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1. Combinatory Algebras In Algebra The Nature Of Elements (henceforth
Nevertheless, it helps to have some notion of object in mind to get a taste of Combinatory Algebra. One such notion considers an object as being any set
Combinatory algebras In algebra the nature of elements (henceforth "objects") is irrelevant. Nevertheless, it helps to have some notion of "object" in mind to get a taste of combinatory algebra. One such notion considers an "object" as being any set of properties expressed in some (first order) language. If that lacks imagery, then simply consider as "object" anything that can be object of thought (cf. Engeler 1995). What can be said about a universe C based on such an abstract notion of "object" and some (as yet entirely unspecified) notion of "multiplication" within C ? A number of things, as it turns out. To make an example, take three particular objects in C , say a b c , and combine them in a certain way, say, . By following the "multiplication" table of C this combination will result in, say, z . Now, any three objects could be combined in this way, and "to combine arbitrary three objects, x x x into (x " is itself a (first order) property. As such it specifies some object that must, by definition, exist in C . Call that object P . The defining property of P is precisely that for its "multiplication" with any three objects x x x the relationship ) = (x holds in C . By this reasoning we conclude that for any arbitrary combination of an arbitrary number of objects in C there must be another object that expresses the possibility of combining things in that way, and there will be a corresponding relationship (an equality) stating that fact. This remarkable property is known as "combinatory abstraction". A series of other properties follow from it, but we won't pursue this matter further. Universes in which it holds are "combinatory complete", and are known as "combinatory algebras".

2. Atlas: Boolean Algebra And Lambda Calculus By Antonino Salibra
The proof of the representation theorem for Combinatory Algebras is based on the Central elements in a Combinatory Algebra constitute a Boolean algebra,
August 5-9, 2007
St Anne's College, University of Oxford
Oxford, England Organizers
Mai Gehrke and Hilary Priestley View Abstracts
Conference Homepage
Boolean algebra and lambda calculus
Antonino Salibra
University of Venice One of the milestones of modern algebra is the Stone representation theorem for Boolean algebras. In [1] Manzonetto and Salibra have shown that Comer's generalization of the Stone representation theorem holds also for combinatory algebras: any combinatory algebra is isomorphic to a weak Boolean product of directly indecomposable combinatory algebras. The proof of the representation theorem for combinatory algebras is based on the fact that every combinatory algebra has central elements, i.e., elements which define a direct decomposition of the algebra as the Cartesian product of two other combinatory algebras. Central elements in a combinatory algebra constitute a Boolean algebra, whose Boolean operations can be defined by suitable combinators. Then it is natural to investigate the semantics of lambda calculus given in terms of models, which are directly indecomposable as combinatory algebras (indecomposable semantics, for short). The indecomposable semantics includes all known models of lambda calculus (i.e., the continuous, stable and strongly stable semantics), and the term models of all semisensible lambda theories (theories which do not equate solvable and unsolvable terms). The class of indecomposible combinatory algebras is a universal class, so that it is closed under subalgebras and ultraproducts. This implies that, if the term model of a lambda theory T is decomposible in a non-trivial way, then every model of T can be also decomposible in a non-trivial way. It follows that the indecomposable semantics is incomplete because there exist lambda theories whose term models admit non-trivial central elements.

3. JSTOR A Sufficient Condition For Completability Of Partial
A Partial Combinatory Algebra is completable if it can be extended to a total one. A Total Combinatory Algebra is a PCA where the application is total,<1209:ASCFCO>2.0.CO;2-U

4. IngentaConnect A Note On Absolutely Unorderable Combinatory Algebras
Plotkin has conjectured that there exists an absolutely unorderable Combinatory Algebra, namely a Combinatory Algebra which cannot be embedded in another
var tcdacmd="dt";

5. Geometry Of Interaction And Linear Combinatory Algebras
We present an axiomatic framework for Girard s Geometry of Interaction based on the notion of linear Combinatory Algebra. We give a general construction on

6. A Sufficient Condition For Completability Of Partial Combinatory
A Partial Combinatory Algebra is completable if it can be extended to a total one. In 1 it is asked (question 11, posed by D. Scott, H. Barendregt,
Log in RSS Title Author(s) Abstract Subject Keyword All Fields FullText more options

