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1. An Introduction To Lambda Calculus And Scheme We can use lambdacalculus to describe such a function We need this to create recursive functions, such as the factorial funtion http://www.jetcafe.org/jim/lambda.html

$Id: lambda.html,v 1.2 2001/02/01 01:43:43 jim Exp jim $ An Introduction to Lambda Calculus and Scheme Jim Larson This talk was given at the JPL Section 312 Programming Lunchtime Seminar. Functions and Lambda Notation A function accepts input and produces an output. Suppose we have a "chocolate-covering" function that produces the following outputs for the corresponding inputs: peanuts -> chocolate-covered peanuts rasins -> chocolate-covered rasins ants -> chocolate-covered ants We can use Lambda-calculus to describe such a function: Lx.chocolate-covered x This is called a lambda-expression. (Here the "L" is supposed to be a lowercase Greek "lambda" character). If we want to apply the function to an argument, we use the following syntax: (Lx.chocolate-covered x)peanuts -> chocolate-covered peanuts Functions can also be the result of applying a lambda-expression, as with this "covering function maker": Ly.Lx.y-covered x We can use this to create a caramel-covering function: (Ly.Lx.y-covered x)caramel -> Lx.caramel-covered x (Lx.caramel-covered x)peanuts -> caramel-covered peanuts Functions can also be the inputs to other functions, as with this "apply-to-ants" function:

2. Perl Contains The Lambda-Calculus In the lambda calculus, a function with formal parameter x and body B is denoted lambda and expressing arbitrary recursive functions on these objects. http://perl.plover.com/lambda/

Perl contains the -calculus -Calculus (pronounced `lambda calculus') is a model of computation invented by Alonzo Church in 1934. It's analogous to Turing machines, but it's both simpler and more practical. Where the Turing machine is something like a model of assembly language, the -calculus is a model of function application. Like Turing machines, it defines a simplified programming language that you can write real programs in. Writing Turing machine programs is like writing in assembly language, but writing -calculus programs is more like writing in a higher-level language, because it has functions. The two legal operations in the -calculus are to construct a function of one argument with a specified body, and to invoke one of these functions on an argument. What can be in the body of the function? Any legal expression, but expressions are limited to variables, function constructions, and function invocations. What can the argument be? It has to be another function; functions are all you have. With this tiny amount of machinery, we can construct a programming language that can express any computation that any other language can express. Unlike most popular programming languages, Perl is powerful enough to express the

3. Lambda Calculus Some Deeper Issues We focus on two questions why there are no primitive functions in lambda calculus and how to write recursive functions. In these notes, we use L for the http://www.cs.ualberta.ca/~mmueller/Courses/325-general/Fun/lambda-primitive.htm

Lambda Calculus: Some Deeper Issues Cmput325 Lecture Notes 1. About "Primitive" Functions Programming is about representation. A problem is given in one form and we encode it in another form that can be computed by the machine. Representation plays a crucial role in the study of computability. Representation of Natural Numbers If-then-else Logic connectives Boolean functions 2. About Recursion The Y combinator

4. Lambda Calculus And Combinators lambda calculus and Combinators. Connections. Related to functions, functional programming; Prerequisite; Requisite for . recursive functions. References. http://cs.wwc.edu/KU/Logic/lambdaCalculus.html

5. Arithmetic In Lambda Calculus - Wolfram Demonstration Project lambda calculus was developed by Alonzo Church and Stephen Kleene in 1930 and OneTerm Nestedly recursive functions The Chow-Ruskey Order 5 Venn http://demonstrations.wolfram.com/ArithmeticInLambdaCalculus/

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.

6. Recursive Functions (Stanford Encyclopedia Of Philosophy) In this entry, we provide an account of the class of recursive functions, By the reduction rules of the lambda calculus,. Cx = N(xx),. and thus http://plato.stanford.edu/entries/recursive-functions/

Cite this entry Search the SEP Advanced Search Tools ... Please Read How You Can Help Keep the Encyclopedia Free Recursive Functions First published Thu 24 Mar, 2005 The recursive functions, which form a class of computable functions, take their name from the process of "recurrence" or "recursion". In its most general numerical form the process of recursion consists in defining the value of a function by using other values of the same function. In this entry, we provide an account of the class of recursive functions, with particular emphasis on six basic kinds of recursion: iteration, primitive recursion, primitive recursion with parameters, course-of-value recursion, and double recursion. We then examine some theorems relating to these types of recursion. One recurring theme that motivates the present discussion is the question of how the basic ideas and methods used in recursion theory, which is a defining area of of logic, derive from, or at least interact with, a wider mathematical and intellectual experience. Though there are some historical references, the entry does not attempt a systematic history of the subject. 1. Types of Recursion

7. Readings: Theory Of Computation The lambda calculus Its Syntax and Semantics. NorthHolland (Amsterdam, 1981). . 5.6 Turing-Computability of the µ-recursive functions http://miser-theory.info/readings/theory.htm

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. Content Preface 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.

8. Constructive Logic And Lambda Calculus lambda calculus and logic of combinators. ChurchRosser theorem. Fixed point theorem. Representability of recursive functions. Undecidability Results. http://www.cs.swan.ac.uk/~csetzer/foerelaesning/constrmath/index.html

Course on Constructive Logic and Lambda Calculus D, 6 points Lecturer Anton Setzer House 2, Room 138, Tel. 018 4713284, Schedule Weekly (with some exceptions to be announced), Monday, 15.15 - 17.00, House 2, Room 314. Friday, 13.15 - 15.00, House 2, Room 315. First lecture: August 31, 1998. Continuation after Christmas: from January 8 till (at the latest) January 18, 1999. Topics covered Intuitionistic Logic. Brouwerian counterexamples. Elementary constructive analysis and algebra. Relationship between classical and constructive logic: double-negation translation. Properties of disjunction and existence. Realizability. Kripke-models and completeness theorem. Proof theory for intuitionistic logic. Normalization. Literature Troelstra, A. S., van Dalen, D.: Constructivism in mathematics, vol. 1. North Holland, 1988. Hindley, J. R., Seldin, J. P.: Introduction to combinators and lambda calculus. Cambridge University Press, 1986. (This book is not available any longer, but we are allowed to copy parts of it for this course). Reference Literature Troelstra, A. S., van Dalen, D.: Constructivism in mathematics, vol. 2. North Holland, 1988.

