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1. John's Combinatory Logic Playground
Online article by John Tromp. Kolmogorov complexity is a recursion theoretic characterisation of randomness.
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.

2. Combinatory Logic - Wikipedia, The Free Encyclopedia
Combinatory Logic is a notation introduced by Moses SchÃ¶nfinkel and Haskell Curry to eliminate the need for variables in mathematical logic.
http://en.wikipedia.org/wiki/Combinatory_logic
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##### 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

3. 20th WCP: Dual Identity Combinators
Article by Katalin BimbÂ³ presented at the 20th World Congress of Philosophy. Investigates the addition of identity combinators, in the formulaeas-types sense, to Combinatory Logic.
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:

4. Combinatory Logic -- From Wolfram MathWorld
The system of Combinatory Logic is extremely fundamental, in that there are a relatively small finite numbers of atoms, axioms, and elementary rules.
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.

5. Mathematical Sciences, Richard Statman
Carnegie Mellon University Theory of computation, lambda calculus, Combinatory Logic.
http://www.math.cmu.edu/people/fac/statman.html
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##### Richard Statman
Professor
Ph.D., Stanford University Office: Wean Hall 7214
Phone: (412) 268-8475
E-mail: statman@cs.cmu.edu
##### Research
My principal research interests lie in the theory of computation with special emphasis on symbolic computation. In particular, my current research involves lambda calculus and combinatory algebra. This area underwent extensive development in the first half of this century, and then lay dormant until Dana Scott's fundamental work in the 1970's. Part of what has emerged from Scott's work is that lambda calculus forms the foundation of functional programming at both the semantic and syntactic levels. As a result, the area has been revived by an influx of theoretical problems directly related to design and implementation issues.
##### Selected Publications
The omega rule is sigma-zero-three hard (with Benedetto Intrigilia), LICS'04 On the lambda Y calculus, LICS '02 Church's lambda delta calculus, LPR '00 The word problem for combinators, RTA '00

Barendregt Barendregt81 defines Combinatory Logic as an equational system satisfying the combinators S and K with (( Sx)y)z = (xz)(yz) and (Kx)y = x.
http://www-fp.mcs.anl.gov/~lusk/papers/equality/node5.html
 Mathematics and Computer Science Division Projects Software Vita Books ... HTML versions of older papers Ewing ("Rusty") Lusk Argonne Distinguished Fellow Division Director Mathematics and Computer Science Division Argonne National Laboratory 9700 South Cass Avenue Argonne, Illinois 60439 Phone Fax email lusk at mcs.anl.gov Biographical Sketch Ewing Lusk received his B.A. in mathematics from the University of Notre Dame in 1965 and his Ph.D. in mathematics from the University of Maryland in 1970. He is currently a senior computer scientist in the Mathematics and Computer Science Division at Argonne National Laboratory. His current projects include implementation of the MPI Message-Passing Standard, parallel performance analysis tools, and system software for clusters. He is a leading member of the team responsible for MPICH implementation of the MPI message-passing interface standard. He is the author of five books and more than a hundred research articles in mathematics, automated deduction, and parallel computing. Current Research Interests Parallel performance tools including Jumpshot and Upshot Cobalt Scalable component-based systems software Some Recent Publications 1. N. Desai, A. Lusk, R. Bradshaw, and E. Lusk, "MPISH: A Parallel Shell for MPI Programs," in Proc. of the 12th European PVM/MPI Users' Group meeting, Recent Advances n Parallel Virtual Machine and Message Passing Interface, Lecture Notes in Computer Science 3666, Springer, 2005, pp. 333-342

 7. Topics In Logic And Proof Theory Brief introductions to Combinatory Logic, the incompleteness theorems and independence results, by Andrew D Burbanks.http://www.maths.bris.ac.uk/~maadb/research/topics/logic/

8. MainFrame: The Lambda-calculus, Combinatory Logic, And Type Systems
Combinatory Logic shows that bound variables can be eliminated without loss of expressiveness. It has applications both in the foundations of mathematics
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.

9. J Roger Hindley
University of Wales, Swansea Lambda-calculus, Combinatory Logic and type-theory.
http://www-maths.swan.ac.uk/staff/jrh/
Swansea University Physical Sciences Mathematics Department Contact
##### J Roger Hindley
Text Only Swansea University Physical Sciences ... Contact 16 October 2007 Webmaster

