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1. BCK And BCI Logics, Condensed Detachment And The $2$-property. BCK and BCI logics, condensed detachment and the $2$property. J. Roger Hindley. Source Notre Dame J. Formal Logic Volume 34, Number 2 (1993), 231-250. http://projecteuclid.org/handle/euclid.ndjfl/1093634655

Log in RSS Title Author(s) Abstract Subject Keyword All Fields FullText more options Home Browse Search ... next BCK and BCI logics, condensed detachment and the $2$-property. J. Roger Hindley Source: Notre Dame J. Formal Logic Volume 34, Number 2 (1993), 231-250. Primary Subjects: Secondary Subjects: Full-text: Access granted (open access) PDF File (1792 KB) Links and Identifiers Permanent link to this document: http://projecteuclid.org/euclid.ndjfl/1093634655 Zentralblatt Math identifier: Mathematical Reviews number (MathSciNet): Digital Object Identifier: doi:10.1305/ndjfl/1093634655 back to Table of Contents previous next Notre Dame Journal of Formal Logic Current Issue All Issues Search this Journal Editorial Board ... Log in RSS Cornell University Library

2. List KWIC DDC22 510 And MSC+ZDM E-N Lexical Connection linear logic, Lambek calculus, BCK and BCI logics) substructural logics (including relevance, entailment, 03B47 linear mappings, matrices, determinants, http://www.math.unipd.it/~biblio/kwic/msc-cdd/dml2_11_33.htm

linear integral equations # systems of linear integral equations # systems of nonsingular linear integral equations # systems of singular linear logic and other substructural logics linear logic, Lambek calculus, BCK and BCI logics) # substructural logics (including relevance, entailment, linear mappings, matrices, determinants, theory of equation) # linear algebra. multilinear algebra. (vector spaces, linear models # generalized linear operators linear operators # equations and inequalities involving linear operators # equations with linear operators # general theory of linear operators # groups and semigroups of linear operators # special classes of linear operators (operator and matrix valued functions, etc., including analytic and meromorphic ones) # functions whose values are linear operators as elements of algebraic systems # individual linear operators) # linear relations (multivalued linear operators, their generalizations and applications # groups and semigroups of linear operators, with operator unknowns # equations involving

3. JSTOR The Number Of Proofs For A BCK-Formula 6 ~, BCK and BCI logics, condensed detachment and the 2property, Notre Dame Journal of Formal Logic, to appear. 7 S. HIROKAWA, Principal types of http://links.jstor.org/sici?sici=0022-4812(199306)58:2<626:TNOPFA>2.0.CO;2-H

4. Wieslaw A. Dudek Rough subalgebras of some binary algebras connected with logics, A computer method of computation of BCK and BCIalgebras of small orders, http://www.im.pwr.wroc.pl/~dudek/alg-log.htm

Wies³aw A. Dudek Selected papers on algebras induced by logics in print Filter theory of BL algebras , (with M. Kondo), Soft Computing Papers Filter theory of BL algebras, (with M. Kondo), Soft Computing 12 (2008), 419-423. Available online Ultra LI-ideals in lattice implication algebras and MTL-algebras, (with X. H. Zhang and K. Qin), Czechoslovak Math. J. 57(132) (2007), 591-605. Available online On hyper BCC-algebras, (with R.A.Borzooei and N.Koohestani), International J. Math. and Math. Sciences (2006), Article ID 49703 Available online Topological structures of rough sets induced by equivalence relations, (with M. Kondo), J. Advanced Computational Intelligence and Intelligent Informatics 10 (2006), 621-624. Available online On multiplicative fuzzy BCC-algebras, (with Y.B.Jun), J. Fuzzy Math. 13 (2005), 929-939. On the transfer principle in fuzzy theory, (with M. Kondo), Mathware and Soft Computing 12 (2005), 41-55. Rough subalgebras of some binary algebras connected with logics, (with Y.B.Jun), International J. Math. and Math. Sciences 2005, No.3, 437-443. Characterization theorem of lattice implication algebras

5. 03Bxx 03B05 Classical propositional logic; 03B10 Classical firstorder logic linear logic, Lambek calculus, BCK and BCI logics) {For proof-theoretic aspects http://www.ams.org/msc/03Bxx.html

Home MathSciNet Journals Books ... Contact Us 201 Charles Street Providence, RI 02904 USA Phone: 401-455-4000 or 800-321-4AMS Or email us at ams@ams.org Open Positions General logic 03B05 Classical propositional logic 03B10 Classical first-order logic 03B15 Higher-order logic and type theory 03B20 Subsystems of classical logic (including intuitionistic logic) 03B22 Abstract deductive systems 03B25 Decidability of theories and sets of sentences [See also 03B30 Foundations of classical theories (including reverse mathematics) [See also 03B35 Mechanization of proofs and logical operations [See also 03B40 Combinatory logic and lambda-calculus [See also 03B42 Logic of knowledge and belief 03B44 Temporal logic ; for temporal logic see ; for provability logic see also 03B48 Probability and inductive logic [See also 03B50 Many-valued logic 03B52 Fuzzy logic; logic of vagueness [See also 03B53 Logics admitting inconsistency (paraconsistent logics, discussive logics, etc.) 03B55 Intermediate logics 03B60 Other nonclassical logic 03B65 Logic of natural languages [See also 03B70 Logic in computer science [See also 68-xx 03B80 Other applications of logic 03B99 None of the above, but in this section

6. J Roger Hindley : Research MR1231287 J R Hindley, BCK and BCI logics, condensed detachment and the 2property, Notre Dame Journal of Formal Logic 34 (1993), 231250. http://www-maths.swan.ac.uk/staff/jrh/JRHresearch.html

Swansea University Physical Sciences Mathematics Department J Roger Hindley J Roger Hindley : Research fields of interest main publications career 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

7. Notre Dame Journal Of Formal Logic, Volume 34 223230 BibTeX J. Roger Hindley BCK and BCI logics, Condensed Detachment and the 2-Property. 231-250 BibTeX M. Gordon Beavers Extensions of the http://www.informatik.uni-trier.de/~ley/db/journals/ndjfl/ndjfl34.html

