from propositional and first order logic, complexity theory and computer algebra, the areas of equational reasoning, deduction, proof theory, abduction and induction, non-monotonic reasoning, logic-based program development, natural language processing and machine learning as well as logic and connectionism are covered.
    Courses in 2007/2008:

    33. Cigars, Lambda Expressions, And .NET | Dev Source | Find Articles At
    According to Wikipedia, lambda calculus can be called the smallest universal programming language, and any computable function can be expressed and
    @import url(/css/us/pub_page_article.css); @import url(/css/us/template_503.css); @import url(/css/us/tabs_503.css); @import url(/css/us/fa_bnet.css); @import url(; BNET Research Center Find 10 Million Articles Advanced Search Find in free and premium articles free articles only premium articles only this publication Arts Autos Business Health News Reference Sports Technology
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    Cigars, Lambda Expressions, and .NET
    Dev Source June, 2007 by Paul Kimmel Introduction Recently I began smoking Rocky Patel's cigars. If you check the Rocky Patel website there is a quote about the Hollywood lawyer turned cigar maker: "He loves the quality of Padron, the construction of Davidoff, and the consistency of Fuentes." Patel wants to incorporate all of these admirable qualities into his cigars and make them affordable. They're great cigars, so Patel seems to be succeeding.
    Most Popular Articles
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    34. J Logic Computation -- Sign In Page
    This covers an application of rewriting techniques to computation via equational reasoning. Chapter 10, by Bethke, is on lambda calculus.
    @import "/resource/css/hw.css"; @import "/resource/css/logcom.css"; Skip Navigation Oxford Journals
    This item requires a subscription* to Journal of Logic and Computation Online. * Please note that articles prior to 1996 are not normally available via a current subscription. In order to view content before this time, access to the Oxford Journals digital archive is required. If you would like to purchase short-term access you must have a personal account. Please sign in below with your personal user name and password or register to obtain a user name and password for free.
    Full Text
    Book Review
    Banach J Logic Computation.
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    35. Pietro Di Gianantonio Publications
    A functional approach to Computability on real numbers. Game semantic for untyped lambda calculus. Pietro Di Gianantonio, Gianluca Franco,
    Papers by Pietro Di Gianantonio
    Real Number Computability
    A certified, corecursive implementation of exact real numbers.
    Alberto Ciaffaglione, Pietro Di Gianantonio.
    Theoretical Computer Science,
    2006, vol. 351, pp. 39-51.
    pdf copy

    A tour with constructive real numbers.
    Alberto Ciaffaglione, Pietro Di Gianantonio.
    Proc. of the workshop Types for Proofs and Programs - Types 2000; LNCS 2277, pp. 41-52.
    pdf copy

    A co-inductive approach to real numbers.
    Alberto Ciaffaglione, Pietro Di Gianantonio.
    Proc. of the workshop ``Types 1999''; LNCS 1956, pp. 114-130.
    pdf copy

    An abstract data type for real numbers.
    Pietro Di Gianantonio.
    Theoretical Computer Science, 1999, vol. 221, n. 1-2, pp. 295-326, extended version of a paper presented at ICALP-97. pdf copy
    A golden notation for real numbers.
    Pietro Di Gianantonio. CWI technical report. pdf copy
    Real number computability and domain theory.
    Pietro Di Gianantonio. Information and Computation, 1996, vol. 127, n. 1, pp. 12-25, extended version of a paper presented at MFCS-93. pdf copy
    A functional approach to computability on real numbers.

    36. The Lambda Calculus Mail Series
    The set of functions that are definable in lambda calculus are exactly those functions that are computable! In fact, lambda calculus and Turing Machines\lambda.html
    The Lambda Calculus Mail Series
    This page is just a mail series (with some editing) that I started while I was trying to read up on lambda calculus from various websites and papers. This, I think, will make nice reading to anybody interested in the subject. But bear in mind that this was just done for fun, and an experiment in a group of friends using mails to learn something new. A lot of things could be missing, wrong or stupid, please let me know if this is the case.
    Series on Lambda Calculus
    Tell me how you like the idea and if you have any suggestion. Also, start thinking about doing something yourself too. In fact, I think its even OK, to send a huge article (somebody else's) on a subject, bit by bit everyday (just copy and paste), as long as we all read it and discuss it, its worth the time and the effort (or so I think). Lets hope all this works out well, and reaches its completion.