9. On Evolving Of Recursive Functions Using {lambda}-abstraction And Higher-order F On Evolving of recursive functions using {lambda} Key Words automatic programming, lambda calculus, higherorder functions http://jigpal.oxfordjournals.org/cgi/content/abstract/13/5/515

@import "/resource/css/hw.css"; @import "/resource/css/igpl.css"; Skip Navigation Oxford Journals Contact Us My Basket ... Volume 13, Number 5 Pp. 515-524 Logic Journal of IGPL 2005 13(5):515-524; doi:10.1093/jigpal/jzi039 This Article Full Text (PDF) Alert me when this article is cited Alert me if a correction is posted Services Email this article to a friend Similar articles in this journal Alert me to new issues of the journal Add to My Personal Archive ... Request Permissions Google Scholar Search for Related Content Original Articles On Evolving of Recursive Functions using -abstraction and Higher-order Functions Dept. Computer Science, Palack University Olomouc, Czech Republic. E-mail: This article introduces an application of higher order functions as structure abstractions of recursive functions as a method to reduce a search effort needed for the automatic programming (evolving) of common types of recursive functions. We will shortly describe a functional paradigm based language specially designed for automatic programming. Henceforward we introduce the structure

10. A Lambda Calculus Interpreter With Arithmetic NEM 10 Sept 2006 The lambda calculus is one of the most elegant, . The simplicity and power of this pair of mutually recursive functions is truly one of http://wiki.tcl.tk/16695

@import url(/wikit.css); @import url(/dtree.css); wiki.tcl.tk A lambda calculus interpreter with arithmetic Updated 2006-09-10 20:57:49 by NEM NEM 10 Sept 2006 : The lambda calculus is one of the most elegant, and earliest, models of computing around. Its simplicity makes it ideal as a language to implement a little interpreter, while it is equivalent in power to a universal Turing machine. The original lambda calculus consists of just three syntactic constructs: Variables : x, y, z etc Abstraction Application Abstraction creates a lambda term, which denotes a single-parameter anonymous function (similar to a proc with no name and only one parameter). Application supplies an argument to a lambda term. Variables in the body of a lambda term are replaced by the argument to that lambda. We will omit the details of how evaluation takes place (Beta-reduction and Alpha-conversion), and instead use a slightly different evaluation strategy that works for what we want. We will also add primitive support for integers and arithmetic into our interpreter. You can actually encode integers using just lambda abstractions (via Church numerals ) and basic arithmetic, but the encoding is very inefficient and a bit clumsy to use.

11. Factasia Logic Originally called recursion theory after Kleene s recursive functions, The connections between the lambdacalculus and programming languages are http://www.rbjones.com/rbjpub/logic/

Factasia Logic (for noframe browsers)

12. CSc 520 Principles Of Programming Languages - Schedule Lecture 12, Mon, Feb 21 lambda calculus 3. Combinators; recursive functions; Read Slonneger and Kurtz, Syntax and Semantics of Programming Languages, pp. http://www.cs.arizona.edu/~collberg/Teaching/520/2005/schedule.html

CSc 520 Principles of Programming Languages - Schedule Introduction Week 1 Lecture #1, Wed, Jan 12 Administrivia, Introduction Grading policy Course content Lecture note #0 (Administrivia) handed out. Lecture note #1 (Introduction) handed out. Functional Programming Week 2 Lecture #2, Mon, Jan 17 MLK Martin Luther King Holiday, no class. Lecture #3, Wed, Jan 19 Introduction to Functional Programming Introduction to functional programming. Lecture note #2 (Functional Programming) handed out. Scheme Week 3 Lecture #4, Mon, Jan 24 Introduction to Scheme Numbers. Lecture note #3 (Scheme - Introduction) handed out. Read HTDP, Section 2 Read HTDP, Section 3 Download and install Dr Scheme on your home machine, if you have one. Play with Dr Scheme , either on lectura or one of the Windows machines in the lab. On lectura, run /usr/local/plt/bin/drscheme Read Take the Dr Scheme Tour! Lecture #5, Wed, Jan 26 Scheme Functions Tracing function applications Tests and conditional execution Recursive function definitions Symbols and Structures Nested data structures History of LISP and Scheme Deep recursion over lists of lists List processing Lecture note #4 (Scheme - History) handed out.

13. To Dissect A Mockingbird: A Graphical Notation For The Lambda Calculus With Anim The lambda calculus, and the closely related theory of combinators, to define functions by means of recursive equations such as that for f above. http://users.bigpond.net.au/d.keenan/Lambda/index.htm

To Dissect a Mockingbird: A Graphical Notation for the Lambda Calculus with Animated Reduction David C Keenan, 27-Aug-1996 last updated 10-May-2001 116 Bowman Parade, Bardon QLD 4065, Australia http://users.bigpond.net.au/d.keenan Abstract The lambda calculus, and the closely related theory of combinators, are important in the foundations of mathematics, logic and computer science. This paper provides an informal and entertaining introduction by means of an animated graphical notation. Introduction In the 1930s and 40s, around the birth of the "automatic computer", mathematicians wanted to formalise what we mean when we say some result or some function is "effectively computable", whether by machine or human. A "computer", originally, was a person who performed arithmetic calculations. The "effectively" part is included to indicate that we are not concerned with the time any particular computer might take to produce the result, so long as it would get there eventually. They wanted to find the simplest possible system that could be said to compute. Several such systems were invented and for the most part looked entirely unlike each other. Remarkably, they were all eventually shown to be equivalent in the sense that any one could be made to behave like the others. Today, the best known of these are the Turing Machine, of the British mathematician Alan Turing (not to be confused with his touring machine, the bicycle of which he was so fond), and the Lambda Calculus, of the American logician Alonzo Church (1941). The Turing machine is reflected in the Von Neumann machine which describes the general form of most computing hardware today. The Lambda calculus is reflected in the programming language Lisp and its relatives which are called "functional" or "applicative" languages.

14. Abstracts Of The Lectures At The School In Logic And Computation Such programs compute the primitive recursive functions. secondorder lambda-calculus can be extended by Church-style recursive types in various ways. http://www.cee.hw.ac.uk/~fairouz/eefschool1999/abstracts2.html

15. Partial Recursive Functions In Lambda Calculus How to represent partial recursive functions in lambda calculus. http://www.safalra.com/science/lambda-calculus/partial-recursive-functions/

Partial Recursive Functions In Lambda Calculus This article is in: Science Lambda Calculus google_ad_client = "pub-0769213970530029"; google_ad_width = 120; google_ad_height = 600; google_ad_format = "120x600_as"; google_ad_type = "text"; //2007-10-28: Right column google_ad_channel = "6490446537"; google_color_border = "A8A1AA"; google_color_bg = "D4D0D4"; google_color_link = "514254"; google_color_text = "514254"; google_color_url = "514254"; google_ui_features = "rc:10"; Integers, projection, zero and successor The integers can be represented in lambda calculus by the standard Church numerals. n i x x x n x i x x x n x i n x x x n x x x n Composition The composition of the function f with the functions g , g n , applied to the tuple ( x x x m ) is f(g x x x m ),g x x x m n x x x m f g g g n x x x m f g x x x m g x x x m g n x x x m Primitive rescursion n (f,g) is the function h such that: h( x x x n ,0) = f( x x x n h( x x x n x +1) = g( x x x n x ,h( x x x n x f g x x x n h x .iszero x f x x x n g x x x n (predecessor x h (predecessor x Minimisation x x x n ) returns the smallest x such that f( x x x n x )=0 (note that such an x may not exist).