7. History Of Mathematics Blog » Blog Archive » Combinatory Algebra
Combinatory Algebra. December 11, 2007 – 420 pm. jonjayray. Do languages reminiscent of combinatory logic, eg. CAM expressions, Joy, Cat, have first class
History of Mathematics Blog
History of Mathematics.
Combinatory algebra
jonjayray Share and Enjoy:
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8. FLoC 2006 - LICS
In this paper we show that the Stone representation theorem for Boolean algebras can be generalized to Combinatory Algebras. In every Combinatory Algebra
LICS 2006 Twenty First Annual IEEE Symposium on Logic in Computer Science
Seattle, August 12 - 15, 2006 FLoC Home About FLoC MEETINGS CAV ICLP IJCAR LICS ... Workshops (by conf.) PROGRAM Room Assignments FLoC at a glance Social Events Invited Talks ... Workshop Proceedings FACILITIES Conference Hotel Event Space Internet Access SEATTLE Travel to/in Seattle Dining Guide Sightseeing in Seattle ORGANIZATION Steering Committee Program Committee Organizing Committee Sponsors MISCELLANEOUS Related Events Site Design OUT-OF-DATE Registration Visa Information Student Travel Support
LICS on Monday, August 14th
Chair: Patrice Godefroid
Location: Metropolitan A
Orna Kupferman (Hebrew University, Israel)
Avoiding Determinization [ppt]
Automata on infinite objects are extensively used in system specification, verification, and synthesis. While some applications of the automata-theoretic approach have been well accepted by the industry, some have not yet been reduced to practice. Applications that involve determinization of automata on infinite words have been doomed to belong to the second category. This has to do with the intricacy of Safra's optimal determinization construction, the fact that the state space that results from determinization is awfully complex and is not amenable to optimizations and a symbolic implementation, and the fact that determinization requires the introduction of acceptance conditions that are more complex than the Buchi acceptance condition. Examples of applications that involve determinization and belong to the unfortunate second category include model checking of omega-regular properties, decidability of branching temporal logics, and synthesis and control of open systems.

9. Fundamenta Informaticae, Volume 30, Abstracts
Like Combinatory Algebras they can be defined by true identities and thus form a variety in the sense of universal algebra.
Abstracts of Fundamenta Informaticae Volume 33.
Number 1
Minimal-Maximal Time Cause-Effect Structures pages 1-16

It is shown that minimal-time singly-valued (counterparts of 1-safe Petri nets) cause-effect (c-e) structures are of the same expressive power as no-time singly-valued c-e structures, while maximal-time multi-valued c-e structures are of essentially greater expressive power than no-time multi-valued c-e structures.
Cause-Effect Structures - Structural and Semantic Properties Revisited pages 17-42

On Scott Consequence Systems pages 43-70

The notion of Scott consequence system (briefly, S-system) was introduced by D. Vakarelov in [32] in an analogy to a similar notion given by D. Scott in [26]. In part one of the paper we study the category SSyst
In part two of the paper we prove that the separation theorem for S-systems is equivalent in ZF to some other separation principles, including the separation theorem for filters and ideals in Boolean algebras and separation theorem for convex sets in convexity spaces.

10. Publications, Lecture Notes Etc. - Thomas Streicher
We show that realizability models over a typed partial Combinatory Algebra (à la J. Longley) are impredicative, i.e. allow for quantification over
Publications, Lecture Notes and some Unpublished Notes
  • Semantics of Type Theory.
    in the series Progress in Theoretical Computer Science. Basel: Birkhaeuser Verlag. XII, 298 p. (1991).
  • Dependence and independence results for (impredicative) calculi of dependent types.
    Math. Struct. Comput. Sci. 2, No.1, 29-54 (1992).
  • Independence of the induction principle and the axiom of choice in the pure calculus of constructions.
    Theor. Comput. Sci. 103, No.2, 395-408 (1992).

  • Math. Struct. Comput. Sci. 4, No.1, 111-115 (1994).
  • The groupoid interpretation of type theory
    with M. Hofmann
    in Sambin, Giovanni (ed.) et al., Twenty-five years of constructive type theory. Proceedings of a congress, Venice, Italy, October 1921, 1995. Oxford: Clarendon Press. Oxf. Logic Guides. 36, 83-111 (1998). ps.gz
  • Induction and recursion on the partial real line with applications to Real PCF.
    with M. Hoetzel Escardo
    Theor. Comput. Sci. 210, No.1, 121-157 (1999).
  • Classical Logic, Continuation Semantics and Abstract Machines ps.gz

11. Homepage For Prof. Erwin Engeler
Equations in Combinatory Algebras. In Logics of Programs eds E. Clarke et al., Springer Lecture Notes in Computer Science 164 (Springer New York,
Prof. Erwin Engeler
Curriculum Vitae: My address:
Department of Mathematics, HUT
Federal Institute of Technology
8092 Zurich, Switzerland
Phone: + 41 1 632 22 25
How to contact me be email:
Click here to visit the home page of my wife Dr. phil. Margaret Engeler.
Dates and Stations
Born in Schaffhausen, Switzerland on the 13th February 1930.
Diploma in mathematics at the ETH, Zurich ETH, Zurich (Prof. P. Bernays)
Assistant Professor at the University of Minnesota
Assistant Professor at the University of California, Berkeley
Associate Professor and Full Professor at the University of Minnesota
Professor of Logic and Computer Science, Mathematics Department, ETH, Zurich
Activities and Offices
  • Author of various books on Logic, Mathematics and Computer Science, translated into Russian, Japanese and Chinese
  • Editor of scientific journals, book series and symposia
  • Collected works 1993
  • Active interest in music, art and various outdoor sports

12. Combinatory Algebra For Sequential Computation
The following paper is available at the address http//www.math.ruu. nl/publications/preprints/ A Combinatory Algebra FOR SEQUENTIAL FUNCTIONALS OF
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combinatory algebra for sequential computation
The following paper is available at the address

13. Baztech Informacja O Publikacji
Like Combinatory Algebras they can be defined by true identities and thus from a variety in the sense if universal algebra.

