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1. MainFrame: The Lambda-calculus, Combinatory Logic, And Type Systems
Pure Combinatory logic is so closely related to Church s lambdacalculus that it is best studied alongside the lambda-calculus, for which the most
http://rbjones.com/rbjpub/logic/cl/cl017.htm
##### Overview:
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.

2. J Roger Hindley : Research
Mathematical logic; particularly lambdacalculus, Combinatory logic and type-theories, with a current bias towards historical aspects.
http://www-maths.swan.ac.uk/staff/jrh/JRHresearch.html
Swansea University Physical Sciences Mathematics Department J Roger Hindley
##### Fields of Interest
Mathematical logic; particularly lambda-calculus, combinatory logic and type-theories, with a current bias towards historical aspects. Lambda-calculus and combinatory logic are formal systems, to some extent rivals, used in the construction and study of programming languages which are higher-order (i.e. in which programs may change other programs). These two systems were invented in the 1920s by mathematicians for use in higher-order logic, and came to be applied in programming theory from the 1970s onward, when that theory expanded to cover higher-order computations. In a type-theory, types are labels which may be attached to certain programs to show what other programs they can change. A type-system is a particular set of rules for attaching types; the rules themselves are usually reasonably simple, but such questions as what programs are typable, what set of types a program may receive, and whether a typable computation can continue indefinitely, are not always easy to answer and have occupied many researchers.
##### Main Publications
J R Hindley, J P Seldin

 3. The Mixed Lambda-Calculus And Combinatory Logic (an Overview) The Mixed lambdacalculus and Combinatory logic (an overview). H. Riis Nielson, F. Nielson. Type, Conference paper With refereehttp://www2.imm.dtu.dk/pubdb/views/publication_details.php?id=1601

 4. JSTOR Introduction To Combinators And $\lambda$-Calculus. lambdacalculus and Combinatory logic (pure and typed) have been rapidly increasing in importance during the last twenty years because of their applicationshttp://links.jstor.org/sici?sici=0022-4812(198809)53:3<985:ITCA>2.0.CO;2-#

5. EWSCS 2004/EATTK 2004: Sergei Artemov, Abstract
The logic of Proofs became both an explicit counterpart of modal logic and a reflexive Combinatory logic (reflexive lambdacalculus) thus providing a
http://cs.ioc.ee/yik/schools/win2004/artemov.php
CIDEC
Estonian Winter Schools in Computer Science

EWSCS 2004 ...
EATTK 2004
##### Sergei Artemov
City University of New York

USA
##### Abstract
According to Brouwer, the truth in intuitionistic logic means provability. On this basis Heyting and Kolmogorov introduced an informal Brouwer-Heyting-Kolmogorov (BHK) semantics for intuitionistic logic. The ideas of BHK led to a discovery of computational semantics of intuitionistic logic, in particular, realizability semantics and Curry-Howard isomorphism of natural derivations and typed lambda-terms. However, despite many efforts the original semantics of intuitionistic logic as logic of proofs did not meet, until recently, an exact mathematical formulation. GÃ¶del in 1933 suggested a mechanism based on modal logic S4 connecting classical provability (represented as S4-modality) to intuitionistic logic. This did not solve the BHK problem, since S4 itself was left without an exact provability model. In 1938 GÃ¶del suggested using the original BHK format of proof carrying formulas to build a provability model of S4. This GÃ¶del's program was accomplished in 1995 when proof polynomials and the Logic of Proofs (LP) were discovered, shown to enjoy a natural provability semantics, and to be able of realizing all S4-modalities by proof polynomials. The Logic of Proofs became both an explicit counterpart of modal logic and a reflexive combinatory logic (reflexive lambda-calculus) thus providing a uniform mathematical model of knowledge and computability.

 6. Deff - Search Service - Research Database The twolevel version of the Combinatory logic therefore is a mixed lambda-calculus and Combinatory logic. They extend the mixed lambda-calculus andhttp://forskningsbasen.deff.dk/ddf/rec.external?id=dtu199779

7. People
Research interests. Mathematical logic; particularly lambdacalculus, Combinatory logic and type-theory. Books published. Combinatory logic Vol.
http://www.wolfengagen.mephi.ru/people.htm
Web
Research interests

J.Roger Hindley
##### Research interests
• Mathematical logic; particularly lambda-calculus, combinatory logic and type-theory.
Books published
• Combinatory Logic Vol. II, North-Holland Co., 1972 (with H. B. Curry, J. P. Seldin). Introduction to Combinatory Logic , Cambridge Univ. Press, 1972 (with B. Lercher, J. P. Seldin). To H. B. Curry , Academic Press, 1980 (edited, with J. Seldin). Introduction to Combinators and Lambda-calculus , Cambridge Univ. Press 1986 (with J. P. Seldin). Basic Simple Type Theory, Cambridge Univ. Press 1995.
• 8. Springer Online Reference Works
In pure Combinatory logic there is a set of terms built by application from for the whole of logic by a complicated combination of lambdacalculus and
http://eom.springer.de/i/i110060.htm
 Encyclopaedia of Mathematics I Article refers to Illative combinatory logic Combinatory logic , which began with a paper by , was developed by H.B. Curry and others with the intention of providing an alternative foundation for mathematics. Curry's theory is divided into two parts: pure combinatory logic ( ), concerning itself with notions like substitution and other (formula) manipulations; and illative combinatory logic ( ), concerning itself with logical notions such as implication, quantification, equality, and types. In pure combinatory logic there is a set of terms built by application from variables and two constants and , and there are two conversion rules: and . In the presence of the rule of extensionality, the theory is equivalent with untyped lambda-calculus (cf. also -calculus ) with -conversion. contains all of , but the alphabet is extended by extra logical constants, and there are derivation rules capturing the intended meaning of these constants. Also, in the early papers A. Church