    37. Pure And Applied Logic At Carnegie Mellon
    Richard Statman Professor of Computer Science and Mathematical Sciences mathematical logic, theory of computation, lambda calculus, combinatory logic
    Pure and Applied Logic
    The Pure and Applied Logic (PAL) program is an interdisciplinary Ph.D. program at Carnegie Mellon University with faculty from: The program builds upon Carnegie Mellon's unique strengths in logic and its applications to computer science. Internationally recognized faculty, frequent workshops, colloquia, seminar series, and excellent computing facilities contribute to an ideal environment for both theoretical and applied research. Graduates of the program have gone on to prominent positions in industry and academe. Carnegie Mellon ranks highly in logic and related fields; see the university-maintained summary of rankings of various departments and programs at Carnegie Mellon. Areas of strength include:
    • automated theorem proving category theory and categorical logic constructive mathematics foundations of decision theory foundations of programming languages logics of programs lambda calculus learning theory model theory proof theory set theory temporal and modal logics theory of computing type theory
    Related research at Carnegie Mellon includes algorithms, artificial intelligence, combinatorial optimization, computational complexity, computational linguistics, operations research, and programming systems. See also the

    38. ComSci 319, U. Chicago
    The lambda calculus is a formal system for studying the definitions of of logic and computation that arise from the ability to interpret a lambda term
    Com Sci 319
    Lambda Calculus
    Winter 2000
    A course in the Department of Computer Science
    The University of Chicago
    Online discussion using HyperNews
    • [8 Feb] Assignment #4 is due on Monday, 14 February, at the beginning of class. (O'D)
      [17 Jan] The HyperNews discussion is set up. Please read the instructions, and jump in. The system reports an error whenever you post, but the only error is the error report itself. I'm trying to get that fixed. In the meantime, please don't post the same message repeatedly in response to the erroneous error message. (O'D)
      [29 Dec 1999] The Web materials for ComSci 319 are under construction. Some links are broken. (O'D)
    • Venue: MW 9:00-10:20, Ryerson 257
      Instructor: Michael J. O'Donnell
      • Office: Ryerson 257A. email: Office hours: by appointment. Contact me by email, phone at the office (312-702-1269), or phone at home (847-835-1837 between 9:30 and 5:30 on days that I work at home). You may drop in to the office any time, but you may find me out or busy if you haven't confirmed an appointment. Check my personal schedule before proposing an appointment.

    39. Foundations Of Computer Science
    Research seminar devoted to problems related to asymptotic densities in logic, Computability theory, computational logic, typed lambda calculus,
    Theoretical Computer Science
    Faculty of Mathematics and Computer Science

    Jagiellonian University

    Foundations of Computer Science guest
    TCS - home algorithmics cs foundations news ... links events: computer science on trail (pl) UZI - January 12, 2008(pl) CLA 2007 past events seminars: Computer Science Foundations faculty: Jakub Kozik Stanisław Sędziwy Edward Szczypka Paweł Waszkiewicz ... Marek Zaionc secretary: Monika Gillert phd students: Katarzyna Grygiel Jarosław Karpiak Mikołaj Pudo graduates: phd thesis msc thesis login: password: Computer Science Foundations Seminar Wednesday: 12:15 - 14:00, room 117 Research seminar devoted to problems related to asymptotic densities in logic, computability theory, computational logic, typed lambda calculus, logic programming, logics of programs, functional programming. table edited by: Marek Zaionc
    Szymon W³jcik Parallel reductions in lambda calculus
    The notion of parallel reduction is extracted from the simple proof of the Church-Rosser theorem by Tait and Martin-L¶f. Intuitively, this means to reduce a number of redexes (existing in a lambda term) simultaneously. During the talk, after reevaluating the significance of the notion of parallel reduction in Tait-and-Martin-L¶f type proofs of the Church-Rosser theorems, we show that the notion of parallel reduction is also useful in giving short and direct proofs of some other fundamental theorems in reduction theory of lambda calculus.
    Dominika Majsterek UJ Behavioural differential equations: a coinductive calculus (część 2)