10. Combinatory Logic Elements
In the last chapter we met several truth tables of single logic elements. But we can also use Truth Tables to represnt Combinatory Logic.
http://www.rz.uni-hohenheim.de/hardware/basics/csc102/ch8.html
##### Contents
• Disjunctive Form Expressions
• De Morgan's Law in Combinatory Logic
• Karnaugh Maps In the last chapter we met several truth tables of single logic elements. But we can also use Truth Tables to represnt combinatory logic. We saw a simple example of one such Truth table in Chapter 7 . But note, this only applies to logic without memory.
##### Disjunctive Form Expressions
We can represent any output bit of some combinatory logic in an expression using just the inputs and Boolean operators. The following is an example. Here is the diagram for the logic:
And here is the Truth Table and the Boolean expression for the logic: It is easy to see that the Boolean expression can quite simply be read off from either the Truth Table or the logic diagram (as we will see later, we actually make the logic diagram from the Boolean Expression). Looking at the diagram, we see that the result is an OR from 3 other expressions (hence the '+' signs). For the top bit, a TRUE output is obtained if A is FALSE, B is TRUE, and C is FALSE (hence the first part of the expression is Â¬A.B.Â¬C). If we go through the rest of the diagram we see that it all matches. The method is even simpler from the Truth Table. All you need to do is say when a TRUE output (X) occurs and convert the words to symbols. To illustrate:

Albeit having precursors, it was Moses SchÃ¶nfinkel who explored first Combinatory Logic. Later it has been continued by Haskell B. Curry.
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##### Combinatory logic
Categories Combinators Theoretical foundations
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##### 1 General
Albeit having precursors, it was Moses SchÂ¶nfinkel who explored first combinatory logic. Later it has been continued by Haskell B. Curry. It was developed to be a theory for the foundation of mathematics [Bun:NatICL], and it has also relevance in linguistics too. Its goal was to understand paradoxons better, and to establish fundamental mathematical concepts on simpler and cleaner principles. Especially, to understand the concept of substitution by replacing it with simpler notions. The Ã¢ÂÂlack of (bound) variablesÃ¢ÂÂ relates combinatory logic to pointfree frameworks. (As a contrast, see a very different approach which also enables full elimination of variables: recursive function theory General materials:

12. Syntactic Pattern-Matching And Combinatory Logic -- From Wolfram Library Archive
Given a pattern and a string, this work addresses the problem of finding all instantiations of the variables forming the pattern to match those appearing in
http://library.wolfram.com/infocenter/Articles/2603/

13. Binary Lambda Calculus And Combinatory Logic | 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/node/2458
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##### Binary Lambda Calculus and Combinatory Logic
While Anton was waxing about , I figured that Occam's Razor would be the type of proof one would postulate when giving the nod to Lambda Calculus over Universal Turing Machines. This leads inexorably to the question of what is the smallest (as measured in binary bits) Turing Machine that can possibly be constructed. John Tromp provides an answer to this question in his always fun Lambda Calculus and Combinatory Logic Playground Pictured you can see the 210 bit binary lambda calculus self-interpreter, and 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.

14. Class: Combinatory Logic - PHP Classes
Calculate and generate array element combinations.
http://www.phpclasses.org/browse/package/1901.html
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##### Class: Combinatory Logic
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• Name Andrea Giammarchi e-mail contact Published packages Country Italy PHP professionals from Italy looking for PHP jobs Home page http://www.3site.it/ Age All time rank Week rank Browse this author's classes
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• Algorithms Numerical and statistical algorithms View top rated classes
• ##### Detailed description
• This class calculates and generates all combinations of array elements for n variables with a k class.
The class takes an array elements and returns another array with values set to the specified type of combination of elements.

15. Combinator Birds
The javascript program to do the combinator reduction was borrowed from the Combinatory Logic Tutorial by Chris Barker. The calculator requires that the
http://www.angelfire.com/tx4/cus/combinator/birds.html
 Combinator Birds Function Abstraction Symbol Bird Combinator SK Combinator abc.a(bc) B Bluebird S(KS)K ((S(KS))K) abcd.a(bcd) B Blackbird BBB ((S(K((S(KS))K)))((S(KS))K)) abcde.a(bcde) B Bunting B(BBB)B ((S(K((S(K((S(KS))K)))((S(KS))K))))((S(KS))K)) abcd.a(b(cd)) B Becard B(BB)B ((S(K((S(K((S(KS))K)))((S(KS))K))))((S(KS))K)) abc.acb C Cardinal S(BBS)(KK) ((S((S(K((S(KS))K)))S))(KK)) abcd.ab(cd) D Dove BB (S(K((S(KS))K))) abcde.abc(de) D Dickcissel B(BB) (S(K(S(K((S(KS))K))))) abcde.a(bc)(de) D Dovekies BB(BB) ((S(K((S(KS))K)))(S(K((S(KS))K)))) abcde.ab(cde) E Eagle B(BBB) (S(K((S(K((S(KS))K)))((S(KS))K)))) abcdefg.a(bcd)(efg) Bald Eagle B(BBB)(B(BBB)) ((S(K((S(K((S(KS))K)))((S(KS))K))))(S(K((S(K((S(KS))K)))((S(KS))K))))) abc.cba F Finch ETTET ((S(K((S((SK)K))(K((S(K(S((SK)K))))K)))))((S(K((S(K((S(KS))K)))((S(KS))K))))((S(K(S((SK)K))))K))) abcd.ad(bc) G Goldfinch BBC ((S(K((S(KS))K)))((S((S(K((S(KS))K)))S))(KK))) abc.abcb H Hummingbird BW(BC) ((S(K((S(K(S((S(K((S((SK)K))((SK)K))))((S(K((S(KS))K)))((S(K(S((SK)K))))K))))))K)))(S(K((S((S(K((S(KS))K)))S))(KK))))) a.a