History of lambdacalculus and Combinatory logic. In Gabbay D and Woods J (Eds.). Handbook of the History of logic(Elsevier, Amsterdam) to appear.
http://logcom.oxfordjournals.org/cgi/content/full/exm001v1

10. Lambda-Calculus And Computer Science Theory 1975
Corrado BÃ¶hm (Ed.) lambdacalculus and Computer Science Theory, Proceedings of the for identifying two elements of whatever model of Combinatory logic.
http://www.informatik.uni-trier.de/~ley/db/conf/lambda/lambda1975.html
##### 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

11. Major Scientific Accomplishments 2000. Reflexive Lambda-calculus
It may be regarded as the pure lambda version of the logic of Proofs LP (cf. and Combinatory logic, typed lambdacalculus and modal lambda-calculus.
http://www.cs.gc.cuny.edu/~sartemov/accom.html
 Major scientific accomplishments . Reflexive lambda-calculus. The Curry-Howard isomorphism converting intuitionistic proofs into typed lambda-terms is a simple instance of an internalization property of a our system lambda-infinity which unifies intuitionistic propositions (types) with lambda-calculus and which is capable of internalizing its own derivations as lambda-terms. We establish confluence and strong normalization of lambda-infinity. The system lambda-infinity is confluent and strongly normalizable (a joint result with J.Alt) , considerably extends the expressive power of each of its major components: typed lambda-calculus, intuitionistic and modal logic. It may be regarded as the pure lambda version of the Logic of Proofs LP (cf. below). Reflexive lambda-calculus is likely to change our conception of the appropriate syntax and semantics for ambda-calculus based programming languages, systems of automated deduction and formal verification. . An explicit provability model for verification. The traditional (implicit) provability model leaves a certain loophole in the foundations of formal verification. Namely, by the Goedel Incompleteness Theorem an extension of a verification system by a verified rule generally speaking is not equivalent to the original system. This model leads to a "reflection tower" of metatheories of increasing proof-theoretical strength which itself cannot be verified formally. One has to believe in correctness of a verification system the way we believe in consistency of a set theory.

12. Longo Symposium
in concerns syntactic and semantic properties of the logical base of functional languages Combinatory logic, lambdacalculus and their extensions.
http://www.pps.jussieu.fr/~gc/other/rdp/talks.html
 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 MÂ©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: "GÂ©omÂ©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 "ComplexitÂ© et information morphologiques" (CIM), centered on foundational problems in the interface between Mathematics, Physics and Biology.

Research interests Theory of computing, Constructive Mathematics, Combinatory logic, lambdacalculus, Semantics and Implementation of Functional
http://www.dsi.uniroma1.it/~boehm/
 Professor Born in Milan (Italy), 1923 Electronic Engineering at EPUL (Lausanne, Switzerland), 1946 "Honoris causa" Degree in Computer Science, University of Milan, 1994 First appointment at University of Rome "La Sapienza" Research interests : Theory of computing, Constructive Mathematics, Combinatory Logic, Lambda-Calculus, Semantics and Implementation of Functional Programming Languages. Before joining "La Sapienza", he was full time researcher at the INAC-CNR in Rome, from 1953 to 68, doing research on Turing machines and on Structured Programming. From 1959 to 69 he taught Numerical Analysis, Programming Techniques and Mathematical Logic at the Universities of Pisa and Rome. In 1970 he joined the University of Turin to cover the first professorship assigned in Italy in Computer Science. His research activity began with the construction of the first compiler written in its own language. Then his studies on Turing machines and Von Neumann machines ended with a theorem (in collaboration with Jacopini) of great interest for the development of Structured Programming. The investigations on lambda-calculus and combinatory logic were centered on the notion of normal form, and a theorem, very important for the semantics of programming languages, was delivered. The more recent research is addressed to the implementation of a pure functional programming system, based on the discovery that not only data and results may be conveniently represented by normal forms, but also functions and functionals. He is a member of IFIP WG 2.2 on concept formation and its formalization and of IFIP WG 2.8 on functional programming. He is member of the Editorial Board of

14. Historical... Data Retrieval [EX Re: [PT] Question On ND And Sequents]
was already aware of something similar to typed lambdacalculus / Combinatory logic (modulo Curry-Howard , so to speak)! only scarce evidence survived
http://osdir.com/ml/science.mathematics.prooftheory/2005-06/msg00001.html
##### historical... data retrieval [EX Re: [PT] question on ND and sequents]
Subject historical... data retrieval [EX Re: [PT] question on ND and sequents] On May 31, 2005, at 12:28 PM, Adrian Rezus wrote: You are possibly right about my absent-mindedness: I can't remember. But please tell me what I absent-mindedly overlooked in stating that Gentzen's first consistency proof in effect involves a formulation of natural deduction using sequents. Your earlier posting on Lukasiewicz's invention of natural deduction is interesting. Do you mean that he explicitly wrote down a complete set of rules of inference governing sequents? (Studia Logica 1934 is not easily available to me now.) Your reference to Hertz does not reassure me on this point. TruthI should probably say credit for truthis ultimately in the details one presents. http://www.vlab.pub.ro/equivalences/_lambda-and-proofs/_ar/beyond.pdf and only an extended abstract'' is in print so far; online copy @ http://www.vlab.pub.ro/equivalences/_lambda-and-proofs/_ar/bbhk93ea.pdf