Notre Dame Journal of Formal Logic , Volume 34 Volume 34, Number 1, Winter 1993 James E. Baumgartner Saharon Shelah Simon Thomas : Maximal Subsets of Infinite Symmetric Groups. 1-11 BibTeX Brian Macpherson : Is It Possible that Belief Isn't Necessary? 12-28 BibTeX Xizhong Zheng : On the Maximality of Some Pairs of p-t Degrees. 29-35 BibTeX Daniel J. Dougherty : Closed Categories and Categorial Grammar. 36-49 BibTeX Margarita Otero : The Amalgamation Property in Normal Open Induction. 50-55 BibTeX Patrick Blackburn : Nominal Tense Logic. 56-83 BibTeX Friedemann Tuttas : An Arithmetical Completeness Theorem for Pre-permutations. 84-89 BibTeX Ernest Adams : Formalizing the Logic of Positive, Comparative, and Superlative. 90-99 BibTeX Paul E. Howard Jeffrey Solski : The Strenght of the Delta-system Lemma. 100-106 BibTeX : The Grounds for the Model-theoretic Account of the Logical Properties. 107-131 BibTeX William McCune : Single Axioms for the Left Group and the Right Group Calculi. 132-139 BibTeX Philip L. Peterson : Intermediate Quantifiers for Finch's Proportions. 140-149 BibTeX Volume 34, Number 2, Spring 1993

8. Logic Jnl IGPL -- Sign In Page Roger Hindley J. (1993) ‘BCK and BCI logics, Condensed Detachment and the 2Property’. Notre Dame Journal of Formal Logic 34231–250. Humberstone Lloyd. http://jigpal.oxfordjournals.org/cgi/content/full/jzm005v1

@import "/resource/css/hw.css"; @import "/resource/css/igpl.css"; Skip Navigation Oxford Journals Contact Us My Basket ... Volume 0, Number 2007 10.1093/jigpal/jzm005 This item requires a subscription* to Logic Journal of IGPL Online. * Please note that articles prior to 1996 are not normally available via a current subscription. In order to view content before this time, access to the Oxford Journals digital archive is required. If you would like to purchase short-term access you must have a personal account. Please sign in below with your personal user name and password or register to obtain a user name and password for free. Full Text The Relevant Equivalence Property Humberstone and Meyer Logic Jnl IGPL. To view this item, select one of the options below: Sign In User Name Sign in without cookies. Can't get past this page? Help with Cookies. Need to Activate? Password Forgot your user name or password? Purchase Short-Term Access Pay per View - If you would like to purchase short-term access you must have a personal account. Please sign in with your personal user name and password or register to obtain a user name and password for free. You will be presented with Pay per Article options after you have successfully signed in with a personal account.

9. 03Bxx 03B47, Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics) {For prooftheoretic aspects see 03F52} http://www.impan.gov.pl/LIB/MSC/03Bxx.html

General logic Classical propositional logic Classical first-order logic Higher-order logic and type theory Subsystems of classical logic (including intuitionistic logic) Abstract deductive systems Decidability of theories and sets of sentences [See also Foundations of classical theories (including reverse mathematics) [See also Mechanization of proofs and logical operations [See also Combinatory logic and lambda-calculus [See also Logic of knowledge and belief Temporal logic Modal logic ; for temporal logic see ; for provability logic see also Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics) Probability and inductive logic [See also Many-valued logic Fuzzy logic; logic of vagueness [See also Logics admitting inconsistency (paraconsistent logics, discussive logics, etc.) Intermediate logics Other nonclassical logic Logic of natural languages [See also Logic in computer science [See also 68-XX Other applications of logic None of the above, but in this section

10. HeiDOK 03B45 Modal logic ( 0 Dok. ) 03B47 Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics) ( 0 Dok. http://archiv.ub.uni-heidelberg.de/volltextserver/msc_ebene3.php?anzahl=0&la=de&

11. Substructural Logic Page We give an implicational formula X such the results of strengthening each of the substructural logics BCI, BCIW and BCK by the addition of X is a distict http://users.rsise.anu.edu.au/~jks/substructural.html

Papers on (mostly distributive) Substructural Logics John Slaney More Proofs of an Axiom of Lukasiewicz This is the abstract of the paper: John Slaney. More Proofs of an Axiom of Lukasiewicz. Journal of Automated Reasoning This paper reports results and some new problems in one of the domains to which automatic first-order theorem provers have been most successfully applied: axiomatics of non-classical propositional logics. It is well known that one of the standard axioms of the denumerable-valued pure implication logic of Lukasiewicz becomes derivable from the remainder in the presence of negation. Here it is shown that the same axiom is similarly derivable using conjunction and disjunction instead of negation. This closes a problem left open by Harris and Fitelson Journal of Automated Reasoning 27, 2001). Related problems are discussed, and five such open problems are hereby proposed as challenges to the automated reasoning community. Logic for Two This is the abstract of the paper: John Slaney and Robert Meyer.

12. BCK-AND-BCI-LOGICS-CONDENSED-DETACHMENT-AND-THE-$2$-PROPERTY Translate this page BCK and BCI logics, CONDENSED DETACHMENT and THE $2$-PROPERTY. http://biblioteca.universia.net/html_bura/ficha/params/id/980008.html

Publicidad Publicidad document.write(''); Biblioteca.Net Buscar recursos: Buscador Google Lógica Matemáticas Astronomía y Astrofísica Física ... COLECCIONES Usuarios registrados Regístrese Usuario: España Argentina Brasil Chile Colombia México Perú Portugal Puerto Rico Venezuela Contraseña: Colecciones destacadas PubMed Central (PMC3 - NLM DTD) arXiv Library of Congress Open Archive Initiative Repository 1 más...] Menú Solicitados Datos del recurso Ver BCK and BCI logics, condensed detachment and the $2$-property. Hindley, J. Roger Localización: http://ProjectEuclid.org/getRecord?id=euclid.ndjfl/1093634655 euclid.ndjfl/1093634655 Citation: Notre Dame J. Formal Logic 34 (1993), Number 2, 231-250 Pertenece a: Project Euclid (Hosted at Cornell University Library) Descargar SCORM ¡Sea el primero en solicitar este recurso! Para poder descargar este recurso debe identificarse como usuario de la biblioteca Valoración de los usuarios No hay ninguna valoración para este recurso. Sea el primero en valorar este recurso Detalles del recurso BCK and BCI logics, condensed detachment and the $2$-property.