    40. Publications
    Completeness of continuation models for lambdamu-calculus. Joint with Thomas Streicher. To appear in Information and Computation This is a journal version
    Publications and preprints
    • The strength of non size-increasing computation ps pdf
    • Realizability models for BLL-like languages ps pdf
      with Phil Scott
      Accepted for the LICS affiliated workshop Implicit Computational Complexity (ICC), Santa Barbara, 28-29 June 2000.
    • A new "feasible arithmetic" ps pdf
      with Steve Bellantoni
      To appear in JSL.
      A classical quantified modal logic is used to define a ``feasible'' arithmetic whose provably total functions are exactly the polynomial-time computable functions. The crucial restrictions are (1) that induction is limited to modality-free formulas and (2) that an induction hypothesis may be used at most once (in the sense of linear logic). The logic is defined without any reference to bounding terms, and admits induction over formulas having arbitrarily many alternations of unbounded quantifiers.
    • Implementing a Program Logic of Objects in a Higher-Order Logic Theorem Prover ps.gz pdf
      with Francis Tang
      We present an implementation of a program logic of objects, extending that (AL) of Abadi and Leino. In particular, the implementation uses higher-order abstract syntax (HOAS) and - unlike previous approaches using HOAS - at the same time uses the built-in higher-order logic of the theorem prover to formulate specifications. We give examples of verifications, extending those given in [Abadi, Leino 1997], that have been attempted with the implementation. Due to the mixing of HOAS and built-in logic the soundness of the encoding is nontrivial. In particular, unlike in other HOAS encodings of program logics, it is not possible to directly reduce normal proofs in the higher-order system to proofs in the first-order object logic.

    41. 2006 August « Reperiendi
    Quantum lambda calculus, symmetric monoidal closed categories, and TQFTs. 2006 August 22 This set is only computably enumerable, not computable.
    Figure it out
    Archive for August, 2006
    Quantum lambda calculus, symmetric monoidal closed categories, and TQFTs
    2006 August 22 and The typical CCC in which models of a lambda theory are interpreted is Set . The typical SMCC in which models of a quantum lambda theory will be interpreted is Hilb , the category of Hilbert spaces and linear transformations between them. Models of lambda theories correspond to functors from the CCC arising from the theory into Set . Similarly, models of a quantum lambda theory should be functors from a SMCC to Hilb Two-dimensional topological quantum field theories (TQFTs) are close to being a model of a quantum lambda theory, but not quite. Set TQFTs are functors from , the theory of a commutative Frobenius algebra, to Hilb . We can look at as defining a data type and a TQFT as a quantum implementation of the type. When we take the free SMCC over , we ought to get a full-fledged quantum programming language with a commutative Frobenius algebra data type. A TQFT would be part of the implementation of the language. Posted in Category theory Math Programming Quantum ...
    CCCs and lambda calculus
    2006 August 22 I finally got it through my thick skull what the connection is between lambda theories and cartesian closed categories.

    42. Théorie De La Démonstration
    P. Baillot and Terui, K., Light types for polynomial time computation in lambdacalculus, Proceedings of LICS 2004, pp. 266–275 (2004).(PS).
    Liste (non exhaustive) d'articles Voir le planning provisoire des soutenances Articles
    • Kanovitch, M. (1992). Horn-programming in linear logic is NP-complete . In Proceedings of the 7th Annual IEEE Symposium on Logic in Computer Science, pp. 200-210. ( P. Baillot and Terui, K., Light types for polynomial time computation in lambda-calculus PS Jean-Marc Andreoli. Focussing and Proof construction Abstract paper.pdf (238KB). (R©serv© par Etienne Miret Andrea Asperti. Light affine logic . In Proc. Symp. Logic in Comp. Sci. (LICS). IEEE, 1998. ( PS A.Asperti, L.Roversi. Intuitionistic Light Affine Logic . ACM Transactions on Computational Logic (TOCL), Volume 3 , Issue 1, January 2002, pp.137 - 175. Y. Lafont, Soft Linear Logic and Polynomial Time , Theoretical Computer Science 318 (special issue on Implicit Computational Complexity) p. 163-180, Elsevier (2004). ( PS Benton, Bierman, de Paiva.