15. John's Combinatory Logic Playground
Pictured above you can see on the left the 210 bit binary lambda calculus selfinterpreter, and on the right the 272 bit binary Combinatory logic
http://www.cwi.nl/~tromp/cl/cl.html
##### John's Lambda Calculus and Combinatory Logic Playground
Pictured above you can see on the left the 210 bit binary lambda calculus self-interpreter, and on the right the 272 bit binary combinatory logic self-interpreter Both are explained in detail in my latest paper available in PostScript and PDF . This design of a minimalistic universal computer was motivated by my desire to come up with a concrete definition of Kolmogorov Complexity , which studies randomness of individual objects. All ideas in the paper have been implemented in the the wonderfully elegant Haskell language, which is basically pure typed lambda calculus with lots of syntactic sugar on top. The implementation also offers a Main module that lets you run the universal binary lambda computer on the command-line argument joined to standard input as a list of bytes: main = do [program] The first example is the half-byte program for identity=^x.x, while the second is the ten-and-a-half-byte program for stutter=^l l(^h^t^n^z z h (^z z h (stutter t)))nil. This online course at Oberlin College provides a very readable introduction to combinators.

16. Combinatory Logic - Wikipedia, The Free Encyclopedia
(1972) survey the early history of Combinatory logic. For a more modern parallel treatment of Combinatory logic and the lambda calculus, see Barendregt
http://en.wikipedia.org/wiki/Combinatory_logic
var wgNotice = ""; var wgNoticeLocal = ""; var wgNoticeLang = "en"; var wgNoticeProject = "wikipedia";
##### Combinatory logic
Jump to: navigation search Not to be confused with combinational logic , a topic in digital electronics. Combinatory logic is a notation introduced by Moses SchÂ¶nfinkel and Haskell Curry to eliminate the need for variables in mathematical logic . It has more recently been used in computer science as a theoretical model of computation and also as a basis for the design of functional programming languages . It is based on combinators . A combinator is a higher-order function which, for defining a result from its arguments, solely use function application and earlier defined combinators.
##### edit Combinatory logic in mathematics
Combinatory logic was originally intended as a 'pre-logic' that would clarify the role of quantified variables in logic, essentially by eliminating them. Another way of eliminating quantified variables is Willard Van Orman Quine 's predicate functors . While most systems of combinatory logic exceed the expressive power of first-order logic The original inventor of combinatory logic, SchÂ¶nfinkel, published nothing on combinatory logic after his original

17. ComSci 319, U. Chicago
This is the bible for Combinatory logic, including lambda calculus and prooftheoretic applications. It is hard to read, and only fluent mathematicians
http://www.classes.cs.uchicago.edu/classes/archive/2000/winter/CS319/
##### 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)
##### Logistics
• Venue: MW 9:00-10:20, Ryerson 257
Instructor: Michael J. O'Donnell
• Office: Ryerson 257A. email: odonnell@cs.uchicago.edu 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.

 18. [Some Citations For Lambda Calculus Books From MathSciNet Djr This paper is a short history of the lambda calculus and Combinatory logic by a participant in that history. After a brief introduction to both systems andhttp://www.math.niu.edu/~rusin/known-math/99/lambdacalc_refs

19. Equivalences Between Pure Type Systems And Systems Of Illative
Systems of illative Combinatory logic or lambda calculus, ICLs, were introduced by Curry and Church as a foundation for logic and mathematics.
http://projecteuclid.org/handle/euclid.ndjfl/1117755149
• Home Browse Search ... next
##### Equivalences between Pure Type Systems and Systems of Illative Combinatory Logic
M. W. Bunder and W. J. M. Dekkers Source: Notre Dame J. Formal Logic Volume 46, Number 2 (2005), 181-205.
##### Abstract
Pure Type Systems, PTSs, were introduced as a generalization of the type systems of Barendregt's lambda cube and were designed to provide a foundation for actual proof assistants which will verify proofs. Systems of illative combinatory logic or lambda calculus, ICLs, were introduced by Curry and Church as a foundation for logic and mathematics. In an earlier paper we considered two changes to the rules of the PTSs which made these rules more like ICL rules. This led to four kinds of PTSs. Most importantly PTSs are about statements of the form M A , where M is a term and A a type. In ICLs there are no explicit types and the statements are terms. In this paper we show that for each of the four forms of PTS there is an equivalent form of ICL, sometimes if certain conditions hold. Primary Subjects: Keywords: pure type systems; illative combinatory logic

 20. Good Book On Combinatory Logic - Docendi.org typed lambda calculus and intuitionistic logic. We are not dealing with typed lambda calculus here, but with untyped Combinatory logic).http://www.docendi.org/good-t29079.html

 21. DROPS - Document We introduce binary representations of both lambda calculus and Combinatory logic terms, and demonstrate their simplicity by providing very compacthttp://drops.dagstuhl.de/opus/frontdoor.php?source_opus=628

22. Good Book On Combinatory Logic - Sci.logic | Google Groups
typed lambda calculus and intuitionistic logic. We are not dealing with typed lambda calculus here, but with untyped Combinatory logic).
 Help Sign in sci.logic Discussions ... Subscribe to this group This is a Usenet group - learn more Message from discussion good book on combinatory logic The group you are posting to is a Usenet group . Messages posted to this group will make your email address visible to anyone on the Internet. Your reply message has not been sent. Your post was successful translogi View profile More options Dec 15, 1:39 pm Newsgroups: sci.logic From: Date: Sat, 15 Dec 2007 16:39:17 -0800 (PST) Local: Sat, Dec 15 2007 1:39 pm Subject: Re: good book on combinatory logic Reply Reply to author Forward Print ... Find messages by this author > > >> I read Smullyans "how to mock a mockingbird" is interesting but is a > > >> Am primarily interested how to "simulate" First order logic (predicate Did read Variables Explained Away Willard V. Quine Proceedings of the American Philosophical Society, Vol. 104, No. 3. (Jun. 15, 1960), pp. 343-347. And that does "full" FOL in combinatory logic. (First order Logic including relations) But it is hopelessly complicated.