13. List KWIC DDC And MSC Lexical Connection logics linear logic and other substructural 03F52 logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics) http://www.mi.imati.cnr.it/~alberto/dml_11_29.htm

linear integral equations # systems of singular linear logic and other substructural logics linear models # generalized linear operators linear operators # equations and inequalities involving linear operators # equations with linear operators # general theory of linear operators # groups and semigroups of linear operators # special classes of linear operators (operator and matrix valued functions, etc., including analytic and meromorphic ones) # functions whose values are linear operators as elements of algebraic systems # individual linear operators) # linear relations (multivalued linear operators, their generalizations and applications # groups and semigroups of linear operators, with operator unknowns # equations involving linear operators, with vector unknowns # equations and inequalities involving linear operators; topological tensor products; approximation properties # spaces of linear optimal control problems linear parabolic PDE; boundary value problems for nonlinear parabolic PDE # nonlinear boundary value problems for linear programming linear programming linear programming (transportation, multi-index, etc.) # special problems of

14. MSC 2000 : CC = Ore 0399 Mathematical logic and foundations (not classified at a more specific linear logic, Lambek calculus, BCK and BCI logics) For proof-theoretic http://portail.mathdoc.fr/cgi-bin/msc2000.py?L=fr&T=Q&C=msc2000&CC=Ore

15. IngentaConnect Substructural Implicational Logics Including The Relevant Logic E Substructural Implicational logics Including the Relevant Logic E EW and S4-W); BCI-logic, BCK-logic, relevant logic R, and the intuitionistic logic. http://www.ingentaconnect.com/content/klu/stud/1999/00000063/00000002/00239710

var tcdacmd="dt"; Home About Ingenta Ingenta Labs Ingenta Blog ... Check our FAQs Or contact us to report problems with: Subscription access Article delivery Registration Library administrator tasks ... Other problems For Publishers Why go online? Why choose IngentaConnect? Beyond Print: enhancing your service Access and authentication ... Contact us For Researchers For Authors About IngentaConnect Search and browse Publications available ... Register For Librarians Resource Zone Why choose IngentaConnect? Why choose IngentaConnect Complete? Activating Subscriptions ... Register CM8ShowAd("HorizontalBanner"); Substructural Implicational Logics Including the Relevant Logic E Authors: Kashima, Ryo ; Kamide, Norihiro Source: Studia Logica , Volume 63, Number 2, September 1999 , pp. 181-212(32) Publisher: Springer view table of contents Key: - Free Content - New Content - Subscribed Content - Free Trial Content CM8ShowAd("Skyscraper"); Abstract: We introduce several restricted versions of the structural rules in the implicational fragment of Gentzen's sequent calculus LJ. For example, we permit the applications of a structural rule only if its principal formula is an implication. We investigate cut-eliminability and theorem-equivalence among various combinations of them. The results include new cut-elimination theorems for the implicational fragments of the following logics: relevant logic E, strict implication S4, and their neighbors (e.g., E-W and S4-W); BCI-logic, BCK-logic, relevant logic R, and the intuitionistic logic.

16. Bibliography On Linear Logic BCKcombinators and linear lambda-terms have types. Theoretical Computer Science, 6497-105, 1989. J. R. Hindley. BCK- and BCI-logics, condensed detachment http://www.cs.cmu.edu/~carsten/linearbib/llb.html

HTML bibliography on linear logic Whenever available, URL links to the referenced papers are provided. Any corrections, updates, comments, suggestion, new entries, new URLs for papers, etc. are very much appreciated. This file is also available in BibTeX (the source) Postscript DVI Other pages of interest the Linear Logic home page the linear logic bibliography home page What's new in Linear Logic Authors index Last modified: Thu May 7 10:05:50 EDT 1998 Fast links ... A B C D ... Z Stål Aanderaa and Hermann R. Jervell. Recursive inseparability in linear logic. In E. Börger, G. Jäger, H. K. Bünig, S. Martini, and M. M. Richter, editors, Proceedings of the Sixth Workshop on Computer Science Logic , pages 5-13, San Miniato, Italy, September 1992. Springer-Verlag LNCS 702. Martin Abadi, George Gonthier, and Jean-Jacques Levy. The geometry of optimal lambda reduction. In Proceedings of the Nineteenth Annual Symposium on Principles of Programming Languages , pages 15-26, Albuquerque, New Mexico, January 1992. ACM Press. Martin Abadi, George Gonthier, and Jean-Jacques Levy. Linear logic without boxes. In

17. Condensed Detachment - Wikipedia, The Free Encyclopedia BCK and BCI logics, condensed detachment and the $2$property by J. Roger Hindley, Notre Dame Journal of Formal Logic 34, no. 2 (1993), 231–250 http://en.wikipedia.org/wiki/Condensed_Detachment

var wgNotice = ""; var wgNoticeLocal = ""; var wgNoticeLang = "en"; var wgNoticeProject = "wikipedia"; Condensed Detachment From Wikipedia, the free encyclopedia Jump to: navigation search Condensed Detachment (Rule D) is a method of finding the most general possible conclusion given two formal logical statements. It was developed by the Irish Logician Carew Meredith in the 1950s and inspired by the work of à ukasiewicz edit Informal description A rule of detachment says: Given that implies , and given produces The condensed detachment goes a step further and says: Given that implies , and given an produces the most general possible result of substitutions into and followed by a standard rule of detachments. For substitution A that when applied to produces , and substitution B that when applied to produces , are called the Unifiers of and Various unifiers may produce expressions with varying numbers of free variables. Some possible unifying expression are substitution instances of others. If one expression is a substitution instance of another (and not just a variable renaming), then that other is called "more general". If the most general unifier is used in Condensed Detachment, then the logical result is the most general conclusion that can be made in the given inference with the given second expression. (And since any weaker inference you can get is a substitution instance of the most general one, nothing less than the most general unifier is ever used in practice.)