 16. Combinatory Logic From FOLDOC Nearby terms colour theories of Â« combination Â« combinator Â« Combinatory Logic Â» communalism Â» communication system Â» communism.http://www.swif.uniba.it/lei/foldop/foldoc.cgi?combinatory logic

17. Discrete Mathematics/Combinatory Logic - Wikibooks, Collection Of Open-content T
Two of the basic principles of Combinatory Logic in discrete mathematics are the Sum principle and the Multiplication principle.
http://en.wikibooks.org/wiki/Discrete_mathematics/Combinatory_logic
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##### From Wikibooks, the open-content textbooks collection
Discrete mathematics Jump to: navigation search Two of the basic principles of combinatory logic in discrete mathematics are the Sum principle and the Multiplication principle. The sum principle holds true in a given partitioned set X where partition Xi intersected with Xj is the empty set unless i is equal to j. The principle states that in such a partitioned set, the sum of the elements of each partition is equal to the number of elements in the set X. Retrieved from " http://en.wikibooks.org/wiki/Discrete_mathematics/Combinatory_logic Subject Discrete mathematics (book) Views Personal tools Navigation Community Search Toolbox

18. Binary Combinatory Logic - Esolang
Binary Combinatory Logic (BCL) is a complete formulation of Combinatory Logic (CL) using only the symbols 0 and 1, together with two termrewriting rules.
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

19. Combinatory Logic
Combinatory Logic. A system for reducing the operational notation of logic, mathematics or a functional language to a sequence of modifications to the input
http://burks.bton.ac.uk/burks/foldoc/91/21.htm
The Free Online Dictionary of Computing ( http://foldoc.doc.ic.ac.uk/ dbh@doc.ic.ac.uk Previous: combinator Next: Comdex
##### combinatory logic
A system for reducing the operational notation of logic , mathematics or a functional language to a sequence of modifications to the input data structure. First introduced in the 1920's by Schoenfinkel. Re-introduced independently by Haskell Curry in the late 1920's (who quickly learned of Schoenfinkel's work after he had the idea). Curry is really responsible for most of the development, at least up until work with Feys in 1958. See combinator

20. Ultra-Formal Verification Of A Result In A System Of Combinatory Logic
This result is a main technical lemma of M. R. Holmes 2, who found a system of Combinatory Logic, which he calls TRC, equivalent to Quine s set theory New
http://www.cityauditorphilwood.com/warren/sectionfr1.html
 Here is the non-frame based version of the document.

21. Combinatory Logic - Wikipedia
Combinatory Logic is a simplified model of computation, used in computability theory (the study of what can be computed) and proof theory (the study of what
http://facetroughgemstones.com/wikipedia/co/Combinatory_logic.html
Contents
##### Combinatory logic
Combinatory logic is a simplified model of computation , used in computability theory (the study of what can be computed) and proof theory (the study of what can be mathematically proven .) The theory, despite its simplicity, captures many essential features of the nature of computation. Combinatory logic is a variation of the Lambda calculus , in which lambda expressions (used to allow for functional abstraction) are replaced by a limited set of primitive functions. Note that the "combinatorial logic" used in electronics is different; see combinatorial logic (electronics) Table of contents showTocToggle("show","hide") 1 Summary of the Lambda Calculus
2 Combinatory Calculi

2.1 Combinatory Terms

2.2 Examples of Combinators
...
6 References
##### Summary of the Lambda Calculus
For complete details about the lambda calculus , see the article under that head. We will summarize here. The lambda calculus is concerned with objects called lambda-terms , which are strings of symbols of one of the following forms:
• v
where v is a variable name drawn from a predefined infinite set of variable names, and

22. Combinatory Logic - Wiktionary
edit English. Wikipedia has an article on. Combinatory Logic Combinatory Logic (uncountable). (computer science) Model of computation based on
http://en.wiktionary.org/wiki/combinatory_logic
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##### edit English
Wikipedia has an article on: Combinatory logic Wikipedia
##### edit Noun
Singular
combinatory logic Plural
uncountable
combinatory logic uncountable
• computer science Model of computation based on combinators
• ##### edit Translations