23. A Hierarchy Of Languages, Logics, And Mathematical Theories - Cogprints
Keywords, Chomsky hierarchy, evolution of language, Combinatory logic, lambda calculus, category theory. Subjects, Linguistics Historical Linguistics
http://cogprints.org/2875/
@import url(http://cogprints.org/style/auto.css); @import url(http://cogprints.org/style/print.css); @import url(http://cogprints.org/style/nojs.css); Cogprints

24. Research/Lambda Calculus And Type Theory - Foundations
The lambda calculus was originally conceived by Church in 1932 as part of a Illative Combinatory logic. In this project we study relations between
http://www.fnds.cs.ru.nl/fndswiki/Research/Lambda_calculus_and_Type_Theory
Search:
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
Projects::
• 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.

25. Partial Applicative Theories And Explicit Substitutions Ã¢ÂÂ IAM
In the literature, they are either presented in the form of partial Combinatory logic or the partial lambda calculus, and sometimes these two approaches are
http://www.iam.unibe.ch/publikationen/techreports/1993/iam-93-008
Direkt zum Inhalt Startseite Publikationen Technische Berichte Partial Applicative Theories and Explicit Substitutions PersÂ¶nliche Werkzeuge Home Forschungsgruppen CGG FKI ... Systemadministration
##### Partial Applicative Theories and Explicit Substitutions
Author(s): Thomas Strahm Download: PostScript document Abstract: Search Suchen Nachrichten Neuer Preis fÂ¼r Informatiker an der UniversitÂ¤t Bern Mehr ... Zuletzt verÂ¤ndert:
Institut fÂ¼r Informatik und angewandte Mathematik Philosophisch-naturwissenschaftliche FakultÂ¤t UniversitÂ¤t Bern Impressum

26. Lambda Calculus | Lambda The Ultimate
Pictured you can see the 210 bit binary lambda calculus selfinterpreter, and the 272 bit binary Combinatory logic self-interpreter.
http://lambda-the-ultimate.org/taxonomy/term/20
@import "misc/drupal.css"; @import "themes/chameleon/ltu/style.css";
##### Lambda the Ultimate
Home Feedback FAQ ... Archives
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##### Natural Deduction for Intuitionistic Non-Commutative Linear Logic
Natural Deduction for Intuitionistic Non-Commutative Linear Logic , Jeff Polakow and Frank Pfenning. TLCA 1999. Intuitionistic logic captures functional programming in a logical way, as can be seen from the Curry-Howard isomorphism between constructive proofs and functional programs. However, there are many structural properties of programs that are not captured within the intuitionistic framework, such as resource usage, computational complexity, and sequentiality. Intuitionistic linear logic can be thought of as a refinement of intuitionistic logic in which the resource consumption properties of functions can be expressed internally. Here, we further refine it to allow the expression of sequencing of computations. We achieve this by controlling the use of the structural rule of exchange to arrive at intuitionistic non-commutative linear logic My earlier post on linguistics reminded me of the Lambek calculus, which is an ordered logic invented in 1958(!) to model how to parse sentences. So I wanted to find a paper on ordered logic (ie, you can't freely swap the order of hypotheses in a context) and link to that.

 27. Recent Publications Curry s program , in To H. B. Curry Essays on Combinatory logic, Lambda Calculus and Formalism edited by J. P. Seldin and J. R. Hindley (Academic Press,http://www.cs.uleth.ca/~seldin/publications.shtml

28. Atlas: From Proof Polynomials To Reflexive Combinators By Sergei Artemov
In this talk we will introduce Reflexive Combinatory logic RCL built on the of \lambdacalculus with arbitrary nesting of type assertions reflexive
http://atlas-conferences.com/c/a/j/y/40.htm
 Atlas home Conferences Abstracts about Atlas Second St.Petersburg Days of Logic and Computability August 24-26, 2003 Petersburg Department of Steklov Institute of Mathematics St. Petersburg, Russia Organizers Sergei ADIAN (Russia), Sergei ARTEMOV (Russia/USA), Nikolai KOSSOVSKI (Russia), Maurice MARGENSTERN (France), Grigori MINTS (USA), Yuri MATIYASEVICH (Russia), the chairman, Nikolai NAGORNY (Russia), Vladimir OREVKOV (Russia), Anatol SLISSENKO (France) View Abstracts Conference Homepage From Proof Polynomials to Reflexive Combinators by Sergei Artemov Moscow University and the Graduate Center of the City University of New York The Logic of Proofs LP RCL built on the basis of LP RCL substantially extends the traditional Combi natory Logic by new combinators capable of handling iterated type assertions. The Logic of Proofs LP LP can be found in [1, 2]. The system LP contains axioms and rules: A0. Axioms (and rules) of the classical propositional logic, reflection application proof checker union The typed combinatory logic may be regarded as a fragment of LP IP if A , A , ... , A

29. IngentaConnect Combinatory Weak Reduction In Lambda Calculus
Combinatory logic claims to do the same work as lambda calculus but with a simpler language and a simpler reduction process. In a sense this claim is true
http://www.ingentaconnect.com/content/els/03043975/1998/00000198/00000001/art002
var tcdacmd="dt";