18. General General Mathematics Mathematics For Nonmathematicians for provability logic see also 03F45} Substructural logics (including linear logic, Lambek calculus, BCK and BCI logics) {For prooftheoretic http://amf.openlib.org/2001/msc2000.xsd

General mathematics Mathematics for nonmathematicians (engineering, social sciences, etc.) Problem books Recreational mathematics [See also 97A20] Bibliographies External book reviews Dictionaries and other general reference works Formularies Philosophy of mathematics [See also 03A05] Methodology of mathematics, didactics [See also 97Cxx, 97Dxx] Theory of mathematical modeling General methods of simulation Dimensional analysis Physics (use more specific entries from Sections 70 through 86 when possible) Miscellaneous topics Collections of abstracts of lectures Collections of articles of general interest Collections of articles of miscellaneous specific content Proceedings of conferences of general interest Proceedings of conferences of miscellaneous specific interest Festschriften Volumes of selected translations Miscellaneous volumes of translations Collections of reprinted articles [See also 01A75] General histories, source books Ethnomathematics, general Paleolithic, Neolithic Indigenous cultures of the Americas Other indigenous cultures (non-European) Indigenous European cultures (pre-Greek, etc.)

19. 363/369 (Total 5522) NO 92 03B50 Many-valued Logic Translate this page 90, 03B47, Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics) {For proof-theoretic aspects see 03F52} http://www.mathnet.or.kr/mathnet/msc_list.php?mode=list&ftype=&fstr=&page=363

20. AUTHOR INDEX A Constructive Negation Defined with a Negation Connective for logics Including Bp+ Classical versions of BCI, BCK and BCIW logics, 6165 23/2 http://www.filozof.uni.lodz.pl/bulletin/s.html

BULLETIN OF THE SECTION OF LOGIC Author Index To find any word: author, title or key word; please, press "Ctrl-F". S SAAD S., Figallo A. V. and Ramon G. iH-Propositional Calculus SAARINEN, Esa On the logic of perception sentences SABALSKI, Bogdan Some remarks about continuous Boolean algebras SAGI, Gabor A model theoretic characterization of complexity of theories Non-Computability of the Equational Theory of Polyadic Algebras SAGI, Gabor and Nemeti, Istvan and Madarasz, Judit On the finitization problem of relational algebras SAIN, Ildiko Total correctness in nonstandard dynamic logic Amalgamation, Epimorphisms and Definability Properties in Algebraic Logic Computer sciences temporal logics need their clocks On finitizing first order logic SAIN, Ildiko and Nemeti, Istvan Fork algebras in usual and in non-well-founded set theories. Part I Fork algebras in usual and in non-well-founded set theories. Part II SALONI, Zygmunt The sequent Gentzen system for m-valued logic SALTO, Francisco and Mendez, Jose More on Peirce's law SALTO, Francisco and Mendez, Jose and Robles, Gemma

21. Notre Dame Journal Of Formal Logic BCK and BCI logics, Condensed Detachment and the 2Property. by J. Roger Hindley v. 34 i. 2 p. 231 - 250. Extensions of the aleph0-Valued Lukasiewicz http://wotan.liu.edu/docis/dbl/ndjofl/index.html

The Digital Librarian's Digital Library search D O CIS  Do cuments in  C omputing and I nformation  S cience Home Journals and Conference Proceedings Notre Dame Journal of Formal Logic Bolzano's Theory of Ground and Consequence by:  Armin Tatzel v. 43 i. 1 p. 1 - 25 Logics of Relative Identity by:  Pawel Garbacz v. 43 i. 1 p. 27 - 50 Realizing Levels of the Hyperarithmetic Hierarchy as Degree Spectra of Relations on Computable Structures by:  Denis R. Hirschfeldt, Walker M. White v. 43 i. 1 p. 51 - 64 Definability of Initial Segments by:  Saharon Shelah, Akito Tsuboi v. 43 i. 2 p. 65 - 73

22. MathGuide - OPAC Subject Catalog 03XX Mathematical logic and foundations 03B47 Substructural logics (including relevance, entailment, linear logic, Lambek calculus, BCK and BCI logics) http://www.mathguide.de/cgi-bin/ssgfi/navigator2.pl/db=math/type=gok/pattern=all

Browse the GBV OPAC by MSC 2000 This is a browse interface to the union catalogue of the Common Library Network GBV based on the MSC 2000 classification. You can browse down to the individual notation, the links available will direct you to the appropriate place in the GBV OPAC, which uses a notation related to, but different from MSC. Note: Not all books available are contained in the online catalogue. Please use the Goettingen State University Library's Alphabetical Catalogue to search for monographs, dissertations and journals missing in the OPAC. Open all categories Close all categories Foundations 00-XX General Instructional exposition (textbooks, tutorial papers, etc.) Research exposition (monographs, survey articles) General mathematics General and miscellaneous specific topics General mathematics Mathematics for nonmathematicians (engineering, social sciences, etc.) Problem books Recreational mathematics Bibliographies External book reviews Dictionaries and other general reference works Formularies Philosophy of mathematics Methodology of mathematics, didactics

23. Brouwer: A Proof Finding Program The BCI mode is like BCK , but restricts proofs to terms where each In the BCK and BCI modes the restriction to only one variable of each type is http://members.ozemail.com.au/~dekker/BROUWER.HTML