 23. JSTOR Elements Of Combinatory Logic. Elements of Combinatory Logic. Yale University Press, New Haven and London 1974, viii + 162 pp. What is Combinatory Logic about?http://links.jstor.org/sici?sici=0022-4812(197612)41:4<789:EOCL>2.0.CO;2-E

 24. Reversible Combinatory Logic Combinatory Logic is a variant of the $\lambda$calculus that maintains irreversibility. Recently, reversible computational models have been studied mainlyhttp://portal.acm.org/citation.cfm?id=1166047

25. The Church-Rosser Property In Dual Combinatory Logic
J. M. Dunn and R. K. Meyer Combinatory Logic and structurally free logic, Logic Journal of IGPL, vol. 5 (1997), pp. 505537.
http://projecteuclid.org/handle/euclid.jsl/1045861508
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##### The Church-Rosser property in dual combinatory logic
Source: J. Symbolic Logic Volume 68, Issue 1 (2003), 132-152.
##### Abstract

26. Springer Online Reference Works
In Combinatory Logic one chooses as basic the concepts of a oneplace One of the first problems in Combinatory Logic was that of the reduction of the
http://eom.springer.de/C/c023310.htm
 Encyclopaedia of Mathematics C Article referred from Combinatory logic A branch of logic devoted to the study and analysis of such concepts and methods as a variable, a function, the substitution operation, the classification of objects into types or categories, and related matters. In combinatory logic one chooses as basic the concepts of a one-place function and the operation of applying a function to an argument (application). Here the concept of a function is regarded as primitive, instead of that of a set, and is generalized in such a way that a function can be applied to objects at the same level with it. In particular, a function can serve as an argument of itself. Since functions can be used both as arguments and values, the notion of a many-placed function reduces to that of a one-placed function. The result of applying a function to an argument is denoted by . For simplicity the brackets are often omitted, so that the expression stands for . A function satisfying the equation where , are arbitrary functions and is an object constructed from these functions (perhaps not from all of them), is called a combinator. (The existence of combinators is tacitly postulated.) Any combinator can be expressed in terms of the two combinators

 27. Combinatory Logic From FOLDOC Combinatory Logic. logic A system for reducing the operational notation of logic, mathematics or a functional language to a sequence of modifications tohttp://foldoc.org/?combinatory logic

 28. A SYSTEM OF COMBINATORY LOGIC Title A SYSTEM OF Combinatory Logic. Corporate Author YALE UNIV NEW HAVEN CT INTERACTION LAB. Personal Author(s) FITCH, FREDERIC B.http://stinet.dtic.mil/oai/oai?verb=getRecord&metadataPrefix=html&identifier=AD0

29. [math/9409201] OTTER Experiments In A System Of Combinatory Logic
This paper describes some experiments involving the automated theoremproving program OTTER in the system TRC of illative Combinatory Logic.
http://arxiv.org/abs/math/9409201
##### arXiv.orgmath
Search or Article-id Help Advanced search All papers Titles Authors Abstracts Full text Help pages
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##### Title: OTTER Experiments in a System of Combinatory Logic
Authors: Thomas Jech (Submitted on 2 Sep 1994) Abstract: This paper describes some experiments involving the automated theorem-proving program OTTER in the system TRC of illative combinatory logic. We show how OTTER can be steered to find a contradiction in an inconsistent variant of TRC, and present some experimentally discovered identities in TRC. Subjects: Logic (math.LO) Report number: Logic E-prints September 02, 1994 Cite as: arXiv:math/9409201v1 [math.LO]
##### Submission history
From: Thomas Jech [ view email
Fri, 2 Sep 1994 00:00:00 GMT (8kb)
Which authors of this paper are endorsers?
Link back to: arXiv form interface contact

 30. Functionality In Combinatory Logic Functionality in Combinatory Logic*. H. B. Curry. Department of Mathematics, The Pennsylvania State College. Full text. Full text is available as a scannedhttp://www.pubmedcentral.nih.gov/articlerender.fcgi?artid=1076489

 31. Brainwagon Â» John’s Combinatory Logic Playground There is much pleasure in useless knowledge.Â Â Bertrand Russell.http://brainwagon.org/?p=2499

 32. Combinatory Logic @ Computer-Dictionary-Online.org Combinatory Logic @ Computer Dictionary Online. Computer terminology definitions including hardware, software, equipment, devices, jargon abbreviations andhttp://www.computer-dictionary-online.org/?q=combinatory logic

 33. Combinatory Logic Combinatory Logic. See EssAndKayCombinators. EditText of this page (last edited November 21, 2003) FindPage by searching (or browse LikePages or take ahttp://c2.com/cgi/wiki?CombinatoryLogic

34. Download Combinatory Logic - Combinatory Logic Class Calculates And Generates Al
Download Combinatory Logic Combinatory Logic class calculates and generates all combinations of array elements for n variables with ak class.
http://webscripts.softpedia.com/script/PHP-Clases/Combinatory-Logic-12449.html