30. Nour
keywords, Combinatory logic, lambdacalculus, Propositional classical logic. abstract, Combinatory logic shows that bound variables can be eliminated
http://www.dmtcs.org/dmtcs-ojs/index.php/proceedings/article/view/dmAF0107/1653
 Home About Login Register ... Nour Computational Logic and Applications, CLA '05 DMTCS Conference Volume AF (2005), pp. 87-96 author: Karim Nour title: Classical Combinatory Logic keywords: Combinatory logic, Lambda-calculus, Propositional classical logic abstract: Sym Prop of Barbanera and Berardi. If your browser does not display the abstract correctly (because of the different mathematical symbols) you may look it up in the PostScript or PDF files. reference: Karim Nour (2005), Classical Combinatory Logic, in Computational Logic and Applications, CLA '05 Discrete Mathematics and Theoretical Computer Science Proceedings AF, pp. 87-96 bibtex: For a corresponding BibTeX entry, please consider our BibTeX-file ps.gz-source: dmAF0107.ps.gz (0 K) ps-source: dmAF0107.ps (0 K) pdf-source: dmAF0107.pdf (0 K) The first source gives you the gzipped' PostScript, the second the plain PostScript and the third the format for the Adobe accrobat reader. Depending on the installation of your web browser, at least one of these should (after some amount of time) pop up a window for you that shows the full article. If this is not the case, you should contact your system administrator to install your browser correctly. Due to limitations of your local software, the two formats may show up differently on your screen. If eg you use xpdf to visualize pdf, some of the graphics in the file may not come across. On the other hand, pdf has a capacity of giving links to sections, bibliography and external references that will not appear with PostScript.

31. Constructive Logic And Lambda Calculus
The lambda calculus (1); Combinatory logic (2); The fixed point theorem (3A, 3B; some remarks on 3D); Representing the recursive functions (4)
http://www.cs.swan.ac.uk/~csetzer/foerelaesning/constrmath/index.html
##### 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.

 32. A FUNCTIONAL APPROACH TO COMPUTING Notes 2000-03-15 S.Whitney J 1980 Seldin Hindley, To H.B.Curry Essays on Combinatory logic, Lambda Calculus and Formalism 1980 Henderson, Functional Programming 1982?http://kx.com/technical/contribs/stephen/talk1.txt

33. Binary Combinatory Logic - Esolang
http://www.esolangs.org/wiki/Binary_combinatory_logic
##### From Esolang
Jump to: navigation search Binary combinatory logic (BCL) is a complete formulation of combinatory logic (CL) using only the symbols and , together with two term-rewriting rules. BCL has applications in the theory of program-size complexity ( Kolmogorov complexity
##### edit Semantics
Rewriting rules for subterms of a given term (parsing from the left): where x y , and z are arbitrary terms. (Note, for example, that because parsing is from the left, is not a subterm of The terms and correspond, respectively, to the K and S basis combinators of CL, and the "prefix " acts as a left parenthesis (which is sufficient for disambiguation in a CL expression). There are four equivalent formulations of BCL, depending on the manner of encoding the triplet (left-parenthesis, K, S). These are (as above), , and
##### edit External resources

34. Science Links Japan | Treatment Of Types In The Framework Of Combinatory Logic A
Abstract;In the framework of Lambda Calculus or Combinatory logic, as for treatment of types there exist two models, that is, one includes types and the
Sitemap Home Opinions Press Releases ... IEIC Technical Report (Institute of Electronics, Information and Communication Engineers)(2001)
##### Treatment of Types in the framework of Combinatory Logic and its calculation using Graph Transformation System.
Accession number; Title; Treatment of Types in the framework of Combinatory Logic and its calculation using Graph Transformation System. Author; SUGITO YOSHIO(Electrotechnical Lab.) Journal Title; IEIC Technical Report (Institute of Electronics, Information and Communication Engineers)
Journal Code:
ISSN: VOL. NO. PAGE. FIG.3, REF.7 Pub. Country; Japan Language; Japanese Abstract; In the framework of Lambda Calculus or Combinatory Logic, as for treatment of types there exist two models, that is, one includes types and the other has no type. In this paper, especially forcusing in case of typed Combinatory Logic, we try to execute calculation on types via our graph transformation system(GMS98) as the case study of using and estimating the system. (author abst.) BACK About J-EAST How to use List of Publications ... FAQ