Brouwer: A Proof Finding Program for Intuitionistic, BCI, BCK, and Classical Logic The Brouwer theorem-prover for Intuitionistic, BCI BCK Anthony Dekker 1992-1995) operates by enumerating lambda-calculus terms of a given type. The initial Brouwer 0.5.0 release [Macintosh SEA/HQX file: 283K] is available for download and testing on request Using Brouwer Version 0.5.0 This program was written by Anthony Dekker to construct terms in long form of a given type. It is based on work conducted by Martin Bunder, drawing on earlier work by Ben-Yelles. The program is written in the CLEAN Functional Programming Language. The program uses about 3MB of RAM. For complex types, this may not be sufficient, since the algorithm uses exponential space and time. On memory overflow, the program may terminate with a 'Heap Full' message, or simply crash with an 'Error of Type 1'. The heap size cannot be increased without recompiling the program. Select Apple-D (from the 'Brouwer' menu) to open the dialog box if necessary, enter a type, and click one of the proof buttons. The 'One Proof' button produces a single term of the given type, while the 'All Proofs' button generates all terms of the given type. The 'Return' key has the same effect as the 'One Proof' button. While entering a type, Apple-C, Apple-X, Apple-V, etc. can be used. The data-entry field scrolls to allow large types to be entered. An alternative way to enter types is to open a file of strings using Apple-O (from the 'File' menu), and click 'Next' to read the next string from the file into the data-entry field. Types have the form:

24. LIACC --- Report 1996--1998 And Plan 1999--2001 -- D.1.1 NCC: Plan For 1999-2001 Here the goal is to explore the relationship between the two systems and the BCK and BCI-logic respectively, also taking into account the investigation http://193.136.27.97/liacc-info/reps/hrep98/report9698_9.html

D.1.1 NCC: Plan for 1999-2001 D.1.1 NCC: Plan for 1999-2001 The Computer Science group at LIACC has 10 PhD researchers (including 5 senior researchers), 11 PhD students (holding a MSc) and 2 MSc students, corresponding to a total effort of 16.5 men years. The main objective is the study of declarative programming based on formal systems and their practical implementation in sequential and distributed computer environments. Besides the logic programming framework, where the research group has substantial past work and experience, work is envisaged in other systems such as functional programming, constraint based programming and object oriented systems for distributed environments. In a more narrow field of application, research is also envisaged in the study of formal systems for Natural Language Proccessing. In any case the kind of work ranges from the formal study of these systems to the state of the art practical implementations in an effort to obtain declarative programming systems with strong formal properties, as well as field experiments with the development of real world applications. D.1.1.1 Declarative Programming and Parallelism

25. Atlas: On N-fold Ideals In BCK-algebras Based On T-norme And Fuzzy Point By Cele In 1991, Xi 7 applied the concept of fuzzy sets to BCI, BCK, MValgebras . in other ways, a BCI-algebra arose from algebra of non- classical logic as http://atlas-conferences.com/c/a/k/e/23.htm

Atlas home Conferences Abstracts about Atlas 65th Workshop on General Algebra, 18th Conference for Young Algebraists March 21-23, 2003 University of Potsdam Potsdam, Germany Organizers Klaus Denecke, Jörg Koppitz View Abstracts Conference Homepage On n-fold ideals in BCK-algebras based on t-norme and fuzzy point by Celestin Lele Dept of Mathematics, Faculty of science, University of Yaounde I, Box 812 Cameroon In this paper, using the concept of fuzzy point, we develop a general theory of fuzzy ideals in BCK-algebra with respect to a t-norm T which includes many known concepts and results as its special cases. Date received: January 28, 2003 Atlas Conferences Inc. Document # cake-23.

26. Logic And Computer Vol.4 Logic.ru Extensions of logic BCK commutative BCK and an implicational fragment Lw® of logic Lw. BCI and BCK-algebras. Algebraic versions of logic Lw®. http://logic.ru/en/node/48

27. Sachgebiete Der AMS-Klassifikation: 00-09 topics 03XX Mathematical logic and foundations 03-00 General reference . See also {06B30, 22A26, 54F05, 54H12} 06F35 BCK-algebras, BCI-algebras, http://www.math.fu-berlin.de/litrech/Class/ams-00-09.html

Sachgebiete der AMS-Klassifikation: 00-09 nach 90-99 Weiter nach 10-19 Suche in allen Klassifikationen 01-XX 03-XX 04-XX 05-XX 06-XX 08-XX nach 90-99 Weiter nach 10-19 Suche in allen Klassifikationen

28. Logicandlanguage.net: Tea With Lloyd And Su BCI is the implicational fragment of Linear Logic. instead of W gets us BCK (I follows from K and C.) BCK (under that name, at least) was first studied http://www.logicandlanguage.net/archives/2005/07/tea_with_lloyd_1.html

logicandlanguage.net a weblog in logic, philosophy of logic, and the philosophy of language Main July 25, 2005 Tea with Lloyd and Su I went along to the philosophy department at Monash yesterday to hang out with logicians Su Rogerson and Lloyd Humberstone. They have been teaching a rather advanced logic course to a small group of brave undergraduates, and so I got to see them in action before tea. Here's just one of the things that I learned from Su and Lloyd yesterday: Some famous Axioms The system which takes B, C and I as axioms (with uniform substitution and modus ponens as the rules) is known as BCI. BCI is the implicational fragment of Linear Logic. If we add contraction (W) we get BCIW, which is the implicational fragment of the best known relevant logic, R . Adding K (known to relevant logicians as "irrelevance"(!)) instead of W gets us BCK (I follows from K and C.) BCK (under that name, at least) was first studied by Meredith. If we add both K and W we get BCKW, the implicational fragment of intuitionistic logic. To get the implicational fragment of classical logic we need only add Peirce's Law: So what we have looks like this: All these axioms possess a confusing multitude of names, so I should mention that B is also known as "prefixing" and C gets called "permutation."