35. Combinatory Logic Tutorial
Combinatory Logic (CL), invented by Shoenfinkel and developed by Curry and others in the 1920 s (note before the calculus!), is equivalent in expressive
http://homepages.nyu.edu/~cb125/Lambda/ski.html
##### Combinatory Logic Tutorial
(((SK)K)(yS)) By convention, combination is left-associative, so the above can be written equivalently as SKK(yS) Instead of alpha, beta, and nu reduction, there are two essential rules of inference: KXY ==> X SXYZ ==> XZ(YZ) where "X", "Y", and "Z" are metavariables ranging over arbitrary expressions. SKKX ==> KX(KX) ==> X Since this derivation works for any expression X, SKK is the identity function.
Done. Any expression in the lambda calculus can be converted. Logic with just one paren symbol and just one operator.

36. IngentaConnect A Combinatory Logic Approach To Higher-order E-unification
A translation of higherorder E-unification problems into a Combinatory Logic framework is presented and justified. The case in which E admits presentation
http://www.ingentaconnect.com/content/els/03043975/1995/00000139/00000001/art002
var tcdacmd="dt";

37. DSpace At RU: Combinatory Logic And The W-rule
Title, Combinatory Logic and the wrule. Authors, Barendregt, HP. Issue Date, 1974. URI, http//hdl.handle.net/2066/17238
http://hdl.handle.net/2066/17238
 About DSpace Software Search DSpace Advanced Search Home Browse Communities Titles Authors Subjects ... By Date Sign on to: Receive email updates My DSpace authorized users Edit Profile Help About DSpace DSpace at RU ... Cream of Science Please use this identifier to cite or link to this item: http://hdl.handle.net/2066/17238 Title: Combinatory logic and the w-rule Authors: Barendregt, H.P. Issue Date: URI: http://hdl.handle.net/2066/17238 Appears in Collections: Cream of Science Files in This Item: There are no files associated with this item. DSpace Software MIT and Hewlett-Packard Feedback

38. J Roger Hindley : Research
Mathematical logic; particularly lambdacalculus, Combinatory Logic and Lambda-calculus and Combinatory Logic are formal systems, to some extent rivals,
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

39. CJO - Abstract - Unique Decomposition Categories, Geometry Of Interaction And Co
Unique decomposition categories, Geometry of Interaction and Combinatory Logic. ESFANDIAR HAGHVERDI Mathematical Structures in Computer Science 100202,
http://journals.cambridge.org/abstract_S0960129599003035
@import url("/css/users_abstract.css"); close
##### Cambridge Journals Online
Skip to content Mathematical Structures in Computer Science (2000), 10: 205-230 Cambridge University Press doi:10.1017/S0960129599003035 Copy and paste this link: http://journals.cambridge.org/action/displayAbstract?aid=44867 Research Article
##### Unique decomposition categories, Geometry of Interaction and combinatory logic
ESFANDIAR HAGHVERDI
Department of Mathematics, University of Ottawa, 585 King Edward St., Ottawa, ON, K1N 6N5, Canada. Email ehaghver@mathstat.uottawa.ca
Abstract
In another paper (Abramsky et al . 1999), we have developed Abramsky's analysis of Girard's Geometry of Interaction programme in detail. In this paper, our goal is to study the data ow based computational aspects of that analysis. We introduce unique decomposition categories that provide a suitable categorical framework for such computational analysis. The current study also serves to establish connections with the work on proof nets and paths by Girard and Danos and Regnier in this categorical setting. The latter goal is partially achieved here by the presentation of categorical models for dynamic algebras.

40. Good Book On Combinatory Logic - Sci.logic | Google Groups
Does anybody know a simple book on Combinatory Logic? I read Smullyans how to mock a mockingbird is interesting but is a bit to much about combinators and

41. Combinators And Structurally Free Logic -- Dunn And Meyer 5 (4): 505 -- Logic Jo
A Kripkestyle semantics is given for Combinatory Logic using frames with a ternary accessibility relation, much as in the Tourley-Meyer semantics for
http://jigpal.oxfordjournals.org/cgi/content/abstract/5/4/505
@import "/resource/css/hw.css"; @import "/resource/css/igpl.css"; Skip Navigation Oxford Journals Logic Journal of IGPL 1997 5(4):505-537; doi:10.1093/jigpal/5.4.505
Oxford University Press

##### Combinators and structurally free logic
JM Dunn and RK Meyer Departments of Philosophy and Computer Science, Indiana University, Bloomington, IN 47405, USA Automated Reasoning Project, The Australian National University, Canberra, ACT 2604, Australia A 'Kripke-style' semantics is given for combinatory logic using frames with a ternary accessibility relation, much as in the Tourley-Meyer semantics for relevance logic. We prove by algebraic means a completeness theorem for combinatory logic, by proving