35. Lambda Calculus: Blogs, Photos, Videos And More On Technorati
Binary Lambda Calculus and Combinatory logic While Anton was waxing about Church Turing, I figured that Occam s Razor would be the type of proof one would
http://technorati.com/tag/Lambda+Calculus
Get the buzz on film, TV, music, and celebs now in Entertainment
##### 16 posts tagged Lambda Calculus
Subscribe search in entire post tags only of blogs with any authority a little authority some authority a lot of authority in language all languages Arabic (ÃÂ§ÃÂÃÂ¹ÃÂ±ÃÂ¨ÃÂÃÂ©) Chinese (Ã¤Â¸Â­Ã¦ÂÂ) Dutch (Nederlands) English French (FranÂ§ais) German (Deutsch) Greek (ÃÂÃÂ»ÃÂ»ÃÂ·ÃÂ½ÃÂ¹ÃÂºÃÂ¬) Hebrew (ÃÂ¢ÃÂÃÂ¨ÃÂÃÂª) Italian (Italiano) Japanese (Ã¦ÂÂ¥Ã¦ÂÂ¬Ã¨ÂªÂ) Korean (Ã­ÂÂÃªÂµÂ­Ã¬ÂÂ´) Norwegian (Norsk) Persian (ÃÂÃÂ§ÃÂ±ÃÂ³ÃÂ) Polish (Polski) Portuguese (PortuguÂªs) Russian (ÃÂ ÃÂÃÂÃÂÃÂºÃÂ¸ÃÂ¹) Spanish (EspaÂ±ol) Swedish (Svenska) Turkish (TÂ¼rkÂ§e) Vietnamese (TiÃ¡ÂºÂ¿ngÃÂ ViÃ¡Â»Ât)
• ##### Natural Deduction for Intuitionistic Non-Commutative Linear Logic
http://lambda-the-ultimate.org/ node/ 2523 Natural Deduction for Intuitionistic Non-Commutative Linear Logic Natural Deduction for Intuitionistic Non-Commutative Linear Logic, Jeff Polakow and Frank Pfenning. TLCA 1999. Intuitionistic logic captures functional programming in a logical way, as can be seen from the Curry-Howard isomorphism between constructive proofs and functional programs. 45 days ago in Authority: 333
##### GÂ¶del, Nagel, minds and machines
• 36. Cdiggins.com Â» Language Theory
This list has been indispensible to me for understanding the relationship between Combinatory logic and the lambda calculus, as well as the relationship
http://cdiggins.com/category/language-theory/page/2/
@import url(http://cdiggins.com/wp-content/themes/pool/style.css);
##### cdiggins.com
• Blog About Christopher Diggins Blogroll ... discuss at reddit.com Combinatory logic is explained using the interesting analogy of birds living in an enchanted forest in the book To Mock a Mockingbird . In it, classical combinators are mapped to bird names (S = starling, K = kestrel, etc.). These are useful mnemonics for classical combinators. Chris Rathman has a very usefulÃÂ  list of all of the combinator birds , and their implementations as combinators, in the SK calculus, and in the lambda calculus. This list has been indispensible to me for understanding the relationship between combinatory logic andÃÂ the lambda calculus, as well as the relationship between the Haskell semantics and types and Cat semantics and types. viewed here . As with any Cat program the type annotation serves only as a sanity check, the types are always inferred by the compiler.ÃÂ The only type signature which is clearly wrong is the W combinator, while a couple of other typesÃÂ have a slight bugs, but are still compatible. These bugs should be ironed out in the next major release of Cat. In Cat:ÃÂ  In Haskell This work is a major step forwards toward my current goal of Scheme to Cat translator, which should allow me to infer types for Scheme programs.

37. ScienceDirect - Journal Of Applied Logic : Editorial
In P. Jonathan Seldin and J. Roger Hindley, Editors, To H.B. Curry Essays on Combinatory logic, Lambda Calculus, and Formalism, Academic Press,
 Athens/Institution Login Not Registered? User Name: Password: Remember me on this computer Forgotten password? Home Browse My Settings ... Help Quick Search Title, abstract, keywords Author e.g. j s smith Journal/book title Volume Issue Page Journal of Applied Logic Volume 2, Issue 2 , June 2004, Pages 169-172 Variants of Logics: from HOL to the calculus of constructions to teaching mathematical proofs on computers Abstract Full Text + Links PDF (137 K) Related Articles in ScienceDirect Interpreting HOL in the calculus of constructions Journal of Applied Logic Interpreting HOL in the calculus of constructions Journal of Applied Logic Volume 2, Issue 2 June 2004 Pages 173-189 Jonathan P. Seldin Abstract The purpose of this paper is to consider a representation of the HOL theorem-prover in the calculus of constructions with the property that consistency results from the calculus of constructions imply such results in HOL. This kind of representation is impossible using the propositions-as-types representation of logic and equality, but it is possible if a different representation is used. Abstract Full Text + Links PDF (231 K) The calculus of constructions ... Information and Computation The calculus of constructions Information and Computation Volume 76, Issues 2-3

38. Joy Compared With Other Functional Languages
One group comprises the lambda calculus and the programming languages Lisp, ML and Miranda. Another comprises Combinatory logic and the language FP by
http://www.latrobe.edu.au/philosophy/phimvt/joy/j08cnt.html
Joy compared with other functional languages Global Utilities Search: Global Navigation By Manfred von Thun Joy is a functional programming language which is not based on the application of functions to arguments but on the composition of functions. This paper compares and contrasts Joy with the theoretical basis of other functional formalisms and the programming languages based on them. One group comprises the lambda calculus and the programming languages Lisp, ML and Miranda. Another comprises combinatory logic and the language FP by Backus. A third comprises Cartesian closed categories. The paper concludes that Joy is significantly different from any of these formalisms and programming languages.
##### Introduction
This paper outlines the principal similarities and differences between Joy and other high-level and low-level functional languages. The best known functional languages are the lambda calculus and, based on it, the programming languages Lisp and its descendants. All of them rely heavily on two operations, abstraction and application, which are in some sense inverses of each other. Abstraction binds free variables in an expression, and it yields a function which is a first class value. The bound variables are the formal parameters of the function, and, importantly, they are named. Application of an abstracted function to some actual parameters can be understood as resulting in a substitution of actual for formal parameters and then evaluation of the modified expression. More efficiently application can be implemented using an

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).
##### Theory of Computation
L ast updated 2003-02-10-22:35 -0800 (pst)
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.