29. Mathematics Subject Classification Index 03 MATHEMATICAL LOGIC and FOUNDATIONS. 03B30 Foundations of classical 06 ORDER, LATTICES, ORDERED ALGEBRAIC STRUCTURES. 06F35 BCKalgebras, BCI-algebras http://web.math.hr/glasnik/classindex.html

Glasnik Matematicki, Subject Index Index of papers published in Glasnik Matematicki from 2000 (Volume 35) to 2006 (Volume 41). Papers are listed according to the 2000 Mathematics Subject Classification . Items with * correspond to the earlier versions of the Mathematics Subject Classification (1991 and 1980). 03 MATHEMATICAL LOGIC AND FOUNDATIONS 03B30 Foundations of classical theories V. Pambuccian 03C05 Equational classes, universal algebra G. Cupona 03D40 Word problems G. Cupona 03G25 Other algebras related to logic W. A. Dudek 03E72 Fuzzy set theory S. A. Solovyov 04 SET THEORY* 04A10 Ordinal and cardinal numbers; generalizations* A. Mani 04A72 Fuzzy sets, fuzzy relations* S. A. Solovyov 05 COMBINATORICS 05B05 Block designs D. Held D. Crnkovic M. Essert S. Rukavina ... D. Crnkovic 05B07 Triple systems S. Markovski 05B30 Other designs, configurations V. Krcadinac 05C20 Directed graphs (digraphs), tournaments V. Neumann-Lara 05C70 Factorization, matching, covering and packing D. Vukicevic 05E20 Group actions on designs, geometries and codes D. Held

30. Publications Primitive recursive reducibility, in Selected Papers on BCK, BCIalgebras and Computer Logic Shen Baiying ed., pp173-178, Shanghai Jiaotong University Press http://www-sst.informatik.tu-cottbus.de/~wwwti/zheng/publication.html

PUBLICATIONS Book X. Zheng, Effective Hierarchy of Real Numbers , Shaker Verlag, Aachen, 2007 (ISBN 978-3-8322-6343-0). Journal Papers X. Zheng , Classification of Computably Approximable Real Numbers. Theor. Comput. Syst. (to appear) Q. Chen, K. Su and X. Zheng . Primitive recursive real numbers. Mathematical Logic Quarterly . (to appear) X. Zheng . On the hierarchy of Delta_2 real numbers, Theoretical Informatics and Application , (in press) R. Rettinger and X. Zheng. A hierarchy of Turing degrees for divergence bounded computable real numbers, J. Complexity . Vol. 22 (2006), No. 6, 818-826.. X. Zheng, D. Lu and K. Bao. Divergence bounded computable real numbers, Theor. Comput. Sci., Vol. 351(2006), 27-38. K. Su, Q. Chen, A. Sattar, W. Yue, G. Lv and X. Zheng. Verification of Authentication Protocols for Epistemic Goals via SAT Compilation, Journal of Computer Science and Technology , 21(2006), No. 6, 932 943, Springer-Verlag. X. Zheng, R. Rettinger and R. Gengler. Closure properties of real numbers classes under CBV functions Theory of Comnputing Systems X. Zheng and R. Rettinger.

31. People Whose Names Are Embedded In Math Subject Classifcation The rules are modus ponens and substitution, and the axioms are for BCK B (q r) - ((p BCI has, in place of K, I p - p. A N Prior, Formal Logic, http://www.math.niu.edu/~rusin/known-math/98/MSC.names

Date: Sun, 6 Jun 1999 12:40:49 +0200 From: "Yuri I. Manin" Date: Thu, 28 Mar 2002 12:43:10 -0800 From: "Antreas P. Hatzipolakis" Cc: rusin@math.niu.edu, a_arakelov@yahoo.co.uk To: hyacinthos@yahoogroups.com Subject: Suren Arekelov This list is devoted to Triangle Geometry. However, from time to time, will be allowed discussions on themes of general interest. Especially when the geometric traffic is not too much. (like this day). Some time ago I asked for information (in fact I FWD-ed an e-mail of D. Rusin) about a notable mathematician who disappeared from the math. horizon. The mathematician is the algebraic geometer Suren Arakelov. Andrei Arakelov has kindly sent the following: > > Date: Thu, 28 Mar 2002 11:29:55 +0000 (GMT) > From: Andrei Arakelov > Subject: Suren Arakelov > To: Antreas P. Hatzipolakis Date: Sat, 05 Feb 2005 11:35:53 +0000 Message-ID:

32. Research A temporal logic approach to program verification. In Selected Papers on BCK, BCIAlgebras and Computer Logic, Shen Baiying (Ed.), page 218, http://www.cs.fiu.edu/~yhq/research.htm

Research Statement Here is a summary of my research statement Research Projects (as PI) Modeling and Enforcing Time and Context-Aware Access Control Policies (Grant No. 6 National Science Foundation of China RMB 230,000, January - December 200 Automatic Verification Techniques for Embedded Systems (Grant No. 69703008). National Science Foundation of China RMB 100,000, January 1998 - December 2000 Efficient Verification Algorithms and Tools for Distributed Real-time Systems (Grant No. 98Q16) . Collegiate Science Foundation of Shanghai for Young Scientists. RMB 30,000, January 1999 -December 2000 Verification Methodology for Hybrid Systems (Grant No. A9902). Foundations of the State Key Laboratory for Novel Software Technology. RMB 20,000, January 1999 -December 2000 Hardware /Software Codesign of Embedded Systems. Foundations of the Ministry of Education of China for Backbone Members of High Education. RMB 120,000, January 2000 - December 2001 Publications DBLP List of Publications Papers Huiqun Yu, Dongmei Liu, Li Yang, Xudong He.

33. Pure Maths Seminars - School Of Mathematics & Applied Statistics @ UOW The talk will briefly discuss a number of such logics (intuitionistic logic,paraconsistent logics, relevant logics, BCK, BCI, BCIW logics, modal logics, http://www.uow.edu.au/informatics/maths/research/seminar/UOW027158.html

var g_HttpRelativeWebRoot = "/content/"; var g_HttpServerAddress="cmsprd.uow.edu.au"; websiteId="UOW_Main"; var SSContributor = false; var SSForceContributor = false; var SSHideContributorUI = false; var ssUrlPrefix = "http://www.uow.edu.au/"; Advanced Search UOW Home Faculty of Informatics Research Interests ... Seminars About the School Staff Contact Details Research Students Mathematics Teachers Day Prospective Students Undergraduate Degrees Honours Postgraduate Degrees High School Enrichment Program ... Information Evening Current Students Information for Students Careers Honours Students PAM Program ... Course Handbook Research Research Interests Research Groups MISG Seminars ... Publications Statistical Consulting Service Statistical Consulting Service Courses On Offer Advice about Applying for a Research Grant This navigation menu requires Javascript to activate. If you cannot enable Javascript use the breadcrumbs to navigate this website. Pure maths seminars Seminars will be held in Room 15.206 at 12:30pm unless otherwise noted. Seminars will be 45 minutes long, followed by discussion. The material should be accessible to an audience with an advanced background in Pure Mathematics. Thursday, 20 December 2007: Ben Chad, Oxford University