14. R000700:The Lambda Calculus And Combinatory Algebra
I will demonstrate that programs for the Miser Engine represent combinatory logic, so Combinatory Algebra is useable for discussion of an exploration of
Miser Project Readings
: The Lambda Calculus and Combinatory Algebra
Barendregt, Hendrik Pieter. The Lambda Calculus: Its Syntax and Semantics . North-Holland (Amsterdam, 1981). ISBN 0-444-85490-8. L ast updated 2002-03-13-23:12 -0800 (pst)
2000-07-18: Initial Notes - How Lambda-Calculus and Combinatory Algebra Figure in Miser
2000-07-18 (orcmid) This was the only serious lambda-calculus book that I kept in preparing to move to Italy in September, 1999. So I am mining it here for combinatory logic (CL).
Miser Representation of Combinatory Logic
I will demonstrate that programs for the Miser Engine represent combinatory logic, so combinatory algebra is useable for discussion of an exploration of computational completeness. The CL is equivalent to the closed-form lambda calculus (no free variables) which is equivalent to the set of recursive functions which is equivalent to what is computable. What's cool about Miser is it provides a pretty direct access to CL without presuming CL as primitive. That is, CL is represented, and this raises some interesting things about that and about representing mathematical structures by computer, generally.
Establishing Valid Interpretation of Combinatory Logic in Miser
My concern is as follows. If the Miser Engine evaluation mechanism provides a

15. DBLP: Inge Bethke
8, Inge Bethke, Jan Willem Klop, Roel C. de Vrijer Completing Partial Combinatory Algebras With Unique HeadNormal Forms. LICS 1996 448-454
Inge Bethke
List of publications from the DBLP Bibliography Server FAQ Coauthor Index - Ask others: ACM DL Guide CiteSeer CSB ... Jan A. Bergstra , Inge Bethke, Alban Ponse : Decision problems for pushdown threads. Acta Inf. 44 EE Jan A. Bergstra , Inge Bethke, Mark Burgess : A process algebra based framework for promise theory CoRR abs/0707.0744 EE Jan A. Bergstra , Inge Bethke: An upper bound for the equational specification of finite state services. Inf. Process. Lett. 94 EE Jan A. Bergstra , Inge Bethke: Network algebra in Java. J. Log. Algebr. Program. 62 EE Jan A. Bergstra , Inge Bethke: Polarized process algebra with reactive composition. Theor. Comput. Sci. 343 EE Jan A. Bergstra , Inge Bethke: Polarized Process Algebra and Program Equivalence. ICALP 2003 EE Jan A. Bergstra , Inge Bethke: Molecular dynamics. J. Log. Algebr. Program. 51 Inge Bethke, Jan Willem Klop Roel C. de Vrijer : Descendants and Origins in Term Rewriting. Inf. Comput. 159 Inge Bethke, Jan Willem Klop Roel C. de Vrijer : Extending partial combinatory algebras. Mathematical Structures in Computer Science 9 Inge Bethke

16. EquMath: Math Lessons » Blog Archive » Combinatory Algebra
Combinatory Algebra. December 11th, 2007 by. grigorix. Do languages reminiscent of combinatory logic, eg. CAM expressions, Joy, Cat, have first class
EquMath: Math Lessons
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  • 17. Combinatory
    Combinatory Algebra Truck Factoring Club Algebra factoring trinomials. Algebra factoring trinomials, algebra formulas.
    algebra by Algebra factoring trinomials
    Combinatory algebra by Algebra factoring trinomials
    Sun, 16 Dec 2007 17:17:20 -0600
    Combinatory algebra
    by Algebra factoring trinomials (algebra-factoring-trinomials) @ Sun, 16 Dec 2007 17:17:20 -0600 CAM expressions, Joy, Cat, have first class functions' Probably, yes, because of the abstractions one can use with code blocks. But if one removed these' Then one can bind identifiers to names, but can't do anything directly analogous . Original post: Combinatory algebra by at Google Blog Search: algebra formulas

    18. Abstracts Of The Workshops At The School In Logic And Computation
    Samson Abramsky and Marina Lenisa Edinburgh University; Realisability Models over linear Combinatory Algebras and the full completeness problem for typed

    19. Barendregt: Lambda Calculus
    In the present context, the theory of a Combinatory Algebra may not be . In every Combinatory Algebra we have lambda*x(A)x = A. An applicative structure
    TPS A higher-order theorem proving system The Omega Group This page was created and is maintained by Chad E Brown
    Barendregt, H. P. The Lambda Calculus: Its Syntax and Semantics. North Holland, Amsterdam (1984).
    Part I: TOWARDS THE THEORY Chapter 1: Introduction. Chapter 2: Conversion. Chapter 3: Reduction. Chapter 4: Theories. ... Chapter 5: Models. Part II: CONVERSION Chapter 6: Classical Lambda Calculus. Chapter 7: The Theory of Combinators. Chapter 8: Classical Lambda Calculus (Continued) Chapter 10: Bohm Trees. Part III: REDUCTION Chapter 11: Fundamental Theorems. Chapter 13: Reduction Strategies. Appendix A: Typed Lambda Calculus. Part I: TOWARDS THE THEORY Chapter 1: Introduction. Lambda calculus is a theory of functions as rules instead of graphs. The objects of study (type free lambda terms) can be used as both function and argument. Lambda calculus was originally invented to provide a general theory of functions which could be extended to provide a foundation for mathematics. Church's original system [1932/33] was inconsistent due to the Kleene-Rosser paradox [1935]. One response by Church [1941] was to study the subsystem given by the lambda-I calculus. [Another response was to study a system based on typed lambda calculus.] Curry proposed to extend pure combinatory logic by illative notions to give a foundation for mathematics. This program has not been completed. Another foundational approach related to lambda calculus is Feferman's [1975/80] systems for constructive mathematics based on partial application. These systems are related to Wagner [1969] and Strong's [1968] Uniformly Reflexive Structures. Also, there are relationships between typed lambda calculus, proof theory and category theory.