40. 20th WCP: Dual Identity Combinators
To H.B. Curry Essays on Combinatory logic, Lambda Calculus and Formalism, Academic Press, London c., pp. 479Â490. Meyer, R.K. 1976.
http://www.bu.edu/wcp/Papers/Logi/LogiBimb.htm
 Logic and Philosophy of Logic Dual Identity Combinators Katalin BimbÃ³ Indiana University kbimbo@phil.indiana.edu ABSTRACT: This paper offers an analysis of the effect of the identity combinators in dual systems. The result is based on an easy technical trick, namely, that the identity combinators collapse all the combinators which are dual with respect to them. l l -calculi in which the functions and/or the application operation are bidirectional. The last section of the paper shows the devastating effect the identity combinators have for a dual system: they half trivialize simple combinatory bases, although they are not sufficient to cause real triviality for what cancellative combinators are needed. Introduction R B C W I C S I 1. Dual combinators. Pure combinators operate on left associated sequences of objects. The result of an application of a combinator is a sequence made out of some of the objects on the left (possibly with repetitions) and parentheses scattered across: Q x x n x i x i m where any x i j j m ) is x k k n ) for some k , and the sequence on the right might be associated arbitrarily. The parentheses on the left of the identity are frequently dropped, since left association is taken to be the default. To recall the most familiar combinators as an illustration of the above general statement we have:

Thomas Jech has worked on implementing the system of Combinatory logic TRC defined by Randall Holmes in his Ph. D. thesis and a subsequent paper under the
http://math.boisestate.edu/~holmes/holmes/nf.html
Note, added March 30, 2005 After systematic neglect for some years, I'm about to overhaul the page (done) and update the bibliography. Any comments from NFistes would be useful at this point... For new information about the mailing list, look in the Mailing List and Links to NF Fans section.
##### Introduction
This page is (permanently) under construction by Randall Holmes The subject of the home page which is developing here is the set theory "New Foundations", first introduced by W. V. O. Quine in 1937 . This is a refinement of Russell's theory of types based on the observation that the types in Russell's theory look the same, as far as one can apparently prove. To see Thomas Forster's master bibliography for the entire subject, as updated and HTML'ed by Paul West, click here . References in this page also refer to the master bibliography. We are very grateful to Thomas Forster for allowing us to use his bibliography. An all purpose reference for this field (best for NF) is

 42. ML; Scope (15 Points) What Do The Following ML Expressions betaReduction (15 points) Using the beta-rule of the lambda-calculus, evaluate the following Your job is to write an interpreter for Combinatory logic.http://www.cs.indiana.edu/~sabry/teaching/proglang/sp00/midterm

43. Great Works In Programming Languages
In J. P. Seldin and J. R. Hindley, editors, To H. B. Curry Essays on Combinatory logic, Lambda Calculus, and Formalism, pages 479490.
http://www.cis.upenn.edu/~bcpierce/courses/670Fall04/GreatWorksInPL.shtml
##### Great Works in Programming Languages
Collected by Benjamin C. Pierce In September, 2004, I posted a query to the Types list asking people to name the five most important papers ever written in the area of programming languages. This page collects the responses I received. (A few are missing because I am still tracking down bibliographic information.) Many thanks to Frank Atanassow, David Benson, Nick Benton, Karl Crary, Olivier Danvy, Mariangiola Dezani, Dan Friedman, Alwyn Goodloe, Pieter Hartel, Michael Hicks, Robert Irwin, Luis Lamb, Rod Moten, Rishiyur Nikhil, Tobias Nipkow, Jens Palsberg, and John Reynolds for contributing. Additional suggestions are welcome. (Bibtex format preferred!) BCP
##### The greatest of the great (mentioned by many people):
C. A. R. Hoare. An axiomatic basis for computer programming. Communications of the ACM , 12(10):576-580 and 583, October 1969.
bib
Peter J. Landin. The next 700 programming languages. Communications of the ACM , 9(3):157-166, March 1966.

44. Workshop On Lambda-Calculus, Type Theory, And Natural Language, 2005
This page describes the second workshop on Lambda Calculus, Type Theory and Natural Types in early Combinatory logic . 1055, Coffee Break (25 mins)
http://lcttnl.foxearth.org/

45. Index
We consider the lambdacalculus obtained from the simply-typed calculus by adding products, A Combinatory logic approach to higher-order E-unification.
http://web.cs.wpi.edu/~dd/publications/
 Daniel J. Dougherty, Kathi Fisler, and Shriram Krishnamurthi. Obligations and their interaction with programs. In 12th European Symposium On Research In Computer Security (ESORICS) , volume 4734 of Lecture Notes in Computer Science , pages 375-389, September 2007. [ bib .pdf Obligations are pervasive in modern systems, often linked to access control decisions. We present a very general model of obligations as objects with state, and discuss its interaction with a program's execution. We describe several analyses that the model enables, both static (for verification) and dynamic (for monitoring). This includes a systematic approach to approximating obligations for enforcement. We also discuss some extensions that would enable practical policy notations. Finally, we evaluate the robustness of our model against standard definitions from jurisprudence. Daniel J. Dougherty, Claude Kirchner, Helene Kirchner, and Anderson Santana de Oliveira. Modular access control via strategic rewriting. In

 46. Nekochan Net Ã¢ÂÂ¢ View Topic - Esoteric Languages... Well, lambda calculus and Combinatory logic play indeed an important role in logic itself no CS here yet. But, of course, in CS, they are at the hearthttp://forums.nekochan.net/viewtopic.php?f=1&t=15561&start=45