34. SCMJ Vol.10 Contents Translate this page 10-15 Jun Y. B. Expansions of subalgebras and ideals in $BCK/BCI$-algebras Abstract /Full. 10-14 Perkgoz C., Kato K., Katagiri H., http://www.jams.or.jp/notice/scmjol/10.html

SCMJ Online Version Vol.10 Whole Number 16 Amini M., Medghalchi A.R. Amenability of the algebras $R(S), F(S)$ of a topological semigroup $S$ Abstract Full Vasant P.M., Nagarajan R., Yaacob S. Application of modified S-curve membership function in decision making process using FLP approach Abstract Full Daniel M. Algebraic structures related to the Combination of belief functions Abstract Full Hao J., Li C.X. On ideals of an ideal in a BCI-algebra Abstract Full Yabuuchi Y., Watada J., Toyoura Y. Fuzzy AR model of stock price Abstract Full Avallone A., Basile A. On a linearity theorem for measures Abstract Full Sudo T. CCR $C^*$-algebras as inductive limits Abstract Full On generalized strong $A$-summability Abstract Full Abstract generalized quasilinearization method for coincidences with applications to elliptic boundary value problems Abstract Full Just-in-time scheduling with periodic time slots Abstract Full Fixed point theorems for multivalued contractions of Hicks type in probabilistic metric spaces Abstract Full Abundo M.

35. Home Page Of R. A. Borzooei Hyper BCKalgebras and Hyper K-algebras - BCK and BCI-algebras . on i.p.s. Hypergroup, Multiple-Valued Logic and Soft Computing Journal, to appear. http://www.usb.ac.ir/ijfs/borzooei.htm

Name Rajab Ali Borzooei Date of Birth: Citizenship: I ranian borzooei@hamoon.usb.ac.ir borzooei@yahoo.com Address: Department of Mathematics University of Sisistan and Baluchestan Zahedan, Iran Educations and Academic Employment B. Sc. In Mathematics, University of Tehran Teacher Education, Iran. M. Sc. In Mathematics, University of Kerman, Kerman, Iran. Ph. D. In Mathematics, University of Kerman, Kerman, Iran. Lecturer in Mathematics, University of Sistan and Baluchistan. - to date Assistant Professor in Mathematics University of Sistan and Baluchistan. Activities Head of Department of Mathematics University of Sistan and Baluchistan. Research Council Member of Sistan and Baluchistan University. Scientific Chair: The Third Iranian Seminar on Fuzzy Sets and Applications, 19-20 June, 2002, Sistan and Baluchistan University, Zahedan, Iran. - to date Managing Editor of: Iranian Journal of Fuzzy Systems, published by the Sistan and Baluchistan University , Zahedan, Iran.

36. Hindley. Basic Simple Type Theory. BCKlogic (no multiple discharging allowed) corresponds to closed BCK-lambda-terms. BCI-logic (no multiple or vacuous discharging allowed) corresponds to http://mathgate.info/cebrown/notes/hindley97.php

TPS A higher-order theorem proving system The Omega Group This page was created and is maintained by Chad E Brown J. Roger Hindley, Basic Simple Type Theory. Cambridge University Press, 1997. Chapter 1: The type-free lambda calculus. Chapter 2: Assigning types to terms. Chapter 3: The principal-type algorithm. Chapter 4: Type assignment with equality. ... Chapter 9: Technical details. Introduction: This book studies the system TA (for "Type Assignment") which is lambda calculus (with variables but no constants) and types built from type variables and arrows. The main results discussed are the principal-type algorithm, the converse principal-type algorithm, and the inhabitant-counting algorithm. Chapter 1: The type-free lambda calculus. Lambda terms are built from term-variables by application and lambda abstraction. For each lambda term we have notions of length, subterms, occurrences, components, body, scope, covering abstractors (of a subterm [those abstractors which dominate the subterm]), free and bound non-binding variable-occurrences. We also have substitution, alpha-conversion, bound-variable clashes, and closed terms = combinators. The common combinators (including Curry's fixed point combinator Y) are given. Beta-contraction (one step of a beta-reduction), beta-reduction and beta-conversion are defined. The

37. 堳于BCI逻辑的一种偏序代数 A Partiall Translate this page This well-known assertion, however, are problematical in that BCI-logic is not BCI- BCK- ( ) Pasting a BCI-Algebra to a BCK-Algebra( ) http://scholar.ilib.cn/A-mhxtysx200703011.html

堳于BCI逻辑的一种偏序代数 堳于BCI逻辑的一种偏序代数 A Partially Ordered Algebra Related with BCI-logic 通常,人们认为Kiyoshi Is©ki在20世纪60年代引堥的BCI-代数是组合逻辑中BCI逻辑的代数对等物.然而这种广为人知的断言却是有问题的,因为BCI逻辑堳于BCI代数是不完备的.在本文中,我们引堥一种称为MPE的偏序代数.在MPE中的每个不等式对应BCI逻辑中的一个重言式且反之亦然,从而MPE代数是与BCI逻辑完备的代数类. BCI-algebras were introduced by Kiyoshi Is©ki in 1960's and they usually are regarded as algebraic formulations of BCI-system in the combinatory logic. This well-known assertion, however, are problematical in that BCI-logic is not complete with respect to BCI-algebras. In this paper, we introduce a class of partially ordered algebras called MPE-algebras such that each inequality in MPE-algebras correspondences to a theorem of BCI-logic and vice versa. Thus this algebra can be regarded as a class of complete algebraic counterpart of BCI-logic. BCI-代数 BCI逻辑 MPE-代数 偏序BCH-代数的一种自映射 A Self-mapping of Partially Ordered BCH-algebra ... [江汉大学学报（社会科学版） Journal of Jianghan University(Social Sciences)] iLib推荐的刊物 数学年刊B辑（英文版） Chinese Annals of Mathematics,Series B