    43. Foundational Papers By Henk Barendregt
    Logic, Meaning and Computation, Kluwer, 275285. Volume 97 kb Constructive proofs of the range property in lambda calculus.
    Foundational papers by Henk Barendregt (and co-authors)
    Papers and talks on logic and computer science
    Over welke kwestie zijn bijna al uw vakgenoten het oneens met u?
    Towards the range property for the lambda theory H
    Proofs of Correctness To appear in Wiley Encyclopedia of Computer Science and Engineering.
    Bewijzen: Romantisch of Cool
    (with F. Wiedijk) In: Euclides, 81 (4), 2006, 175-179.
    B"ohm's Theorem, Church's Delta, Numeral Systems, and Ershov Morphisms (with R. Statman)
    In: Processes, Terms and Cycles: Steps on the Road to Infinity,
    Essays Dedicated to Jan Willem Klop on the Occasion of His 60th Birthday,
    Eds. A. Middeldorp, V. van Oostrom and F. van Raamsdonk,
    Springer LNCS, vol. 3838, 2005, 40-54.
    The Challenge of Computer Mathematics
    (with F. Wiedijk)
    In Transactions A of the Royal Society, vol. 363, no. 1835, 2351-2375.
    Foundations of Mathematics from the Perspective of Computer Mathematics
    To appear in the Buchberger Festschrift
    Towards an Interactive Mathematical Proof Language
    in: Thirty Five Years of Automath, Ed. F. Kamareddine, Kluwer, 2003, 25-36. (cartoon via advi)
    Autarkic computations in formal proofs
    (with E. Barendsen)

    44. The Little Calculist: Jewels Of The Lambda Calculus: Abstraction As Generalizati
    LC is a ultrageneral calculus of computable functions, so it captures The point is that the lambda calculus formalizes the metalanguage we use to talk
    The Little Calculist
    Dave Herman's research blog.
    Thursday, April 27, 2006
    Jewels of the lambda calculus: abstraction as generalization
    Since there are few things as likely to inspire in me a sense of religious reverence as the lambda calculus, I've decided to start a new series on some of the deep principles of computer science lurking in the lambda calculus.
    Let's start with the crown jewel, abstraction:
    x e This simple form represents the first rule of programming. Ruby hackers call it the " don't repeat yourself " rule. Perl hackers call it laziness : good programmers avoid repetitive, manual tasks. No matter the size, shape, or form, any time you write a program, you create an abstraction, a piece of code that automates a common task that otherwise would have to be repeated by a human.
    Lambda abstractions represent the generalization of repeated patterns by parameterizing over their differences. If you perform several computations that are similar but not identical:
    then each one of these computations is a special case of a more generalized procedure. The abstraction of the three is represented by parameterizing over the portion where they differ:
    c c From functions to abstract classes to Microsoft Office macros to parametric polymorphism, every form of abstraction is at its heart this same phenomenon.

    45. CS302 2006S : Class 08: Lambda Calculus
    Any function that is computable under a reasonable model of computation is computable by (a Turing machine; the lambda calculus); Any sufficiently powerful
    Programming Languages (CS302 2006S) Skip to Body
    Primary: Front Door Current Glance Honesty ... Links
    Groupings: EBoards Examples Exams Handouts ... Reference
    Misc: SamR
    Class 08: Lambda Calculus
    Back to McCarthy's LISP . On to Continuation Basics Held: Wednesday, February 8, 2006 Summary: Today we consider the basics of the lambda calculus, a notation (and way of thinking about functions) that underlies most modern functional languages. Assignments Notes:
    • A few of you have asked for extensions on the homework, or at least time to talk to me. It is now due a week from Friday.
    • Historical aspects. Notation basics. Currying.
    Historical Aspects
    • As you saw in the papers from McCarthy, lambda notation has been a part of functional programming since the first version of Lisp.
    • Lambda notation predates Lisp by a number of years.
    • Lambda notation was developed by Alonzo Church, a logician, in the early part of the 20th century.