42. Combinatory Logic - Definitions From Dictionary.com
Definitions of Combinatory Logic at Dictionary.com.
http://dictionary.reference.com/browse/combinatory logic
var pid = 125500; var nid = 125542; var mid = 679788; var word = 'combinatory%20logic'; SafeAddOnload(init_near);
##### 1 result for: combinatory logic

43. Math: Logic And Foundations: Computational Logic: Combinatory Logic And Lambda C
Kolmogorov Complexity in Combinatory Logic Online article by John Tromp. Kolmogorov complexity is a recursion theoretic characterisation of randomness.
http://www.spacetransportation.org/Math/Logic_and_Foundations/Computational_Logi
##### Categories
Lambda Calculus
Introduction to the lambda calculus for computer scientists. Shows how the calculus can be formalised in Scheme.
http://www.mactech.com/articles/mactech/Vol.07/07.05/LambdaCalculus/
Dual Identity Combinators

Article by Katalin BimbÂ³ presented at the 20th World Congress of Philosophy. Investigates the addition of identity combinators, in the formulae-as-types sense, to combinatory logic.
http://www.bu.edu/wcp/Papers/Logi/LogiBimb.htm
Kolmogorov Complexity in Combinatory Logic

Online article by John Tromp. Kolmogorov complexity is a recursion theoretic characterisation of randomness.
http://www.cwi.nl/~tromp/cl/cl.html
Lambda

An online introduction to the lambda calculus by Lloyd Allison, complete with a web form that will evaluate lambda expressions. http://www.csse.monash.edu.au/~lloyd/tildeFP/Lambda/

 44. JSTOR The Principal Type-Scheme Of An Object In Combinatory Logic In their book Combinatory Logic 1, Curry and Feys introduced the notion of functional character (here called typescheme ) of an object of combinatoryhttp://dx.doi.org/10.2307/1995158

45. Combinatory Logic On GlobalSpec
GlobalSpec offers a variety of Combinatory Logic for engineers and through SpecSearch the Combinatory Logic can be searched for the exact specifications
http://semiconductors.globalspec.com/Industrial-Directory/combinatory_logic

46. Combinatory Logic - Definition Of Combinatory Logic By The Online Dictionary Fro
Definition of Combinatory Logic in the Online Dictionary. Multiple meanings, detailed information and synonyms for Combinatory Logic.
http://onlinedictionary.datasegment.com/word/combinatory logic
Online Dictionary C : combinatory logic
##### combinatory logic
1 definition found combinatory logic Free On-line Dictionary of Computing (26 May 2007) : logic , mathematics or a functional language to a sequence of modifications to the input data structure. First introduced in the 1920's by Schoenfinkel. Re-introduced independently by Haskell Curry in the late 1920's (who quickly learned of Schoenfinkel's work after he had the idea). Curry is really responsible for most of the development, at least up until work with Feys in 1958. See combinator

47. THE-CHURCH-ROSSER-PROPERTY-IN-SYMMETRIC-COMBINATORY-LOGIC
THE CHURCHROSSER PROPERTY IN SYMMETRIC Combinatory Logic.
http://biblioteca.universia.net/html_bura/ficha/params/id/1445170.html
##### Biblioteca.Net
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##### Datos del recurso
Ver
The Church-Rosser property in symmetric combinatory logic
BimbÃ³, Katalin
LocalizaciÃ³n: http://ProjectEuclid.org/getRecord?id=euclid.jsl/1120224727
euclid.jsl/1120224727

Citation: J. Symbolic Logic 70 (2005), Issue 2, 536-556

Pertenece a: Project Euclid (Hosted at Cornell University Library)
##### Descargar SCORM
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##### Detalles del recurso
The Church-Rosser property in symmetric combinatory logic Id.

48. COMBINATORY LOGIC
Combinatory Logic A system for reducing the operational notation of logic, mathematics or a functional language to a sequence of modifications to the input
http://www.websters-online-dictionary.org/co/combinatory logic.html
##### Specialty Definition: COMBINATORY LOGIC
Domain Definition
##### Computing
Combinatory logic A system for reducing the operational notation of logic, mathematics or a functional language to a sequence of modifications to the input data structure. First introduced in the 1920's by Schoenfinkel. Re-introduced independently by Haskell Curry in the late 1920's (who quickly learned of Schoenfinkel's work after he had the idea). Curry is really responsible for most of the development, at least up until work with Feys in 1958. See combinator. (1995-01-05). Source: The Free On-line Dictionary of Computing Source: compiled by the editor from various references ; see credits. Top
##### Crosswords: COMBINATORY LOGIC
Specialty definitions using "COMBINATORY LOGIC" combinator fixed point combinator Haskell Curry microcode ... Top
Domain Title
##### Books
• Elements of combinatory logic reference
• To Mock a Mocking Bird and Other Logic Puzzles: Including an Amazing Adventure in Combinatory Logic reference (more book examples)
Source: compiled by the editor from various references ; see credits.