16. [Some Citations For Lambda Calculus Books From MathSciNet Djr This paper is a short history of the lambda calculus and combinatory logic by recursive definitions and the lambdacombinator. mutual recursion infinite http://www.math.niu.edu/~rusin/known-math/99/lambdacalc_refs

17. Lambda Calculus lambda calculus is a theory of functions that is central to (theoretical) computer science. It is well known that all recursive functions are representable http://users.comlab.ox.ac.uk/luke.ong/teaching/lambda/

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

18. Teach Untyped Lambda Calculus? Point 1 While it is possible to represent all partial recursive functions in untyped lambda calculus, I think this is almost a red herring, http://www.seas.upenn.edu/~sweirich/types/archive/1988/msg00128.html

[Prev] [Next] [Index] [Thread] teach untyped lambda calculus? To types@theory.LCS.MIT.EDU Subject : teach untyped lambda calculus? From jcm@ra.stanford.edu (John Mitchell) Date : Wed, 5 Oct 88 21:41:11 EDT Sender meyer@theory.LCS.MIT.EDU Prev: Polymorphism and side effects Next: teach untyped lambda calculus? Index(es): Main Thread

19. LAMBDA CALCULUS Introduction The Lambda Calculus Is A Notation For To fully appreciate lambda calculus requires two insights how lambda These are the concepts from which recursive functions definitions are built. http://www.cc.gatech.edu/data_files/classes/cs6390/resources/lambdaNotes.txt

20. Notes: Lambda Calculus lambda calculus as a basis for functional programming languages Relational model (resolution/unification, Prolog basis); Theory of recursive functions http://www.cs.unc.edu/~stotts/723/Lambda/

Lambda Calculus Lambda Calculus Interpreter Fun to try after you learn the basics of the syntax Notes Lambda Calculus Let's examine some of the theoretical foundations of computation, specifically functional computation. You may wish to read early parts of this paper: Cardelli and Wegner, "On Understanding Types, Data Abstraction, and Polymorphism" Notes we will refer to Encoding Lambda calculus in ML Boolean values and operators ML code: booleans Church numerals ... And More Lambda notes Lambda calculus is a formal model of computation. Others include Turing Machine Post production system (phrase-structure, unrestricted, type-0 grammars) Graph/Net models Petri-nets with inhibitor arcs predicate/transition nets debit-nets under forced anihilation Relational model (resolution/unification, Prolog basis) Theory of recursive functions Basics: variables are lambda-expressions (lexp) lambda abstractions (function definitions) are lexp function applications are lexp For example: variables: a, x, foo, bar lambda abstraction: lambda x 2*x function application: ( (lambda x 2*x) 5 ) specifies value 10 Syntax examples for lambda-expressions x a single variable lambda x x a function abstraction with one argument (x) and the body "x" (x y) function application where function lexp "x" is applied to arg lexp "y" (lambda x x y) function "lambda x x" applied to "y" lambda x (x y) function abstraction with one variable "x" and body "(x y)" which is a function application

21. Norman Danner Publications in the same way as the recursive functions are obtained from the primitive We define here a second order lambda calculus in which type abstraction is http://ndanner.web.wesleyan.edu/personal/papers/index.html

Norman Danner Publications Refereed publications Circuit Principles and Weak Pigeonhole Variants (with C. Pollett) To appear in Computing: The Australasian Theory Symposium (Newcastle, Australia, 2005) (PDF) Abstract: This paper considers the relational versions of the surjective and multifunction weak pigeonhole principles for and -formulas. We show that the relational surjective pigeonhole principle for formulas in implies a circuit block-recognition principle which in turn implies the surjective weak pigeonhole principle for formulas. We introduce a class of predicates corresponding to poly-log length iterates of polynomial-time computable predicates and show that over , the multifunction pigeonhole principle for such predicates is equivalent to an ``iterative'' circuit block-recognition principle. A consequence of this is that if proves this circuit iteration principle then RSA is vulnerable to quasi-polynomial time attacks. Minimization and multifunctions (with C. Pollett) Theoretical Computer Science Note: Abstract: The implicit characterizations of the polynomial-time computable functions given by Bellantoni-Cook and Leivant suggest that this class is the complexity-theoretic analog of the primitive recursive functions. Hence it is natural to add minimization operators to these characterizations and investigate the resulting class of partial functions as a candidate for the analog of the partial recursive functions. We do so in this paper for Cobham's definition of

22. Pure Lambda Calculus It is possible to ;; write recursive functions within the lambda calculus, but it is ;; impossible to stop at the base case without relying on knowledge of http://www.cs.cmu.edu/~dst/DeCSS/Gallery/css-descramble-lambda.txt

< subfunctions ((lambda (reversed-join generalized-filter z4 z8 z1920 zxor) ; < more subfunctions ((lambda (stream->bitstream bitstream->stream xor-bitstreams) ; < more subfunctions ((lambda (test-encryption-flag get-sector-key crypt) ; < subfunctions (lambda (key sector-body) (xor-bitstreams (z* z8 z1920) (unmangle sector-body) (cipher-bitstream key)) )) ;; ================ ;; cipher-bitstream ;; ================ ;; This function returns the pseudo-random bitstream generated ;; by the given key. The random number generator uses two ;; LFSRs, one of 17 bits and the other of 25 bits, initialized ;; from the "key" argument. The outputs of the two LFSRs are ;; added together (with carry) to produce the final result. ;; The 17-bit LFSR's output is negated before the addition. ((lambda (tap-lfsr17 tap-lfsr25 make-bitstream negate-bitstream add-bitstreams) ; < subfunctions (lambda (n stream) (generalized-filter stream n (lambda (s tail) (byte->bitstream (zcar s) tail) ) zcdr z1 ))) ;; =============== ;; byte->bitstream ;; =============== ((lambda (shift) ;

23. Lambda Calculus The pure calculus appears to lack recursion (or equivalently iteration) but recursive functions can in fact be defined, as examples also demonstrate. http://www.allisons.org/ll/FP/Lambda/

var REMOTE_HOST='(none)', REMOTE_ADDR='67.18.104.18'; Lambda Calculus LA home FP Intro. Examples ... Interp.(S) Also see Prolog application abstraction Constants including integers, booleans and so on can all be defined using just the syntax above, as some of the examples demonstrate. Because of this it is sometimes convenient to extend the syntax with a fifth option of constants (0, 1, 2, ..., true, false, and, but, if that is done, it is only a convenience and does not increase the power of the language. Note that an abstraction defines an anonymous function. can in fact be defined, as examples also demonstrate NB. The applet above needs Java on. The introduction describes the semantics of and programming techniques, and the interpreter shows how it can be made to work. window on the wide world: The Darwin Awards 4: Intelligent Design Linux free op.-sys. OpenOffice free office-suite, ver. 2.2+. The GIMP ~ free photoshop. Firefox web browser. http://www.allisons.org/ll/ or as otherwise indicated Created with "vi (Linux + Solaris)", charset=iso-8859-1, fetched Sunday, 23-Dec-2007 03:57:58 EST.