    20. Ramón Augusto Pino Pérez
    A Strict Partial Combinatory Algebra which modelizes Partial Lambda Calculus. An Extensional Partial Combinatory Algebra based on terms.
    Experiencia profesional Experiencia docente Publicaciones Comunicaciones a Congresos Capitulos de libro
  • Notes in Computer Science, vol. 520. 1991. pp. 387-396.
    pp 637-646. Morgan-Kaufmann Publishers, 2000.
  • Monografías de tipo pedagógico

    21. Emerald: Article Request
    Combinatory Algebra is explained in much more detail and rigor in Engeler’s (1995) Combinatory Programme. Also, a more formal definition related to QFD is

    22. Speaker Samson Abramsky Title Axiomatics Of No-Cloning And No
    Speaker Robin Cockett Title Itegories and partial Combinatory Algebras Abstract Flow diagrams are used extensively in Computer Science especially in the

    23. Stupid Question. | Lambda The Ultimate
    It is just these natural proof methods that are usually missing from ordinary Combinatory Algebra. (The same could be said about the theory of URS.
    @import "misc/drupal.css"; @import "themes/chameleon/ltu/style.css";
    Lambda the Ultimate
    Home Feedback FAQ ... Archives
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    Stupid Question.
    I realize this may be a dumb question... having had some Abstract/Modern Algebra classes, I understand what an Algebra is, but I have never really understood what defines a "Calculus". What's the common thread between Lambda Calculus, Pi-Calculus, and traditional "Calculus"(and all the other calculi...) that merits them being called a calculus? I realize this may be an obvious thing, but my tendency is to assume that there is some formal definition of Calculus out there which is being used which I am just unaware of. Again, if I'm just being dumb, someone email me and put me out of my misery :) By Matt Estes LtU Forum previous forum topic next forum topic ... other blogs
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    24. A Combinatory Algebra For Sequential Functionals Of Finite
    It is shown that the type structure of finitetype functionals associated to a Combinatory Algebra of partial functions from IN to IN (in the same way as
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    25. Developing Theories Of Types And Computability Via Realizability
    Given a partial Combinatory Algebra A, which we think of as continuous realizers, with a subalgebra a is subset of A which we think of as computable

    26. Co-Com
    Combinatory Algebra, 1172 Comet orbits and Gaussian distribution, 977 Common subexpressions in equation solutions, 945 in multilevel logic, 1096
    A B C D E F G ... Ce-Cm Co-Com Con-Cp Cr-Cz
    and interesting chemicals,
    and defining randomness,
    history of,
    in thermodynamics,
    in cellular structures,
    origin of shapes of,
    Coats of animals patterns on, Cobol and history of computing, Coccolithophorid shape of, Cocconi, Giuseppe (Italy/USA/Switzerland, 1914- ) and SETI, Cochlea and audio perception, Cochrane, Canada and patterns from space, Cockle shell growth of, Codd, Edgar F. (USA, 1923-[2003]) and universal CAs, Code in notes to this book, see also Programs Code (hash) Code (machine) Code 10 and my CA history, Code 12 and Ulam systems, Code 20 attractors in, as candidate for universality, excluded blocks in, halting probability in, persistent structures in, transient lengths in, Code 52 as candidate for universality, phase transition in, Code 52 (2D) domains in, Code 111 (2D) domains in, Code 177 randomness from

    27. USF Mathematics Faculty Research
    Selected Publications A measure of subgroup diversity, J. of Algebra 61 (1979), 308. A glimpse into the paradise of Combinatory Algebra,
    FACULTY RESEARCH Note: The information is provided by each faculty member individually. , Ph.D., SUNY-Albany, 1999; Assistant Professor. Research Interests: Complex analysis, interpolation, extremal problems in Hardy and Bergman spaces. Selected Publications: Jensen-type inequalities and radial null sets (with B. Korenblum), Analysis Some coefficient estimates for H P functions Remarks on the Bohr phenomenon (with A. Dahlner and D. Khavinson), Computational Methods and Function Theory (2004), no. 1, 1-19. Extremal problems for non-vanishing functions in Bergman spaces (with D. Aharonov, D. Khavinson and H. S. Shapiro), Operator Theory: Advances and Applications The isoperimetric inequality via approximation theory and free boundary problems (with D. Khavinson), Computational Methods and Function Theory (2006), no. 2, 253-274. W. Edwin Clark , Ph.D., Tulane University, 1964; Professor Emeritus. Research Interests: I am interested in a variety of algebraic, combinatorial and geometric problems. I have secondary interests in semigroups of matrices, associative rings and algebras, homological algebra, and categorical aspects of rings and semigroups. Selected Publications: On the probability that a t-subset of a finite vector space contains an r-subspace-with applications to short, light codewords in a BCH code