49. Computer Science, Computation Fundamentals, Combinatory Logic.
Activesign Technology of Computer, Computation Fundamentals and Combinatory Logic information from www.activesign.net.
http://www.activesign.net/contactus.php
 Contact Us USA Office Address : 501 Silverside Rd, Suite 105 Wilmington, DE 19809, USA Toll Free No.:- 1-866-249-4291 Fax :- 1-302-351-6292. Home Theory of Computation Resources Diversity of Computer Science ... Hotel Website Hosted, Maintained SEO by Cgsinfotech Ltd.

50. Combinatory Logic - Computing Reference - ELook.org
Previous Terms, Terms Containing Combinatory Logic, Next Terms . COM com COMAL combination combinator, combinator curried function
http://www.elook.org/computing/combinatory-logic.htm
By Letter: Non-alphabet A B C ... Email this page to a friend
##### Combinatory logic
A system for reducing the operational notation of logic , mathematics or a functional language to a sequence of modifications to the input data structure.
First introduced in the 1920's by Schoenfinkel.
Re-introduced independently by Haskell Curry in the late 1920's (who quickly learned of Schoenfinkel's work after he had the idea).
Curry is really responsible for most of the development, at least up until work with Feys in 1958.
See combinator
Terms Containing combinatory logic COM
com

COMAL

combination
... Contact

 51. CAT.INIST In theoretical computer science and mathematics the models of Combinatory Logic are of significance in various ways. In particular within the disciplinehttp://cat.inist.fr/?aModele=afficheN&cpsidt=4382689

52. Good Book On Combinatory Logic - Sci.logic | Google Groups
And that does full FOL in Combinatory Logic. (First order Logic including relations) But it is hopelessly complicated.

53. ComSci 319, U. Chicago
Combinatory Logic volume I by Haskell B. Curry and Robert Feys with two sections by William Craig, NorthHolland, Amsterdam, Studies in Logic and the
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.

54. Combinatory Logic - Definition By Dict.die.net
Combinatory Logic A system for reducing the operational notation of logic, mathematics or a functional language to a sequence of modifications to the input
http://dict.die.net/combinatory logic/
##### Definition: combinatory logic
Search dictionary for Source: The Free On-line Dictionary of Computing (2003-OCT-10) combinatory logic A system for reducing the operational notation of logic , mathematics or a functional language to a sequence of modifications to the input data structure. First introduced in the 1920's by Schoenfinkel . Re-introduced independently by Haskell Curry in the late 1920's (who quickly learned of Schoenfinkel's work after he had the idea). Curry is really responsible for most of the development, at least up until work with Feys in 1958. See combinator
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55. Curry Biography
Curry began working on Combinatory Logic in 1950 when he was awarded a Fulbright Grant Combinatory Logic is concerned with certain basic notions of the
http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Curry.html
##### Born: 12 Sept 1900 in Millis, Massachusetts, USA Died: 1 Sept 1982 in State College, Pennsylvania , USA
Click the picture above
to see a larger version Show birthplace location Previous (Chronologically) Next Main Index Previous (Alphabetically) Next Biographies index
Version for printing
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.

56. A Hierarchy Of Languages, Logics, And Mathematical Theories - Cogprints
Starting from the root tier, the mathematical theories in this hierarchy are Combinatory Logic restricted to the identity I, Combinatory Logic,
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

 57. Combinatory Logic Definition. Define Combinatory Logic. What Is Combinatory Logi DefineThat provides thousands of technical definitions and defines computer related terms. Great for IT professionals and home users.http://www.definethat.com/hitting.asp?ID=3879

58. Science Links Japan | Functional Language Processing Via Combinatory Logic And I
Abstract;As one of the methods of processing functional language, there exists the way which evaluates and executes its corresponding Combinatory Logic
Sitemap Home Opinions Press Releases ... IEIC Technical Report (Institute of Electronics, Information and Communication Engineers)(2000)
##### Functional Language Processing via Combinatory Logic and its relation to Graph Transformation System.
Accession number; Title; Functional Language Processing via Combinatory Logic and its relation to Graph Transformation System. Author; SUGITO YOSHIO(Electrotech. Lab., Agency of Ind. Sci. and Technol.) Journal Title; IEIC Technical Report (Institute of Electronics, Information and Communication Engineers)
Journal Code:
ISSN: VOL. NO. PAGE. REF.7 Pub. Country; Japan Language; Japanese Abstract; As one of the methods of processing functional language, there exists the way which evaluates and executes its corresponding combinatory logic codes by means of combinatory logic's reduction. In this paper, especially forcusing in case of using the framework of category theory(that is, categorical combinatory logic-CCL-), we try to execute its CCL-code using our graph transformation system(GMS-98) as the case study of using and estimating the system. (author abst.) BACK About J-EAST How to use List of Publications ... FAQ