    46. Definitions Of Computable
    Church s Hypothesis on Computable; Turing Machines; lambda calculus; Post Formal Systems; Partial Recursive Functions; Unrestricted Grammars
    Definitions of computable
  • Church's Hypothesis on Computable
  • Turing Machines
  • Lambda Calculus
  • Post Formal Systems ...
  • Other Links
    Church's hypothesis, Church Turing Thesis
    This is a mathematically unprovable belief that a reasonable intuitive definition of "computable" is equivalent to the list provably equivalent formal models of computation: Turing machines Lambda Calculus Post Formal Systems Partial Recursive Functions ... Recursively Enumerable Languages and intuitively what is computable by a computer program written in any reasonable programming language.
    Turing Machines
    Lambda Calculus
    Post Formal Systems
    Partial recursive functions
    A Partial Recursive Function is allowed to have an infinite loop for some input values. A Recursive Function also called a Total Recursive Function always returns a value for all possible input values. Partial Recursive Functions correspond to Turing machines that may not halt. (Total) Recursive Function correspond to Turing machines that always halt. Primitive Recursive Functions are a subset of Total Recursive Functions with the restriction that only primitive recursion is used a finite number of times and recursion uses zero and the successor function. Primitive recursion is defined for f(x1,...,xn) as f(x1,...,xn) = g(x1,...,xn-1) if xn = = h(x1,...,xn,f(x1,...,xn-1, xn -1)) if xn > where g and h are primitive recursive functions. Ackermann's function is not primitive recursive. For technical reasons a projection function, a selector, is often used. Pi(x1,...,xn) returns xi where 1
  • 47. Lambda Calculus Introduction
    Our notion of computation in lambda calculus will be to reduce a term to its normal form. This will make more sense after we define numbers, below.
    Lambda Calculus Introduction
    This page is adapted from The Lambda Calculus: Its Syntax and Semantics , by H. P. Barendregt, 1984, primarily sections 2.1, 3.1, 3.2, 6.1, and 6.2. Contents:
    Lambda Terms

    Beta Conversion

    Normal Forms

    Useful Combinators
    Further Reading
    Lambda Terms
  • Outermost parentheses may be omitted. x ...x n n MN N n )N n ) (association to the left) xx
  • An occurence of a variable x is bound free
    Beta Conversion
    M[x:=N]. We also say syntactic sugar Functions in lambda calculus take only one argument, but we can get the effect of multiple-argument functions:
    Normal Forms
    normal form Examples:
  • x is a normal form.
  • A term M has M , then M also has normal form N. However, this does not mean that any
    Useful Combinators
    A combinator I
    S I is the identity function, and K is the function of two arguments that ignores its second argument and returns its first. K can also be thought of as a function of one argument that returns a constant function: K It can be shown that any combinator can be generated from S and K using only function application. As an exercise, verify that I SKK Define truth values as follows: T
    F It may not be immediately obvious why these expressions should represent the truth values, but notice that

    48. JSTOR Minimal Forms In $\lambda$-Calculus Computations
    The purpose of this work is to consider 2calculus as a computation model. .. 3 A. C~tJRC~, The calcttli of lambda conversion, Annals of Mathematical<165:MFIC>2.0.CO;2-A

    49. FACT! - Multiparadigm Programming With C++ Glossary
    The lambda calculus is a formal mathematical system. It was developed by Alonzo Church in the 1930s as a theoretical model for computation.
    Glossary Algorithmic Skeletons

    Expression Templates

    Functor (Function Object)
    Turing Machine

    Algorithmic Skeletons The Skeletal Parallelism Homepage says: There are many definitions, but the gist of the idea is that useful patterns of parallel computation and interaction can be packaged up as second order constructs (i.e. parameterized by other pieces of code), perhaps presented without reference to explicit parallelism, perhaps not. Implementations and analyses can be shared between instances. Such constructs are skeletons', in that they have structure but lack detail. That's all. The rest is up for investigation and debate.
    Web material:
    Currying Turning an uncurried function into a curried function.
    Every functions with N arguments can be turned into a unary function (a function that takes a single argument) which returns another function with N-1 arguments.