24. CiteULike: Comparing Computational Power For example, one says that the (untyped) lambda calculus is as powerfulcomputationally speakingas the partial recursive functions, because the lambda http://www.citeulike.org/group/12/article/383919

Register Log in FAQ CiteULike News Discussion Journals Browse current issues Groups Search groups Info Library Messages ... Import Comparing Computational Power Authors Boker U Dershowitz N Online Article arXiv (abstract): View article online arXiv (PDF): View article online Note: You or your institution must have access rights to this article. CiteULike will not help you view an online article which you aren't authorized to view. Copy-and-Pasteable Citation (23 Oct 2005) Citation format: Plain APA Chicago Elsevier Harvard MLA Nature Oxford Science Turabian Vancouver Philosophy_of_Information's tags for this article comparing computational power Other users who have posted this article Groups containing this article Everyone's tags for this article comparing computational power Abstract BibTeX Note: You may cite this page as: http://www.citeulike.org/group/12/article/383919 EndNote BibTeX Group Tags All tags in the group Philosophy_of_Information Filter: 4-dimensional aberration about absolute ... zitterbewegung CiteULike organises scholarly (or academic) papers or literature and provides bibliographic (which means it makes bibliographies) for universities and higher education establishments. It helps undergraduates and postgraduates. People studying for PhDs or in postdoctoral (postdoc) positions. The service is similar in scope to EndNote or RefWorks or any other reference manager like BibTeX, but it is a social bookmarking service for scientists and humanities researchers.

25. Introduction To Functional Programming (1996/7) The predecessor operation. Writing recursive functions fixed point combinators. Let expressions. lambda calculus as a declarative language. http://www.cl.cam.ac.uk/teaching/Lectures/funprog-jrh-1996/

Introduction to Functional Programming (1996/7) John Harrison This year's (1997/8) course is here . The present page has more material. Computer Science Tripos, Part II (General) and Diploma in Computer Science Twelve lectures, beginning on Fri 16th Jan 1997, ending on Tue 11th Feb 1997 Hopkinson Lecture Room Lecture Notes Entire notes (160 pages): DVI Postscript Title, preface, table of contents: DVI Postscript Chapter 1 - Introduction: DVI Postscript Chapter 2 - Lambda calculus: DVI Postscript Chapter 3 - Lambda calculus as a programming language: DVI Postscript Chapter 4 - Types: DVI Postscript Chapter 5 - A taste of ML: DVI Postscript Chapter 6 - Further ML: DVI Postscript Chapter 7 - Proving programs correct: DVI Postscript Chapter 8 - Effective ML: DVI Postscript Chapter 9 - Examples: DVI Postscript References: DVI Postscript Copies of slides Introduction and Overview: Colour Postscript Lambda calculus as a formal system: Colour Postscript Lambda calculus as a programming language: Colour Postscript Types: Colour Postscript ML: Colour Postscript Details of ML: Colour Postscript Proving programs correct: Colour Postscript Effective ML: Colour Postscript ML examples I: Symbolic differentiation: Colour Postscript ML examples II: Recursive descent parsing: Colour Postscript ML examples III: Exact real arithmetic: Colour Postscript ML examples IV: Prolog and theorem proving: Colour Postscript Additional material is available from those who have taught the course in previous years: Mike Gordon and Andy Gordon . The theoretical part of the course is also covered by lecture notes from

26. Abstraction (or, Lambda Calculus For Dummies) Previous message Abstraction (or, lambda calculus for Dummies) Things get more complicated if/when you define recursive functions, tho. http://lists.tunes.org/archives/tunes/1997-December/001257.html

Abstraction (or, Lambda Calculus for Dummies) Fare Rideau rideau@nef.ens.fr Tue, 16 Dec 1997 14:05:24 +0100 (MET) Previous message: Abstraction (or, Lambda Calculus for Dummies) Next message: Abstraction (or, Lambda Calculus for Dummies) Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] Why is a meta-operator different from a function? Doesn't higher-order mean functions can be meta-operators? Not exactly. lambda operates on parsed syntax, not on objects. It's a language construct. In expression "(lambda (x) x)", you don't apply a function "lambda" to objects "(x)" and "x"! Functions operate on objects. In expression "(f x)", function "f" operates on "x". The idea is that there is a difference between source code (which might itself be considered as an object in a reflective system), and the objects that it denotes. Surely, there might be somewhere a function that takes source code, and returns a function, but this has to do with reflection (in this case, absorption), not with higher-order functions (functional objects that take as parameter and/or return other functional objects). > Let me just make sure: when you say lambda you are talking about abstraction. When you say abstraction, to do it you would use lambda. I

27. Stephen Wolfram: A New Kind Of Science | Online Notes for The Notion of Computation Universality in Turing Machines and Other Systems. * lambda calculus. Formulations of recursive function theory from http://www.wolframscience.com/nksonline/page-1121c-text

Cookies Required A New Kind of Science See http://www.wolframscience.com/nksonlineFAQs.html for more information or send email to support@wolframscience.com Search site Get the NKSwire newsletter Send a Message

28. The Mellow Musings Of Dr. T : Recursive Lambda Expressions 2) Figure out how to make two mutually recursive functions in this way. Anyway, enough lambda calculus for one day. Now get back to work! http://blogs.msdn.com/madst/archive/2007/05/11/recursive-lambda-expressions.aspx

Welcome to MSDN Blogs Sign in Join Help The Mellow Musings of Dr. T This Blog Email Syndication RSS 2.0 Atom 1.0 Search Go Tags No tags have been created or used yet. Archives May 2007 (1) January 2007 (1) October 2006 (1) August 2006 (1) Recursive lambda expressions This is a very geeky post. The tiny piece of useful information comes right at the bottom. The rest of it is all artifacts of the obscure art of doing lambda calculus in C# , which can also be characterized as doing very much with very little, sacrificing only readability People sometimes complain that you cannot write a lambda expression that is recursive. Good old factorial, for instance, how to write that as a lambda expression? Well the fathers of lambda calculus, who invented the lambda expressions in the 1930’s struggled with that, too, and as you might have guessed they came up with a solution. In this post let us stand on the shoulders of those giants and see how you can get recursion into your own lambda expressions in C#. It may not be practical but it is fun! How to write factorial So what you really want is to be able to write the lambda expression: But that won’t work – we’re using fac to define fac, but we haven’t defined it yet! Usually when you want to use something you don’t know, you abstract over it at let someone pass it to you later. So we could abstract over the factorial function itself and write:

29. Lambda Calculus See the file fanf.lambda for more examples of lambda calculus source. . The `Y combinator is used for implementing recursive functions. http://www.ioccc.org/1998/fanf.hint

30. Re^2: Pissed Off About Functional Programming In 1931, Kurt Godel defined the theory of murecursive functions. generated by lambda calculus was actually the set of general recursive functions. http://www.perlmonks.org/?node_id=451236

31. Re: "Green Card" For Untyped Lambda Calculus? You ll find it hard to write some recursive functions even for this Previous by Thread, RE Green Card for untyped lambda calculus?, Mark P Jones http://osdir.com/ml/lang.haskell.general/2000-11/msg00096.html

var addthis_pub = 'comforteagle'; lang.haskell.general Top All Lists Date Thread Re: "Green Card" for untyped lambda calculus? Subject Re: "Green Card" for untyped lambda calculus? List-id nil :: List a The implementation I'm interested in (one without constructors) is: nil fornil forcons = fornil cons x xs fornil forcons = forcons x xs forlist fornil forcons ls = ls fornil forcons As one might guess, this implementation relies on infinite types, which becomes obvious when offering for example More with this subject... Current Thread Previous by Date: Showing Haskell Koen Claessen Next by Date: Release 2: Mondrian and Haskell for .NET Nigel Perry Previous by Thread: RE: "Green Card" for untyped lambda calculus? Mark P Jones Next by Thread: WoLLIC'2001 - Call for Papers Ruy de Queiroz 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 OSDir.com Red Hat to get new CEO from Delta Air Lines 21/12/07 16:01 from OSDir.com