    28. ( )
    Inner algebras 2. Representation in Combinatory Algebras 3. Varieties as solutions of combinatory equations 4. Normal form theore

    29. Scientific Commons Inge Bethke
    Partial Combinatory Algebras occur regularly in the literature as a framework We introduce typed combinatory process algebra, a system combining process
    YÍCÛj½üP#µ¡µÉóVk <¤K%Lú§3¡æ ÞýŸ+le¾¬¹–x³ <µ˜Y¢© <G£ØrY6¡¶÷wK

    30. Item Subject Baudrillard From
    The same change in fortune as for a strategy of pacification by the mirror of equivalence, all binary oppositions and all Combinatory Algebra.
    Item Subject: baudrillard >From Mon Aug 21 09:08:37 1989 Received: from by; Mon, 21 Aug 89 09:08:33 BST Message-Id:

    31. An Extensional Partial Combinatory Algebra Based On Lamda-Terms
    An Extensional Partial Combinatory Algebra Based on LamdaTerms. Resource URI http//

    32. ODOBS - Publication Page: An Extensional Partial Combinatory Algebra Based On La
    An Extensional Partial Combinatory Algebra Based on LamdaTerms. Authors. Ramón PINO PÉREZ. Year, 1991. Booktitle, MFCS. Pages, 387-396.;jsessionid=2B061467611E6E

    33. From MAILER-DAEMON Sun Mar 30 154127 2003 Date 30 Mar 2003 15
    The somewhat mysterious definition of (partial or full) Combinatory Algebras is really motivated by the fact that it is equivalent to a combinatory
    From MAILER-DAEMON Sun Mar 30 15:41:27 2003 Date: 30 Mar 2003 15:41:27 -0400 From: Mail System Internal Data Subject: DON'T DELETE THIS MESSAGE FOLDER INTERNAL DATA Message-ID: X-IMAP: 1044293014 0000000066 Status: RO This text is part of the internal format of your mail folder, and is not a real message. It is created automatically by the mail system software. If deleted, important folder data will be lost, and it will be re-created with the data reset to initial values. From Mon Feb 3 11:41:21 2003 -0400 Return-path: Envelope-to: Delivery-date: Mon, 03 Feb 2003 11:41:21 -0400 Received: from Majordom by with local (Exim 4.10) id 18fiWo-0004rg-00 for; Mon, 03 Feb 2003 11:28:58 -0400 X-Authentication-Warning: birkedal set sender to using -f To: Subject: categories: Ph.D. scholarships at the IT University of Copenhagen Reply-To: From: Lars Birkedal Date: 03 Feb 2003 09:45:25 +0100 Message-ID: Envelope-to: Delivery-date: Wed, 05 Feb 2003 10:49:48 -0400 Received: from Majordom by with local (Exim 4.10) id 18gQko-0005Fp-00 for; Wed, 05 Feb 2003 10:42:22 -0400 Message-Id:

    34. Libra: A Combinatory Algebra For Sequential Functionals Of Finite
    The Combinatory Algebras. Wb Are. Abstract We will show that in the realisability model over the combinatoryalgebra A wb;e of e ective wellbracketed

    35. Logic Seminar Abstracts
    New applicative systems based on an untyped partial Combinatory Algebra are proposed whose provably recursive functions conicide with the functions

    36. Lars Birkedal / Realizability Bibliography
    A Combinatory Algebra for sequential functionals of finite type. In S.B. Cooper and J.K. Truss, editors, Models and Computability , pages 389406.
    Realizability Bibliography
    This bibliography on realizability has been established in connection with the Workshop on Realizability Semantics and Applications 1999 in Trento, Italy. Additions and corrections are very welcome. Please email them to Lars Birkedal ( BibTeX database: realizability.bib
    Search the Bibliography
    Keywords: Authors: 345 references, last updated Tue Jun 13 8:59:00 2000
    • M . Abadi and G.D. Plotkin. A per model of polymorphism and recursive types . In J. Mitchell, editor, 5th Annual IEEE Symposium on Logic in Computer Science , pages 355-365, Philadelphia, 1990. IEEE Computer Society Press.
    • M . Abadi and L. Cardelli. A Theory of Objects . Springer Verlag, 1996.
    • S . Abramsky. Typed realizability . Talk at the workshop on Category Theory and Computer Science in Cambridge, England, August 1995.
    • P .H.G. Aczel. A note on interpreting intuitionistic higher-order logic , 1980. Handwritten note.
    • T . Altenkirch. Constructions, Inductive Types and Strong Normalization . PhD thesis, University of Edinburgh, 1993. Available as report ECS-LFCS-93-279.