59. Combinatory Logic - Indopedia, The Indological Knowledgebase
http://www.indopedia.org/Combinatory_logic.html
Categories
Logic in computer science Mathematical logic Lambda calculus ... Wikipedia Article
##### Combinatory logic
Ã Â¤ÂÃ Â¥ÂÃ Â¤ÂÃ Â¤Â¾Ã Â¤Â¨Ã Â¤ÂÃ Â¥ÂÃ Â¤Â¶: - The Indological Knowledgebase
Combinatory logic is a simplified model of computation , used in computability theory (the study of what can be computed) and proof theory (the study of what can be mathematically proven .) The theory, despite its simplicity, captures many essential features of the nature of computation. Combinatory logic is a variation of the lambda calculus , in which lambda expressions (used to allow for functional abstraction) are replaced by a limited set of combinators , primitive functions which contain no free variables . It is easy to transform lambda expressions into combinator expressions, and since combinator reduction is much simpler than lambda reduction, it has been used as the basis for the implementation of some non-strict functional programming languages in software and hardware Contents showTocToggle("show","hide")

60. Kolmogorov Complexity In Combinatory Logic - Online Article By John Tromp. Kolmo
Kolmogorov Complexity in Combinatory Logic On the Formulaeas-Types Correspondence for Classical Logic (Popularity ) Doctoral thesis of Charles
http://www.sciencecentral.com/site/495602
 Saturday, 22 December, 2007 Home Submit Science Site Add to Favorite Contact search for Directories Aeronautics and Aerospace Agriculture Anomalies and Alternative Science Astronomy ... Technology Category: Science Math Logic and Foundations Computational Logic ... REPORT BROKEN LINK Kolmogorov Complexity in Combinatory Logic Popularity: Details document.write(''); Online article by John Tromp. Kolmogorov complexity is a recursion theoretic characterisation of randomness. URL Title Description Category: Related sites Dual Identity Combinators (Popularity: ): Article by Katalin BimbÂ³ presented at the 20th World Congress of Philosophy. Investigates the addition ... Lambda (Popularity: ): An online introduction to the lambda calculus by Lloyd Allison, complete with a web form ... Perl Contains the Lambda-Calculus (Popularity: ): Explains why this computer program is well suited to apply to functional application. On the Formulae-as-Types Correspondence for Classical Logic (Popularity: ): Doctoral thesis of Charles Stewart, which investigates foundational aspects of the application of the formulae-as-types ... Automated Reasoning (Popularity: ): Survey of automated deduction and theorem proving; from the Stanford Encyclopedia of Philosophy by Frederic ...

 61. [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

62. Combinatory Logic - ExampleProblems.com
http://www.exampleproblems.com/wiki/index.php/Combinatory_logic
var skin = 'monobook';var stylepath = '/wiki/skins';
##### From ExampleProblems.com
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.
edit
##### Combinatory logic in mathematics
Combinatory logic was intended as a simple 'pre-logic' which would clarify the meaning of variables in logical notation, and indeed eliminate the need for them. See Curry, 1958-72. edit
##### Combinatory logic in computing
In computer science, combinatory logic is used as a simplified model of computation , used in computability theory (the study of what can be computed) and proof theory (the study of what can be mathematically proven .) The theory, despite its simplicity, captures many essential features of the nature of computation.

63. Combinatory Logic In TutorGig Encyclopedia
Combinatory Logic is a notation introduced by Moses SchÃ¶nfinkel and Haskell Curry to eliminate the need for variables in mathematical logic.
http://www.tutorgig.com/ed/Combinatory_logic
Combinatory logic Encyclopedia Search: in Tutorials Encyclopedia Dictionary Entire Web Store Tutorials Encyclopedia Dictionary Web ... Email this to a friend
##### Combinatory logic
This article is about a topic in mathematical logic and theoretical computer science, not to be confused with combinational logic , sometimes known as combinatorial logic , a topic in digital electronics 'Combinatory logic ' is a notation introduced by Moses SchÃ¶nfinkel and Haskell Curry to eliminate the need for variable s 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 ', which are higher-order function s that solely use function application and possibly other, earlier defined combinators for defining a result from their arguments.
##### Combinatory logic in mathematics
Combinatory logic was intended as a simple 'pre-logic' which would clarify the meaning of variables in logical notation, and indeed eliminate the need for them. See Curry, 1958?1972.

64. Combinatory Logic - Spock Search
Haskell Curry, Corrado BÃ¶hm, Moses SchÃ¶nfinkel, Mark Steedman, Robert Feys and other people matching \
http://www.spock.com/q/combinatory-logic