47. Curry (print-only)
After giving a very clear exposition of the fundamentals of Combinatory logic, showing its close relationship to the lambda calculus developed by Church,
http://www-groups.dcs.st-and.ac.uk/~history/Printonly/Curry.html
##### Born: 12 Sept 1900 in Millis, Massachusetts, USA Died: 1 Sept 1982 in State College, Pennsylvania , USA
Haskell Curry 's mother was Anna Baright and his father was Samuel Silas Curry. Samuel was the president of the School of Expression in Boston and Anna was the Dean of the School. Haskell did not show particular interest in mathematics when at high school and when he graduated in 1916 he fully intended to study medicine. He entered Harvard College, the undergraduate school of Harvard University, and took a mathematics course in his first year of study as part of his studies towards a degree in medicine. A major influence on the direction that his studies took was the entry of the United States into World War I in the spring of 1917. Curry wanted to serve his country, and decided that he would be more likely to see action if he had a mathematics training rather than if he continued the pre-medical course he was on. Anyhow he had enjoyed the mathematics course he had taken and had done very well in the course. He changed his major subject to mathematics and then enlisted in the Student Army Training Corps on 18 October 1918. The war, however, ended shortly after this (in November) and on 9 December 1918 Curry left the army. He continued on the mathematics course at Harvard, however, and graduated in 1920 with an A.B. degree. Curry now decided that he would look for a career in electrical engineering and he took a job with the General Electric Company which allowed him to study electrical engineering part-time at the Massachusetts Institute of Technology. However he soon discovered that he had a different attitude to the others taking the courses, for he wanted to know why a result was correct when for everyone else it only mattered that it was correct. Realising that he was more suited to pure science than to applied science, he changed course to study physics in 1922. Also Harvard seemed a better place to study pure science so he returned there, having been appointed to a half-post as a research assistant of P W Bridgeman for the session 1922-23. Curry graduated with a Master's Degree in physics from Harvard in 1924 but by now he realised that the subject for him was not physics but it was mathematics. He began to undertake research for his doctorate in mathematics at Harvard.

48. Combinatory Logic -- From Wolfram MathWorld
conversion or lambda calculus by Church. The system of Combinatory logic is extremely fundamental, in that there are a relatively small finite numbers
http://mathworld.wolfram.com/CombinatoryLogic.html
 Search Site Algebra Applied Mathematics Calculus and Analysis ... General Logic Combinatory Logic A fundamental system of logic based on the concept of a generalized function whose argument is also a function (Schönfinkel 1924). This mathematical discipline was subsequently termed combinatory logic by Curry and " -conversion" or " lambda calculus " by Church. The system of combinatory logic is extremely fundamental, in that there are a relatively small finite numbers of atoms, axioms, and elementary rules. Despite the fact that the system contains no formal variables, it can be used for doing anything that can be done with variables in more usual systems (Curry 1977, p. 119). SEE ALSO: Combinator Lambda Calculus [Pages Linking Here] REFERENCES: Curry, H. B. "Combinatory Logic." §3D5 in Foundations of Mathematical Logic. New York: Dover, pp. 117-119, 1977. Curry, H. and Feys, R. Combinatory Logic, Vol. 1. Amsterdam, Netherlands: North-Holland, 1958. Hindley, J. R.; Lercher, B.; Seldin, J. P. Introduction to Combinatory Logic.

 49. A New Translation Algorithm From Lambda Calculus Into Combinatory Logic A New Translation Algorithm from Lambda Calculus into Combinatory logic. Source, Lecture Notes In Computer Science archive Proceedings of the 7th Portuguesehttp://portal.acm.org/citation.cfm?id=645375.651062

50. Lambda Calculus
History, brief survey, formal theory, fixed point (recursion) theorems, Combinatory logic Combinatory completeness, translations between lambda calculus
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

51. Full Bibliography
Normalization by evaluation for typed lambda calculus with coproducts. In logic in Computer . Combinatory logic, Volume II. NorthHolland, Amsterdam.
http://tlca.di.unito.it/opltlca/opltlcali1.html
Next Previous Up
##### Full Bibliography
Alessi, F., Barbanera, F., and Dezani-Ciancaglini, M. (2006). Intersection types and lambda models. Theoretical Computer Science Alessi, F., Dezani-Ciancaglini, M., and Lusin, S. (2004). Intersection types and domain operators. Theoretical Computer Science [Alessi and Lusin, 2002] Alessi, F. and Lusin, S. (2002). Simple easy terms. In van Bakel, S., editor, Intersection Types and Related Systems Electronic Notes in Computer Science . Elsevier. Altenkirch, T., Dybjer, P., Hofmann, M., and Scott, P. (2001). Normalization by evaluation for typed lambda calculus with coproducts. In Logic in Computer Science [Altenkirch and Uustalu, 2004] Altenkirch, T. and Uustalu, T. (2004). Normalization by evaluation for . In Functional and Logic Programming , volume 2998 of Lecture Notes in Computer Science [Anderson, 1960] Journal of Symbolic Logic , 25:388. (Abstract). [Anderson and Belnap, 1975] Entailment. The Logic of Relevance and Necessity, Volume . Princeton University Press, U.S.A.

52. Combinator Module Users' Manual
Combinatory logic (CL) is one language of mathematics.1 Like the lambda calculus, one expresses a function by combining a set of terms; unlike the lambda
http://www.cotilliongroup.com/man/combinators-man.html
##### Purpose
This document provides information on using the predicates exported from the combinator module. Furthermore, as the predicates, themselves, are derivative (they depend entirely on the syntax and semantics of the combinators and the language used to express them), this document provides an overview of the implementation language and the use of combinators.
##### Background
Combinatory logic (CL) is one language of mathematics. Like the lambda calculus, one expresses a function by combining a set of terms; unlike the lambda calculus, CL has no variables; the terms are combinators, not lambda terms, and the method of combining combinators is composition, not application, as it is for the lambda calculus. Given, or, despite, the similarities and differences of CL to the lambda calculus, the results of both languages is the same: one can build any arbitrary computable expression using a small set of "primitive" terms.
##### Concessions
Pure combinatory logic has normal order evaluation ("lazy"), and the result of composing combinators is a combinatory term. We allow some intrusions of the "real world" into this implementation of CL in order to simplify translating information between the rest of Prolog and the CL domain.

 53. Glossary (n) a function with no free variables; one of the primitive functions on which the variant of lambda calculus known as Combinatory logic is built.http://community.schemewiki.org/?glossary

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