    37. Mathematics And Computation » Publications
    Abstract We compare realizability models over partial Combinatory Algebras by embedding them into sheaf toposes. We then use the machinery of Grothendieck
    @import url( );
    Mathematics and Computation
    April 12, 2007
    Implementing real numbers with RZ
    Filed under: RZ Talks Publications Computation ... Constructive math With Iztok Kavkler Abstract: RZ is a tool which translates axiomatizations of mathematical structures to program specifications using the realizability interpretation of logic. This helps programmers correctly implement data structures for computable mathematics. RZ does not prescribe a particular method of implementation, but allows programmers to write efficient code by hand, or to extract trusted code from formal proofs, if they so desire. We used this methodology to axiomatize real numbers and implemented the specification computed by RZ. The axiomatization is the standard domain-theoretic construction of reals as the maximal elements of the interval domain, while the implementation closely follows current state-of-the-art implementations of exact real arithmetic. Our results shows not only that the theory and practice of computable mathematics can coexist, but also that they work together harmoniously. Presented at Computability and Complexity in Analysis 2007 Download paper: rzreals.pdf

    38. Bibliography.bib
    In every Combinatory Algebra there is a Boolean algebra of central elements Central elements are used to represent any Combinatory Algebra as a Boolean
    bibliography.bib ... This file has been generated by

    39. WoLLIC'2006 - Content
    the language of set theory is here augmented by variables for operations in an untyped partial Combinatory Algebra over sets. Like the earlier approach,
    Content Title and Abstracts of Tutorial Lectures New insights into Probabilistically Checkable Proofs (PCPs) by Eli Ben-Sasson (Comput Sci Dept, Technion Inst of Technology, Israel) We revisit the celebrated PCP Theorem in light of recent simplifications to its proof and discuss new applications to error correcting codes and sub-linear time proof verification.
    Reminder : The celebrated PCP Theorem was proved in the early 1990's [Arora,Safra;Arora,Lund,Motwani,Sudan,Szegedy]. Informally speaking, it says there is a format of writing proofs and a probabilistic method of verifying their correctness. In this method the verifier needs to read only a constant number of random bits from the proof (independent of the proof length) to test its validity. Good proofs of true statements are accepted with probability one, whereas any purported "proof" of a false statement is rejected with probability of at least one third. Operational Theories of Sets by Solomon Feferman (Depts of Mathematics and Philosophy, Stanford University, USA)

    40. Java Combinators
    We have seen that we can map the Lambda Calculus down into a Combinatory Algebra. How might we implement a Combinatory Algebra in the Java environment?
    Computer Science 591i
    Implementing a Combinatory Algebra in Java
    [Edited from material submitted by Mr Schwartz, derived from our class discussion] We have seen that we can map the Lambda Calculus down into a combinatory algebra. How might we implement a combinatory algebra in the Java environment?
    The Carrier Set of the Algebra
    The carrier set of an algebra is the set of objects upon which the operation(s) of the algebra work. Clearly, the carrier set of our Algebra should be implemented as a set of Java objects. One approach would be to treat our carrier set as a Java class , which would thus be a subclass of the generic Java class Object. in turn would have subclasses and , representing numbers and functions respectively. The snag about this approach is that it would not be so easy to use existing Java classes such as the BigInteger class. So, it's probably better to treat the carrier set of our algebra as being simply the Java Object class, and to develop a class, which we really need to implement functions.

    41. MathGroup Archive: April 1999 [00139]
    (Makes use of Dynamic Programming, Stochastic Processes to provide the randomization or probability vectors using things learned in Combinatory Algebra or
    PreloadImages('/common/images2003/link_products_on.gif','/common/images2003/link_purchasing_on.gif','/common/images2003/link_forusers_on.gif','/common/images2003/link_aboutus_on.gif','/common/images2003/link_oursites_on.gif'); Wolfram Forums MathGroup Archive January February ... Give us feedback Sign up for our newsletter: Date Index Thread Index Author Index Re: Scheduling algorithms
    • To : mathgroup at Subject : [mg16995] Re: [mg16922] Scheduling algorithms From : "Hans J.-I. Michel" <hans at> Date : Sat, 10 Apr 1999 02:13:32 -0400 Sender : owner-wri-mathgroup at Operations Research (OR) Library anonymous ftp to Web Optimization and 2LP programming
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  • 42. Fully Complete Models For ML Polymorphic Types
    We show that a special linear Combinatory Algebra of partial involutions induces an hyperdoctrine which satisfies our axiomatization, and hence it provides
    Fully Complete Models for ML Polymorphic Types
    Samson Abramsky and Marina Lenisa Abstract: We present an axiomatic characterization of models fully-complete for ML-polymorphic types of System F. This axiomatization is given for hyperdoctrine models, which arise as adjoint models , i.e. co-Kleisli categories of suitable linear categories . Examples of adjoint models can be obtained from categories of Partial Equivalence Relations over Linear Combinatory Algebras . We show that a special linear combinatory algebra of partial involutions induces an hyperdoctrine which satisfies our axiomatization, and hence it provides a fully-complete model for ML-types. ECS-LFCS-99-414 This report is available in the following formats:

    Given a partial Combinatory Algebra A, which we think of as continuous realizers, with a subalgebra A , which we think of as computable realizers,
    Computer Science Department
    School of Computer Science, Carnegie Mellon University
    CMU-CS-99-173 Developing Theories of Types and Computability via Realizability Lars Birkedal December 1999 Ph.D. Thesis Temporarily Unavailable Electronically

    Keywords: Semantics, logic, type theory, category theory, realizability
    We investigate the development of theories of types and computability via realizability. In the first part of the thesis, we suggest a general notion of realizability, based on weakly closed partial cartesian categories, which generalizes the usual notion of realizability over a partial combinatory algebra. We show how to construct categories of so-called assemblies and modest sets over any weakly closed partial cartesian category and that these categories of assemblies and modest sets model dependent predicate logic, that is, first-order logic over dependent type theory. We further characterize when a weakly closed partial cartesian category gives rise to a topos. Scott's category of equilogical spaces arises as a special case of our notion of realizability, namely as modest sets over the category of algebraic lattices. Thus, as a consequence, we conclude that the category of equilogical spaces models dependent predicate logic; we include a concrete description of this model. modal logic for computability . Moreover, we characterize some interesting subcategories of RT(A,A#) (in much the same way as assemblies and modest sets are characterized in standard realizability toposes) and show the validity of some logical principles in RT(A,A#).

    44. Science/AAAS | Science Magazine: Published E-Letter Responses For Maziak, 308 (5
    Algebra (Arabic for “reduction”) is also an Arab invention. unknown in the Greek world as well as Combinatory Algebra), plain trigonometry, etc.
    Jump to: Page Content Section Navigation Site Navigation Site Search ... Account Information , or Site Tools Note to users. If you're seeing this message, it means that your browser cannot find this page's style/presentation instructions or possibly that you are using a browser that does not support current Web standards. Find out more about why this message is appearing, and what you can do to make your experience of our site the best it can be.
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    • Current Issue Previous Issues Science Express ... Magazine
      E-Letter responses to:
      Wasim Maziak
      Science in the Arab World: Vision of Glories Beyond
      Science 2005; 308: 1416-1418 [Summary] [Full text] [PDF]
      E-Letters: Submit a response to this article
      Published E-Letter responses:
      Arabs Neither Need a Scientific Revolution Nor Are They a Cultural Exception
      Kamal Chaouachi (7 March 2006)
      Arabs Neither Need a Scientific Revolution Nor Are They a Cultural Exception 7 March 2006 Kamal Chaouachi

    45. Npat's Blog
    to all Combinatory Algebra . The fake does not give you a clearer glimpse to the world, but, under the best conditions, a glimpse to a cleaner world.
    entries archive friends userinfo Nick Patavalis website userinfo livejournal userinfo archive journal archive Links Links: RSS lost in tyme 05:34 am Tags culture Current Music Six Feet Under - Inspiration In My Head
    Go visit the lost-in-tyme blog. There you will find showcased, and you can download full releases of, exquisite psychedelic / garage / folk / hippie rarities... Link Leave a comment warning signs for the future 03:29 pm Tags culture thenet
    Warning Signs For Tomorrow
    Source: Chris's Wiki Link 1 comment Leave a comment the song of the sausage creature 08:51 pm Tags culture moto Current Music Sonic Youth - Death Valley '69
    Hunter S. Thompson had a thing for motorcycles, not much unlike the thing he had for drugs, as well as for practically every forbidden pleasure. Once he found himself with a Ducati super-bike and he had to write a review about it for the CycleWorld magazine. The result was a very characteristic document (a classic gonzo-style piece) titled The Song of the Sausage Creature . I came upon it lately and, for no particular reason, I made an attempt to translate it to Greek . I tried to maintain the overall feeling of Thompson's writing without staying too close to the exact words. Regardless, I'm still not sure if it makes any sense in Greek. Read it if you don't have anything better to do, and tell me what you think about it...

    46. TAC Abstracs
    In the second talk, we show that the fully abstract model of PCF a la Milner arises as a realisability model over the partial Combinatory Algebra (pca) as
    The topos of 3-colored graphs The problem of 3 coloring finite undirected graphs is NP-complete and therefore there is interest in the structure of graphs which are and are not 3 colorable. The category of undirected graphs U is a topos. A graph with a given 3 coloring is said to be 3-colored. The category C of 3-colored graphs is is a cocomplete topos, hence also complete. There is an essential geometric morphism between U and C, and the three functors involved have considerable natural appeal. These functors justify two of the heuristics used to quicken algorithms which determine whether or not a 3 coloring exists for argument finite undirected graphs. Indeed, one of these heuristics deserves to be better known. ...
    Finally, we observe that instead of B we might take any pca of the form L/T where T is a theory containing the (obvious) conversion theory for L and itself being containEd in Th(A). This may be considered as a solution of (a variant of) the Longley-Phoa Conjecture claiming that realisability over a term model of untyped lambda-calculus gives rise to a fully abstract model of PCF.