    50. IngentaConnect Constant Time Parallel Computations In Lambda-calculus
    We prove that a function over free algebras is computable in parallel constant time in the pure lambda calculus, iff it is representable in the simply
    var tcdacmd="dt";

    51. Typed Lambda Calculi Publications
    Abstract In this paper I propose a new set of rules for a judgement calculus, i.e. a typed lambda calculus, based on Intuitionistic Linear Logic;
    Type lambda Calculi
    U. Solitro, S. Valentini, Toward Typed lambda-calculi for Linear Logic, proceeding of "The Third Italian Conference on theoretical Computer Science" (ed. Bertoni, Bshm, Miglioli), World Scientific, Mantova 1989, pp. 413-424. Abstract: In this paper it is studied the problem of stating a Curry-Howard isomorphism for Girard's Linear Logic. Two solutions are proposed: one for the intuitionistic variant of the logic and one for a subsytem of the logic which includes the multiplicative and exponential connectives. Different calculi have been developed for the two cases.
    S. Valentini, The judgement calculus for Intuitionistic Linear Logic: proof theory and semantics, Abstract: In this paper I propose a new set of rules for a judgement calculus, i.e. a typed lambda calculus, based on Intuitionistic Linear Logic; these rules ease the problem of defining a suitable mathematical semantics. A proof of the canonical form theorem for this new system is given: it assures, beside the consistency of the calculus, the termination of the evaluation process of every well-typed element. The definition of the mathematical semantics and a completeness theorem, that turns out to be a representation theorem, follow. This semantics is the basis to obtain a semantics for the evaluation process of every well-typed program.
    U. Solitro, S. Valentini

    52. Lecture 12 -- The Lambda Calculus As A Foundation For Computing
    Church is responsible for Church s thesis, which states that every computable function is representable as a term of the lambda calculus (equivalently,
    The Lambda Calculus as a Foundation for Computing
    The Lambda Calculus as a Foundation for Computing
    [Note: I'm going to omit the ""@ symbol for function application in this section, as it just clutters up the formulas.] The lambda calculus was invented / designed by Alonzo Church and his student Stephen Kleene at Princeton in the early 1930's. Church showed the undecidability of validity in the First-Order Predicate Logic in 1936 using the lambda calculus (Turing showed the same result independently shortly thereafter). Turing later showed the equivalence of the lambda calculus and Turing machines in computational power. Church is responsible for Church's thesis, which states that every computable function is representable as a term of the lambda calculus (equivalently, as a Turing machine). The (pure) untyped lambda calculus has the following very simple description as the language generated by: Term v v Term (Term Term) where v represents any variable symbol. Thus terms are built up from variables, lambda abstractions, and function applications. Because any term may be applied to any other term, all terms of the lambda calculus may be interpreted as functions. Because lambda expressions are supposed to represent all computable functions - in particular all computable functions from natural numbers to natural numbers - there must be a way of encoding natural numbers as lambda expressions. Below we provide the simplest way of providing this encoding, but we start first with booleans.

    53. Lambda Calculus
    lcalculus is a calculus which expresses “computation” via anonymous How then does l-calculus express a lambda form that has 2 or more arguments?
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    Volume Number: Issue Number: Column Tag: Lisp Listener
    Lambda Calculus
    Abe Lincoln
    Intro to l-calculus
    LISP 101
    What follows will be a crash course in LISP. or even because a list in any position can have a function position and argument positions of its own, and so on, to arbitrary depth. The next thing we need is a way to abstract out common process patterns into descriptions. This is done via lambda, the anonymous function. For instance, (lambda (x) (+ x 1)) is a function that takes in an evaluated argument, binds it with x, and then computes the body of the lambda form with the understanding that any occurrence of parameter x in the body will refer to the value of x bound by the lambda form. In this case, the returned result will be the argument plus one, and the argument will not be side effected. To invoke an anonymous function, we simply invoke it like any other function. We invoked sine like this: (sin 3). Invoking (lambda (x) (+ x 1)) on the argument 3 would look like this:

    54. Longo Symposium
    We show how an extension to the infinitary lambda calculus, where Böhm trees can . An analogy with computer sciences (numerical computation for partial
    28-29 June 2007 From Type Theory to Morphologic Complexity: A Colloquium in Honor of Giuseppe Longo In conjunction with RDP 2007 Paris, Conservatoire National des Arts et M©tiers , Amphitheaters 3 and A. This colloquium was organised to celebrate the 60th birthday of Giuseppe Longo . Some photos of the meeting can be found here The main research area Giuseppe Longo has been interested in concerns syntactic and semantic properties of the "logical base" of functional languages: Combinatory Logic, Lambda-calculus and their extensions. However, he always investigated these topics in its broadest setting which relates them to Recursion Theory, Proof Theory and Category Theory. In this perspective, Longo worked at some aspects of Recursion Theory, Higher Type Recursion Theory, Domain Theory and Category Theory as part of a unified mathematical framework for the theory and the design of functional languages. In a sense, Longo has always been mostly interested in the "interconnecting results" or "bridges" and applications among different areas and to language design. He also worked at the applications of functional approaches to Object-Oriented programming. He is currently extending his interdisciplinary interests to Philosophy of Mathematics and Cognitive Sciences. A recent interdisciplinary project on Geometry and Cognition (started with the corresponding grant: "G©om©trie et Cognition", 1999 - 2002 with J. Petitot et B. Teissier), focused on the geometry of physical and biological spaces. The developements of this project lead to a new initiative at DI-ENS, in 2002, the setting up of the research team "Complexit© et information morphologiques" (CIM), centered on foundational problems in the interface between Mathematics, Physics and Biology.

    55. Real Analysis In Abstract Stone Duality
    A lambda calculus for Real Analysis if you are interested in (constructive, computable or classical) real analysis;. Interval Analysis Without Intervals if
    Real Analysis in Abstract Stone Duality
    Paul Taylor
    8 September 2006
    Abstract Stone Duality provides a new approach to general topology and real analysis that is based on continuous functions and predicates, and not sets. It is both constructive and computable, but its idioms of reasoning often look like a sanitised form of classical logic.
    Although the applications to analysis are very recent and do not cover very much of that subject, these papers currently provide the best introductions to the technically more developed theory of locally compact and more general spaces in ASD. Depending on your background, you should read Please note, however, that all three papers are still "under construction", and you may find that the associated conference slides provide a more friendly introduction.

    56. Lambda Notation
    We study the operational semantics of lambda calculus, which is based on reducing expressions to their simplest possible forms. That is, a computation is
    Lambda Notation
    At a level of conceptualization, a function doesn't necessarily need a name. Executing a functional program is a process of function applications. An application of a function is about what a function does. What a function does is specified in its definition, the code in the body. Naming provides us a convenient mechanism to refer to that code. It is however not the only mechanism. There are only two things we do in functional programming: 1. function definitions 2. function applications Of course, the purpose of a definition is to apply it. Though related, it's important to see that they refer to different things. You will get confused later on if you don't separate them conceptually. In Lisp, one can write a function without giving it a name by using (lambda (x1 ... xn) BODY) where (x1 ... xn) is the parameter list (usually called formal parameters), and BODY the function definition, i.e. what it does. We call this a lambda function (note that it is a function definition, not a function application.) Important : A lambda function is not an application. An application using a lambda function is of the form