32. Introduction To The Polymorphic Lambda Calculus Thus, the set of functions representable as terms in the polymorphic lambdacalculus (without a recursion operator) does not exhaust the recursive functions http://mathgate.info/cebrown/notes/reynolds89.php

TPS A higher-order theorem proving system The Omega Group This page was created and is maintained by Chad E Brown John C. Reynolds, "An Introduction to the Polymorphic Lambda Calculus." 1989. page TPS A higher-order theorem proving system The Omega Group

33. CS 152 Homework: Lambda Calculus There are three problems on implementing the lambda calculus and three problems . You may not use explicit recursion; if you want a recursive solution, http://www.eecs.harvard.edu/~nr/cs152/homework/lambda.html

CS152 Assignment: Lambda Calculus Due Tuesday, April 3, at 11:59PM. Setup and Instructions In your cs152 directory create a subdirectory called lambda . You will need to copy the files linterp.sml Lhelp.ui Lhelp.uo Makefile , and basis.lam from directory ~cs152/software/lambda Do all six problems below. There are three problems on implementing the lambda calculus and three problems on programming with Church numerals. For problems 1-3, modify the linterp.sml file. Put your answers for the Church numerals problems, 4-6, in a file called church.lam Introduction to the Lambda interpreter You will be working with a small, interactive interpreter for the lambda calculus. This section explains what syntax to use and how to interact with the interpreter. Concrete syntax All toplevel items in the lambda interpreter must be terminated with a semicolon. Comments are C++ style line comments, starting with the string and ending at the next newline. A toplevel item can be a term, a binding, or a use statement. A use statement loads a file into the interpreter as if it had been typed in directly. A

34. 22c:245 Advanced Artificial Intelligence 12/05/06, Defining recursive functions to the simply typed lambda calculus. Comparisons with the untyped calculus. Adding exceptions. http://www.cs.uiowa.edu/~tinelli/classes/185/Fall06/logs.shtml

The University of Iowa 22c:185 Programming Language Foundations Fall 2006 Course Info Announcements Staff and Hours Syllabus Course Work Class Logs Exercises Exams WebCT Resources Readings OCaml Learning Research Class Logs and Required Readings The cited references can be found in the Readings section. Date Topics Readings Introduction and syllabus overview. Main aspects of programming languages: syntax and semantics. Examples of syntax and semantics in natural languages. The case for formal semantics of PLs. Examples of how different semantics affect the equivalence or inequivalence of two program fragments. Perceived disadvantages of formal semantics. Advantages of formal semantics in implementation, verification and design of PLs. Brief overview of three major styles of formal semantics: operational, axiomatic and denotational. - Chap. 1 of [Win] - Chap. 11-16 of [Nis] (optional) Concrete and abstract syntax. Examples of concrete grammars and parse trees. Examples of abstract grammars and abstract syntax trees. IMP, a simple imperative language.

35. An Elementary Fragment Of Second-order Lambda Calculus A fragment of secondorder lambda calculus (System F) is defined that characterizes the elementary recursive functions. Type quantification is restricted to http://portal.acm.org/citation.cfm?id=1055695&dl=acm&coll=&CFID=15151515&CFTOKEN

36. Thinking Inside A Bigger Box » 2006 » January to evaluate recursive functions. It gives the mathematical construct of lambda calculus as much computing power as a programming language, without http://www.brodwall.com/johannes/blog/2006/01/

@import url( http://www.brodwall.com/johannes/blog/wp-content/themes/phoenixblue/style.css ); Thinking Inside a Bigger Box About Top articles Archive for January, 2006 Uncategorized One of the most profound ideas in lambda calculus, is the Y-operator. it extremely hard to understand. This blog is therefore an example of a quixotic undertaking: I want to see whether it is possible for me to explain the Y-operator so that you, gentle reader, can understand it at least a few years of college level mathematics. Background: Why Y? The Y-operator is part of lambda calculus, a field of logic that syntactically (that is, mechanically) transform expressions. A short (lambda (x) (* x x)) argument (called x ) and returns the square of the argument. Here is how you use it: The Y-operator is used to create recursive functions using lambdas. That is, functions that call themselves. That is its purpose . Y is defined as follows: (Y F) = (F (Y F)) For this to work, F better be a function. Read it as follows: An example: Factorial As I said: This allows us to define recursive stuff using just lambdas. To demonstrate, I will need a volunteer recursive function.

37. Discussion 5 - CS164-Fa07 1 Responses to Class Lecture; 2 lambda calculus; 3 Goal; 4 Booleans; 5 Lists; 6 Numbers function application, recursive functions factorial( ) http://play.cs.berkeley.edu/wiki/cs164/fa07/index.php/Discussion_5

Discussion 5 From CS164-Fa07 Jump to: navigation search Contents Responses to Class Lecture Lambda Calculus Goal Booleans ... edit Responses to Class Lecture Can we exploit parallelism without changing the development process? Smarter compilers: STM, static analysis, ... Smarter browser: lexer, parser, renderer What can we change to make code 'more easily' parallel? Compiler hints (type annotations, region annotations, data flow..) Change layout language ('easier' constraints to solve) Can we do a library based or otherwise backwards compatible approach? Is it worth it? Case studies: Flash, Opera edit Lambda Calculus Side note: this is a stolen lecture that I do not know the origin of, but for a good reason: it rocks. Jumping from the edge of computing, we're going back to where it started. This may be familiar to those who wrote Scheme code in 61A, but puts it on its head - as is one of the biggest gems of computer science. Once it was clear there was a limit to the Turing machines and the more mathematical systems, the next question was whether one was better than the other. Oddly enough, they came out the same. Yet, almost 80 years later, programmers seem to still be having this debate. As we're all used to Turing machines (reading and writing memory, albeit in fancier ways), we'll explore the other side of the coin. At the end of the tunnel, in addition to already being able to say "it's all just 1s and 0s", you will also believe "it's all just a bunch of lambdas". This is an essential instinct when writing compilers (programs that transform other programs), and provides insight into what we mean by compositional properties of languages (which mutation typically violates).