    47. Flengyel's Bookmarks Tagged With "partial-combinatory-algebra" On
    flengyel s items tagged partialCombinatory-Algebra view all, popular .. to logic combinator partial-Combinatory-Algebra LP logic-of-proofs .
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    48. Geometry, Algebra, Arithmetics And Combinatory — IRISA
    Geometry, Algebra, Arithmetics and Combinatory HAVEGE The unpredictable random number generator; Geometry, Algebra, Arithmetics and Combinatory
    Intranet Fran§ais Skip to content. Search
    Sections You are here: Home Valorisation and Partnership Software and Patents Software Geometry, Algebra, Arithmetics and Combinatory
    Geometry, Algebra, Arithmetics and Combinatory
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    The unpredictable random number generator
    A library of polyhedral functions
    Phone numbers Fax : +33 299 847 171 Legal informations and credits

    49. List Of Algebraic Structures - Wikipedia, The Free Encyclopedia
    2.4 Ringlike structures; 2.5 Modules and Algebras. 3 Quasivarieties. 3.1 Magmas. 3.1.1 Cancellative; 3.1.2 Combinatory logic. 3.2 Graphs; 3.3 Lattices
    var wgNotice = ""; var wgNoticeLocal = ""; var wgNoticeLang = "en"; var wgNoticeProject = "wikipedia";
    List of algebraic structures
    From Wikipedia, the free encyclopedia
    Jump to: navigation search In universal algebra , a branch of pure mathematics , an algebraic structure is a variety or quasivariety Abstract algebra is primarily the study of algebraic structures and their properties. Some axiomatic formal systems that are neither varieties nor quasivarieties, called nonvarieties below, are included among the algebraic structures by tradition. Other web lists of algebraic structures, organized more or less alphabetically, include Jipsen and PlanetMath. These lists mention many structures not included below, and may present more information about some structures than is presented here.
    • Generalities Varieties
      edit Generalities
      An algebraic structure consists of one or two sets closed under some operations functions , and relations , satisfying a number of axioms , including none. This definition of an algebraic structure should not be taken as restrictive. Anything that satisfies the axioms defining a structure is an instance of that structure, regardless of how many other axioms that instance happens to satisfy. For example, all groups are also semigroups and magmas Structures are listed below in approximate order of increasing complexity as follows:
      • Structures that are varieties precede those that are not;

    50. BibTeX Bibliography Intjcomputinfsci.bib
    Dept., New Mexico Inst. of Mining and Technol., Socorro, NM, USA , keywords = Algebra; combinatorial circuits; hazards and race conditions; fuzzy Algebra; beebe at ack-nhfb ... j-INT-J-COMPUT-INF-SCI Simon:1972:APN j-INT-J-COMPUT-INF-SCI ack-nhfb Biss:1972:DSC j-INT-J-COMPUT-INF-SCI ack-nhfb Hunter:1972:DRC j-INT-J-COMPUT-INF-SCI ack-nhfb Tou:1972:ARH j-INT-J-COMPUT-INF-SCI ack-nhfb Terrenoire:1972:EAI j-INT-J-COMPUT-INF-SCI ack-nhfb Ibaraki:1972:IEP j-INT-J-COMPUT-INF-SCI ack-nhfb Katzan:1972:AAR j-INT-J-COMPUT-INF-SCI ack-nhfb Inselberg:1972:TLS j-INT-J-COMPUT-INF-SCI ack-nhfb Fu:1972:SGL j-INT-J-COMPUT-INF-SCI ack-nhfb Bohm:1972:CAT j-INT-J-COMPUT-INF-SCI ack-nhfb Wells:1972:UDS j-INT-J-COMPUT-INF-SCI ack-nhfb Wile:1972:IBE j-INT-J-COMPUT-INF-SCI ack-nhfb Wilkes:1972:ATD j-INT-J-COMPUT-INF-SCI ack-nhfb Busam:1972:DSP j-INT-J-COMPUT-INF-SCI ack-nhfb Maruyama:1972:PDI j-INT-J-COMPUT-INF-SCI ack-nhfb Mishelevich:1972:CSA j-INT-J-COMPUT-INF-SCI ack-nhfb vanDam:1972:SII j-INT-J-COMPUT-INF-SCI ack-nhfb Wilkes:1972:ALA j-INT-J-COMPUT-INF-SCI ack-nhfb Lipovski:1972:DSC j-INT-J-COMPUT-INF-SCI ack-nhfb Hughes:1972:DUA j-INT-J-COMPUT-INF-SCI ack-nhfb Walk:1973:MSP j-INT-J-COMPUT-INF-SCI ack-nhfb Zalcstein:1973:SLS j-INT-J-COMPUT-INF-SCI ack-nhfb Ivakhnenko:1973:PCS j-INT-J-COMPUT-INF-SCI ack-nhfb Diday:1973:DCM j-INT-J-COMPUT-INF-SCI ack-nhfb Brunnstein:1973:SRI j-INT-J-COMPUT-INF-SCI ack-nhfb Babu:1973:APD j-INT-J-COMPUT-INF-SCI ack-nhfb Rickman:1973:SIO j-INT-J-COMPUT-INF-SCI ack-nhfb Book:1973:SCG j-INT-J-COMPUT-INF-SCI ack-nhfb Payne:1973:PSL j-INT-J-COMPUT-INF-SCI ack-nhfb Bohm:1973:NCA

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