    Church s Thesis says that the meaning of effective computation is equivalent to all computations that can be programmed in Church s lambda calculus
    Number of In text In summary Ratio Sentences Paragraph
    A summary of the physical basis of quantum computation appeared last week in " Science " magazine . In it David Vencenzo points out that " It is evident from this survey of the current state of the art in quantum experimental physics that the construction of quantum computers is presently in the most rudimentary stage , and that to even think about a procedure like Shor factorization , which might require millions of operations on thousands of qubits , might be absurdly premature . Nevertheless , there are some useful things to think about in quantum computation from the perspective of the construction of consciousness . He points out that this algorithm can be implemented on an analog computer , but " do not worry: more subtle uses of quantum interference cannot be explained away so easily with classical thinking. " The Church - Turing Thesis says that the meaning of " effective computation " is equivalent to " all computations that can be performed by a Universal Turing Machine " . Church 's Thesis says that the meaning of " effective computation " is equivalent to " all computations that can be programmed in Church 's lambda calculus " . The BlumShubSmale model is constructed over any ring , including both integers and reals .
    A summary of the physical basis of quantum computation appeared last week in " Science " magazine .

    58. Online Otter-λ
    lambda calculus, or calculus, is a theory of rules . Whether these rules are programs or some kind of more general (noncomputable) function we do not

    59. I. Why Study The Theory Of Programming Languages? A. Arguments For
    Computation by substitution, call by name III. The simply typed lambda calculus A. syntax TYPED LANGUAGE CONCRETE SYNTAX t,s in Type-Expr e
    inc - 4. convertability relation denoted =, cnv, or Y, then there is some Z such that X >* Z and Y >* Z (this property is called *confluence*) - - 2. If X has a normal form Y, there is a normal order reduction from X to Y. corollary: if want to find normal form, use normal order eval. - D. Normalization, comparison of simply typed and untyped calculus - NORMALIZATION def: a system is * weakly normalizing (or church-rosser, confluent): if only one normal form for a term * strong normalizing (terminating): if every term has a normal form -

    60. Peter Selinger: Math 4680/5680, Fall 2007
    PCF stands for programming with computable functions . The language PCF is an extension of simplytyped lambda calculus with booleans, natural numbers,
    Math 4680/5680, Topics in Logic and Computation
    Fall 2007
    Peter Selinger
    Course Information
    See the Course Information Sheet
    Final Exam Location (posted Nov 13). Our final exam will be on Thursday, Dec 6, at 3:30pm. The location will be Dunn 302 Old announcements
    Graduate Student Presentations
    As announced, graduate students in this course are expected to give an in-class presentation (a presentation is optional for undergraduates and auditors). Here is a list of seven suggested topics. You can choose from these topics or make up your own topic; in either case, please see me for approval.
  • Pi-calculus. The pi-calculus is a prototypical programming language for concurrent processes invented by Milner, Parrow, and Walker in the late 1980's. Unlike lambda-calculus, it is not confluent; the behavior of a process is explicitly allowed to be non-deterministic. The presentation will explain the basics of pi-calculus, and show how lambda calculus can be simulated in pi-calculus. Reference: Robin Milner, Joachim Parrow, David Walker, "A Calculus of Mobile Processes, Part I"
  • 61. Searching Lambda Calculus
    Results 1 to 15 for lambda calculus (view as list tiles) . lambdas, lambda calculus, bosnian lesbian pirate cult, direct compositionality calculus
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    62. A Brief History Of Functional Programming
    Church s Thesis Effectively computable functions from positive integers to positive integers are just those definable in the lambda calculus.
    A Brief History of Functional Programming
  • Their common ancester: Lambda Calculus! Early Functional Languages ... References
  • Their common ancester: Lambda Calculus!
    The most fascinating aspect about functional programming is that functional programming languages are all based on a simple and elegant mathematical foundation - Lambda Calculus. As an effort to capture the intuitive computational model based on mathematical functions, Church( ) defined a calculus that can express the behavior of functions. First, lambda calculus can use lambda abstractions to define a function. E.g., an one-argument function f that increments its argument by one can be defined by f = x. x+1 Church chose the symbol to do function abstraction and hence the name lambda calculus. With a function at hand, the next natural thing to do is to apply this function to an argument. E.g., we can apply the above function f to argument 3 and the syntax for this function application is "f 3". Finally, to do computations, lambda calculus has a reduction rule to tell the meaning of function application such as "f 3" (this reduction rule is where the computation happens and it is called the

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