38. A FUNCTIONAL APPROACH TO COMPUTING Notes 2000-03-15 S.Whitney ALGOL 60 (rev.1963) 1960 McCarthy, recursive functions of symbolic expressions. (b) lambda calculus (function abstraction) f = Lxfx (lambda conversion) http://kx.com/technical/contribs/stephen/talk1.txt

39. Definitions Of Computable Church s Hypothesis on Computable; Turing Machines; lambda calculus; Post Formal Systems; Partial recursive functions; Unrestricted Grammars http://www.csee.umbc.edu/help/theory/computable.shtml

Definitions of computable Contents 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 Unrestricted Grammars 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

40. The Lambda Calculus Mail Series I emphasize on this because factorial is a common example of recursive function. recursive functions in lambda calculus, is a very interesting topic, http://www24.brinkster.com/srineet/lambda\lambda.html

Homepage The Lambda Calculus Mail Series Contents Series on Lambda Calculus Lets Start At The Very Beginning Lambda Calculus - The First Glimpse Digression - Russell's Paradox ... Recursive Function Theory - Another Summary 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.

41. Cdiggins.com The lambda calculus is also leftassociative so “a b c” means call the function Tail recursive functions are recursive functions that perform the http://www.cdiggins.com/index.php?s=associative

42. Programming Languages; CS6371 Programming with functions; lambda calculus and ML programming; Lecture 4, Lists and recursion;. Lecture 5, lambda calculus (handout given in class). http://www.utdallas.edu/~gupta/courses/apl/

CS6371: Advanced Programming Languages Spr 2007, Tue,Th 4:00-5:15 PM Classroom: ECSS 2.306 http://www.utdallas.edu/~gupta/courses/apl Click here for Homeworks and Solutions Instructor Gopal Gupta , ECSS 4.907, 883-4107, gupta@utdallas.edu Office hours: Wed 2-4PM Teaching Assistant: Ajay Bansal; ajay.bansal@student.utdallas.edu TA Office Hours and Office: 2-4pm Monday; ECSS 4.404 Goal This course will deal with Advanced Concepts in Programming Languages. The following are the course learning objectives: Lambda Calculus and functional programming Logic Programming and Prolog Denotational Definition of Programming Languages Fixpoints; Program Verification We will first cover functional programming, followed by logic programming. Then we will switch to studying semantics of programming languages (Operational, Denotational and Axiomatic). We will also study basics of program verification. Good understanding of set theory, discrete mathematics, and data structures is required for this course. In the semantics part, significant amount of time will be devoted to denotational semantics. Outcomes This course will help in considerably improving your understanding of the process of programming and the structure of computation. You will also learn how to

43. Lambda Calculus recursive or selfreferential definitions are not needed to write a recursive function in lambda calculus! The function Y gives the effect of recursion. http://www.csse.monash.edu.au/~lloyd/tildeFP/Lambda/Ch/01.Calc.html

Lambda Calculus. The syntax of the lambda calculus is short and simple. There are clauses for variable identifiers, constants, function abstraction, function application and subexpressions. This simplicity provides great power, an example of `less is more'. A function abstraction is an expression for a function. The identifier is the name of the parameter; it is said to be bound . An unbound identifier is free . The function itself has no name. Application is left associative, f x y=(f x)y. The clause for constants can be omitted because constants can be defined with what is left but this is inconvenient and does not aid our purpose. (But see [ints] [bool] and [list] if interested.) The constants certainly include true, false, integers, some functions such as plus, minus, and a conditional function (if). Just what the set of constants contains is not crucial; it is a parameter of the lambda calculus rather than a central part of it, and the set can be varied from application to application. The previous grammar forces the use of prefix notation. It costs nothing to extend the grammar to include prefix and infix operators. The gain is purely in convenience not in power or difficulty of implementation.

44. Fall 99, CSE 520: Lectures We give here the translation in ML and we leave the translation from ML to the lambda calculus as an exercise. Let f be defined by primitive recursion as http://www.lix.polytechnique.fr/~catuscia/teaching/cg520/99Fall/lecture_notes/L0

Fall 99, CSE 520: Lectures 3, 4 and 5 Expressiveness of the Lambda Calculus These lectures will focus on the computational power of the Lambda Calculus. We will show that the Lambda Calculus is computationally complete, in the sense that it is able to express any computable function. This result was proved by Kleene in 1936. Church's numerals In order to show the above result, we first need to encode the natutal numbers as lambda terms. The following is a possible representation: n y where, for generic lambda terms M and N, M n N represents the term M(...(M(M N))...), i.e. the application of M to N repeated n times. These terms [n] are called Church's numerals. Lambda-definability A partial function f : N k -> N is lambda-definable if there exists a lambda term [f] such that, whenever f(n ,...,n k ) is defined, we have [f][n ]...[n k ] = [f(n ,...,n k The equality here is the "lambda-convertibility" (see previous lecture), i.e. the equality in the theory of Lambda Calculus. Note that [f] is curried, i.e. it takes its arguments one by one. Recursive Functions Initial functions Zero: Z(n) = Successor: S(n) = n+1 Projection: U p i (n ,...,n

45. Lambda Calculus Introduction functions in lambda calculus take only one argument, but we can get the . This is starting to look like recursion, since reduces to a longer term that http://sage.mc.yu.edu/kbeen/teaching/toc/resources/lambda.html

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 K 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

46. JSTOR Recursion Theory And The Lambda-Calculus A semantics for the lambdacalculus due to Friedman is used to describe a large and natural class of categorical recursion-theoretic notions. http://links.jstor.org/sici?sici=0022-4812(198203)47:1<67:RTATL>2.0.CO;2-Y

47. Online Otter-λ A fundamental theorem about lambda calculus permits the definition of functions by recursion. The proof is a deep mystery, in the sense that it is very http://michaelbeeson.com/research/otter-lambda/index.php?include=LambdaCalculus

48. Education, Master Class 1988/1999, MRI Nijmegen Course content 1st semester Model Theory W. Veldman lambda calculus - H. Barendregt, E. Barendsen Recursion Theory and Proof Theory - H. Schellinx http://www.math.uu.nl/mri/education/course_9899.html

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 seminar 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.

49. Lambda Calculus And Lambda Calculators Several lambdacalculators and many applications of lambda calculus. We impose an additional constraint of avoiding general recursion. http://okmij.org/ftp/Computation/lambda-calc.html

previous next contents top Lambda Calculus and Lambda Calculators Negative numbers, subtraction and division in lambda-calculus Basic Lambda Calculus terms Numerals and lists in lambda-calculus: P-numerals The bluff combinator ... Predecessor and lists are not representable in simply typed lambda-calculus Lambda-calculators In Haskell In Scheme As a CPS R5RS Scheme macro As a short, direct-style R5RS Scheme macro Negative numbers, subtraction and division in lambda-calculus This article will demonstrate basic arithmetic operations comparison, addition, subtraction, multiplication, and division on non-negative and negative integer numbers. Both the integers and the operations on them are represented as terms in the pure untyped lambda-calculus. The only building blocks are identifiers, abstractions and applications. No constants or delta-rules are used. Reductions follow only the familiar beta-substitution and eta-rules. We also show two approaches of defining a predecessor of a Church numeral. Curiously, none of the arithmetic operations involve the Y combinator; even the division gets by without Y. Addition, multiplication, squaring, and exponentiation are easily expressed in lambda-calculus. We discuss two approaches to their "opposites" subtraction, division, finding of a square root and of a discrete logarithm. The "opposite" operations present quite a challenge. One approach, based on fixpoints, is more elegant but rather impractical. The "dual" approach, which relies on counting, leads to constructive and useful, albeit messy, terms.

50. Harry Mairson How light is safe recursion? Translations between logics of polynomial time (with P.M. Neergaard). Linear lambda calculus and polynomial time. http://www.cs.brandeis.edu/~mairson/

Harry Mairson Ph.D., Computer Science, Stanford University, 1984 mairson at cs dot brandeis dot edu 257 Volen / (781) 736-2724 Research Interests Optimal evaluation Game semantics and proof structures Expressibility and typed lambda calculus ... Other Slides from my invited talk at ICFP 2003, From Hilbert space to Dilbert space: context semantics made simple. (Comments welcome. Sorry this is a big file.) Research Interests My research examines the interaction between mathematical logic and computation theory, with application to the design and analysis of functional programming languages, type systems, and database query languages. A primary current research interest is optimal evaluation in lambda calculus and its computational complexity. I also do fundamental research in algorithmics and algorithm analysis. Optimal evaluation: An evaluator for lambda calculus (or more broadly speaking, a functional programming language) is said to be correct and optimal if it returns a normal form whenever there is one (i.e., it never diverges by choosing to evaluate the wrong redex), and never does inessential work (i.e., copying redexes). Recently, several researchers, including Lamping, Gonthier, Abadi, Levy, and Asperti, have constructed optimal evaluators. But are they efficient? I have examined the computational complexity of this problem via the algorithmic analysis of these proposed evaluators. In addition, this work has addressed what reasonable

51. LAMBDA-CALCULUS MODELS OF PROGRAMMING LANGUAGES. The main result on recursion is an analogue to Kleene s first recursion theorem If A for the lambdacalculus and its semantic assumptions are identified. http://stinet.dtic.mil/oai/oai?verb=getRecord&metadataPrefix=html&identifier=AD0

52. Lambda Calculus | Lambda The Ultimate Just as LCF is a logic for the simplytyped lambda-calculus with recursion and arithmetic, our logic is a logic for System F. The logic permits the formal http://lambda-the-ultimate.org/taxonomy/term/20?from=10

53. IngentaConnect Lambda Calculus With Explicit Recursion lambda calculus with Explicit Recursion. Authors Ariola Z.M.1; Klop J.W.2, 3. Source Information and Computation, Volume 139, Number 2, December 1997 , pp http://www.ingentaconnect.com/content/ap/ic/1997/00000139/00000002/art02651

var tcdacmd="dt"; Home About Ingenta Ingenta Labs Ingenta Blog ... Check our FAQs Or contact us to report problems with: Subscription access Article delivery Registration Library administrator tasks ... Other problems For Publishers Why go online? Why choose IngentaConnect? Beyond Print: enhancing your service Access and authentication ... Contact us For Researchers For Authors About IngentaConnect Search and browse Publications available ... Register For Librarians Resource Zone Why choose IngentaConnect? Why choose IngentaConnect Complete? Activating Subscriptions ... Register CM8ShowAd("HorizontalBanner"); Lambda Calculus with Explicit Recursion Authors: Ariola Z.M. ; Klop J.W. Source: Information and Computation , Volume 139, Number 2, December 1997 , pp. 154-233(80) Publisher: Academic Press view table of contents Key: - Free Content - New Content - Subscribed Content - Free Trial Content CM8ShowAd("Skyscraper"); Abstract: Language: English Document Type: Research article Affiliations: Computer and Information Science Department, University of Oregon, Eugene, Oregon, 97401

54. Harry Mairson Optimal evaluation An evaluator for lambda calculus (or more broadly speaking, beta steps is not bounded by any Kalmarelementary recursive function. http://cs-people.bu.edu/mairson/

Harry Mairson Ph.D., Computer Science, Stanford University, 1984 Visiting Professor of Computer Science mairson@cs.bu.edu 283 MCS / (617) 353-8923 Research Interests Optimal evaluation Game semantics and proof structures Expressibility and typed lambda calculus ... Other Research Interests My research examines the interaction between mathematical logic and computation theory, with application to the design and analysis of functional programming languages, type systems, and database query languages. A primary current research interest is optimal evaluation in lambda calculus and its computational complexity. I also do fundamental research in algorithmics and algorithm analysis. Optimal evaluation: An evaluator for lambda calculus (or more broadly speaking, a functional programming language) is said to be correct and optimal if it returns a normal form whenever there is one (i.e., it never diverges by choosing to evaluate the wrong redex), and never does inessential work (i.e., copying redexes). Recently, several researchers, including Lamping, Gonthier, Abadi, Levy, and Asperti, have constructed optimal evaluators. But are they efficient? I have examined the computational complexity of this problem via the algorithmic analysis of these proposed evaluators. In addition, this work has addressed what reasonable

55. Typed Lambda Calculus And Applications Elementary Affine Logic and the Callby-Value lambda calculus . Recursion from Cyclic Sharing Traced Monoidal Categories and Models of Cyclic lambda http://wotan.liu.edu/docis/dbl/tlcatl/index.html

The Digital Librarian's Digital Library search D O CIS  Do cuments in  C omputing and I nformation  S cience Home Journals and Conference Proceedings Typed Lambda Calculus and Applications Untyped Algorithmic Equality for Martin-L¶f's Logical Framework with Surjective Pairs by:  Andreas Abel, Thierry Coquand A Lambda Calculus for Quantum Computation with Classical Control by:  Peter Selinger, Benoit Valiron Elementary Affine Logic and the Call-by-Value Lambda Calculus by:  Paolo Coppola, Ugo Dal Lago, Simona Ronchi Della Rocca Binding Signatures for Generic Contexts by:  John Power, Miki Tanaka A Feasible Algorithm for Typing in Elementary Affine Logic by:  Patrick Baillot, Kazushige Terui Semantic Cut Elimination in the Intuitionistic Sequent Calculus by:  Olivier Hermant Galois Embedding from Polymorphic Types into Existential Types by:  Ken-etsu Fujita Arithmetical Proofs of Strong Normalization Results for the Symmetric lambda-µ-Calculus by:  Ren© David, Karim Nour Filters on CoInductive Streams, an Application to Eratosthenes' Sieve by:  Yves Bertot Completeness Theorems and lambda-Calculus by:  Thierry Coquand L3: A Linear Language with Locations by:  Greg Morrisett, Amal J. Ahmed, Matthew Fluet

56. Lambda Calculus - Wikipedia, The Free Encyclopedia Recursion is the definition of a function using the function itself; on the face of it, lambda calculus does not allow this. However, this impression is http://en.wikipedia.org/wiki/Lambda_calculus

var wgNotice = ""; var wgNoticeLocal = ""; var wgNoticeLang = "en"; var wgNoticeProject = "wikipedia"; Lambda calculus From Wikipedia, the free encyclopedia Jump to: navigation search This article needs additional citations for verification Please help improve this article by adding reliable references . Unsourced material may be challenged and removed. (October 2007) This article or section includes a list of references or external links , but its sources remain unclear because it lacks in-text citations 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

57. Sketchy LISP - C3 S5 lambda calculus (LC) is the basis of all programming languages of the LISP . When using the Y combinator of LC in Scheme, indefinite recursion occurs http://www.t3x.org/sketchy/vol1/sl21.html

t3x.org sketchy sl21.html Sketchy LISP (C) 2006,2007 Nils M Holm Buy a copy at Lulu.com An Introduction to Functional Programming in Scheme Previous Section Contents Index Next Section 3.5 Lambda Calculus and the Y Combinator Earlier in this book, the self-application combinator S was introduced as a device for the construction of anonymous recursive functions: (lambda (f x) (f f x)) A combinator is a function that does not make use of any free variables, which is why it cannot even use basic operations like or car . So it is basically limited to binding constructs, function applications, and its (bound) variables. While the self-application combinator works fine, it has one limitation. It requires two arguments: the function to be applied and the value the function is to be applied to. It would be much more elegant, if it could just transform a function into a recursive function. Such an operator would turn the following factorial function (which does not work because (lambda (x) ...) closes over the free variable f ) into a working recursive function R Indeed a function performing this mystery transformation exists. It is called the

58. Rubycond: The Birth Of Recursion From Self-application In Lambda Calculus }).call(5). Posted by Rubycond at 448 PM. Labels lambda calculus, Lisp, recursion, Ruby, Scheme, selfapplication, U combinator, Y combinator http://rubycond.blogspot.com/2006/10/birth-of-recursion-from-self.html

skip to main skip to sidebar Rubycond A Passion Play on Ruby. Be led within temptation by this gem of a language and get your daily blood-red aluminum oxide dose. Monday, October 16, 2006 The birth of recursion from self-application in lambda calculus Reading Many faces of the fixed-point combinator and Y overriding self-application by Oleg Kislyov (oleg-at-pobox.com), I came to understand how to express recursive functions in lambda calculus in an elegant way. Oleg's code was written in Scheme, but it was clear that it was easily translated in pure lambda calculus. I used the Lambda calculator by Carl Burch to do this for the (recursive) factorial function Afterwards, I found on the CLiki Factorial page a Common Lisp translation of the above: ((lambda (f) #'(lambda (n) (funcall f f n))) This would probably look much better in Scheme, and I am beginning to understand why Scheme is regarded as a "better Lisp" than Lisp. Just when I was contemplating "porting" the lambda expression to Ruby, here it is... well, almost. Actually, the Y combinator is used here, instead of the U combinator used by Oleg in the above articles. So, here is the true Ruby translation (actually a sample call of it): p lambda f f call f call lambda recurse lambda n if n then else n recurse call recurse call n end call Posted by Rubycond at 4:48 PM Labels: lambda calculus Lisp recursion Ruby ... Y combinator comments: Post a Comment Newer Post Older Post Home Subscribe to:

59. Rotten Cotton » Blog Archive » Bootstrapping From The Pure Lambda Calculus We don’t have Scheme’s define or ML’s letrec to implement recursion in the lambda calculus, so how do we write the ubiquitous recursion example, factorial? http://onebadseed.com/blog/?p=34

60. Lambda Calculus How then does lcalculus express a lambda form that has 2 or more arguments? .. While recursion doesn’t come for free in l-calculus (as you will see) we http://www.mactech.com/articles/mactech/Vol.07/07.05/LambdaCalculus/

MacTech Network: MacForge.net Computer Memory Register Domains ... Mac Book Shelf The journal of Macintosh technology Magazine In Print About MacTech Home Page Subscribe MacTech CD ... Get a copy of MacTech RISK FREE Entire Web mactech.com Mac Community More... MacTech Central by Category by Company by Product MacTech News MacTech News Previous News MacTech RSS Article Archives Show Indices by Volume by Author Source Code FTP Inside MacTech Writer's Kit Editorial Staff Editorial Calendar Back Issues ... Advertising Contact Us Customer Service MacTech Store Webmaster Feedback ADVERTISEMENT Volume Number: Issue Number: Column Tag: Lisp Listener Lambda Calculus Abe Lincoln Introduction 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:

61. Why Oh Why Y In lambda calculus, all iteration is done by recursion. In fact, recursion is a pretty natural way of expressing iteration. It takes a bit of getting used http://science.blogdig.net/archives/articles/August2006/31/Why_oh_why_Y_.html

@import url("http://www.blogdig.net/page/global_css/global.css"); var sc_project=1489011; var sc_invisible=1; var sc_partition=13; var sc_security="996db76a"; //auction ads var sd_keywords = "science, observation, biology, chemistry, astronomy, geology" Method Creepy Texas dentist slathers on the smarm Home add your blog Archives ... Gadget Gift Ideas Cool Stuff Join the Live Discussion News Ticker RSS News Feed Have A Great "science" BLOG? Add Your Blog To this Site ... Marine Engineers Beneficial Today's Posts View All Today's Posts (12) A mystery Mars rovers find new evidence of 'habitable niche'; perilous third winter approaches Ridicule works! ... Creepy Texas dentist slathers on the smarm Links Witches and Scientists Pinoy Explorer science projects for kids Aetiology ... Wolverines Den BlogDig Sites Click here for a list of our sites oswd.org Why oh why Y post from Good Math, Bad Math So in the last few posts, I've been building up the bits and pieces that turn lambda calculus into a useful system. We've got numbers, booleans, and choice operators. The only thing we're lacking is some kind of repetition or iteration. In lambda calculus, all iteration is done by recursion. In fact, recursion is a pretty natural way of expressing iteration. It takes a bit of getting used to, but anyone who's programmed in a language like Scheme, ML, or Haskell for a while gets very used to idea, and feels frustrated coming back to a language like Java, where you need to write loops.

62. Into The Wibble [lambda-calculus] Recursion in the lambda calculus can be done with a fixpoint operator. A nice one is Turing s fix = (\xy.(y (xxy)) \xy.(y (xxy))) http://frmb.org/lambda.html

Fred's Home Page Main About me csh is bad Debian ... XHPD Lambda calculus intro basic lists recursion ... download This page is all about my lambda computer a program that computes in the lambda calculus. For fun, I also did a Turing machine in lambda calculus (fairly trivial representation). Introduction The lambda calculus is basically function theory, there are only three things available in the pure lambda calculus, these are: Variables a the variable a Applications (a b) a applied to b Abstractions function e with argument x Dispite not being very much here, the expressive power is quite suprising. Using these three simple things, numbers, lists, recursion and much more can be defined. This page is not intended to be an introduction to the lambda calculus; there are lots of good books for that. One excellent one is Barendregt's "The Lambda Calculus : its syntax and semantics". ISBN: 0444875085 (paper-back) and 0444867481 (hard-back). It's rather mathematically extreme though... Acknowledgement: Most of the terms here are from the CO611 (types and computation) course at Kent Uni, kindly taught by Stefan Kahrs and Simon Thompson. The type-theory computing program may happen at some point too.. Here is a picture of the program running.

63. Examples Illustrating The Lambda Calculus. Compared To The Pure Compared to the pure lambda calculus, the examples use the Haskell notation n = n (+1) 0 To illustrate the lambda calculus approach to recursion, http://www.cs.bris.ac.uk/Teaching/Resources/COMS21101/examples/lambda.hs