38. CAT.INIST Translate this page (iii) Whether there is a finite axiomatic system for the fuzzy BCK logic? (iv) Whether the logic obtained by adding the axioms Fm,n to the BCI logic is http://cat.inist.fr/?aModele=afficheN&cpsidt=17869577

39. A Sub L III, Preliminary Programme 15.55 16.25, Josep Maria Font, Substructural logics preserving degrees of truth 10.40 - 11.00, Tomasz Kowalski, Self-Implications in BCI http://www.iphils.uj.edu.pl/conf/log/programme.html

Jagiellonian University, Cracow, Poland 6 - 8 November 2006 preliminary programme Monday, 6 November 2006 morning session Janusz Czelakowski Additivity of the commutator and residuation Wojciech Buszkowski Horn Theories of Some Substructural Logics: Interpretation and Complexity Maciej Farulewski Finite Embeddability Property for Residuated Ordered Grupoids - lunch - afternoon session Daniele Mundici Probability in many-valued logic Josep Maria Font Substructural logics preserving degrees of truth - coffee - Felix Bou Recursively Enumerable Prime Theories in Infinite­Valued Lukasiewicz Logic are not Uniformly Decidable Kazimierz Swirydowicz Variety of Relevant Algebras - the Finite Height Problem Mai Gehrke Completing lattices and generalising Kripke semantics Tuesday, 7 November 2006 morning session James Raftery Residuated structures, concentric sums and finiteness conditions Franco Montagna Residuated lattices with conuclei Norbert Preining Linear Kripke frames for some intermediate logics Clint van Alten Implicational subreducts of n-potent residuated lattices 12.40 - Collegium Maius

40. BAA British Airports Authority BAA British Astronomical BCI Business Component Integrator BCJ Blackberry Creek Journal BCK Line BFL Buffered Fieldeffect-transistor Logic BFM Business Flow Manager BFN Bye http://www.xs4all.nl/~jtv/gtf/by_letter/B

41. Kurt Gödel Research Center For Mathematical Logic At The University Of Vienna, Translate this page Proseminar zu Mathematische Logik I (Seminar for Mathematical Logic I), . substrukturale (relevante, BCK, BCI, Lambek-Kalkül, lineare), Quanten-, http://www.logic.univie.ac.at/previouslectures.html

Lectures from previous semesters Summer 2007 Winter 2006/2007 Summer 2006 See also: Official catalogue Grundbegriffe der Mathematischen Logik, Jakob Kellner , VO 2 h Mi 11.00-13.00 (starts 2005-10-05). Alle Informationen unter http://www.logic.univie.ac.at/~kellner/teaching/WS2005_grundbegriffe.html Ort: UZA 4, Seminar Room C207, Fak.f. Mathematik, Nordbergstr. 15 Proseminar zu den Grundbegriffen der Mathematischen Logik, Heike Mildenberger , PS/1 h Di 11.00-12.00 (starts 2005-10-04). Alle Informationen unter http://www.logic.univie.ac.at/~kellner/teaching/WS2005_grundbegriffe.html Axiomatische Mengenlehre I, Heike Mildenberger , VO/3 h Mo 17.00-18.15 Alle Informationen unter http://www.logic.univie.ac.at/~heike/ankW0506.pdf Proseminar zur Axiomatischen Mengenlehre I, Jakob Kellner , PS/2 h Do 11.15-12.45 Sy D. Friedman ,VO/2 h Di, Do 14.00-15.30 Uhr (starts 2005-10-04) Seminar aus der Mathematischen Logik, Sy D. Friedman , SE/2 h Mo, Mi 14.00-15.00 (starts 2005-10-03) Curt Christian (gem. mit Claudia Neubert ), SE/2 h

42. 東京電機大学 教員惠報 Translate this page , On a logic determined by Boolean algebras with conjugate 2006/10 , On transfer principle of Fuzzy BCK/BCI-algebras 2004/05 http://ra-data.dendai.ac.jp/tduhp/KgApp?kyoinId=ymkiyggyggy

43. Teaching Faculty GCU Lahore Algebra, Ring Theory, BCKalgebra, BCI-algebra, C*-algebra, Topology Functional Analysis .. 3- Logic. 4- Philosophy of Science. Mr. Kaiser Mehmood http://www.gclahore.edu.pk/facedu.htm

TEACHING STAFF, FACULTY WISE QUALIFICATIONS AND DESIGNATION Faculty of Science BOTANY DEPARTMENT Ext. No. 227 Area of Interest Dr. Amin ul Haq Khan M.Sc. Ph.D. Prof. and Chairman Environmental Impact Assessment and Plant Conservation Dr. Ikram ul Haq M.Sc. Ph.D. Professor Pathology and Biotechnology Dr. Zaheer ud Din Khan M.Sc. Ph.D. Associate Prof. Ethnobotany, Antomy, Taxanomy Dr. Zaheer ul Hassan Naqvi M.Sc. Ph.D. LLB. Associate Prof. Genetics and Plant Breeding Mr. M. Naeem Akhtar M.Sc. Assistant Prof. Mycology Dr.. Athar Hussain Shah M.Sc. Ph.D. Assistant Prof. Plant Tissue Culture and Biotechnology Mr. Tariq Rizwan

44. Dr. Moharram Ali Khan, Centre For Interdisciplinary Research In Basic Sciences Differential Equations; Logic Set Theory; Linear AlgebraI, II. Abstract Algebra- I II; (i) On the ideal theory of BCK (BCI) –Algebra (1995). http://jmi.nic.in/cirbs/mali.cirbs.htm

Centre for Interdisciplinary Research in Basic Sciences Jamia Millia Islamia New Delhi-110025 Dr. Moharram Ali Khan Professor of Mathematics Office Address: Centre for Interdisciplinary Research in Basic Sciences (CIRBSc), Jamia Millia Islamia, Jamia Nagar, New-Delhi-25 Office Phone: + 26980164 / 26981717 Ext. 4494, 4492 Residential Address: D-8, Okhla Vihar, New Delhi-110025 Mobile: Mobile No. + 9990669773 Email: moharram_a@yahoo.com Date of Joining J.M.I.: th January, 2007 Date of Birth: st January, 1957 Field of Specialization: Primary Secondary : Field Theory: Derivations on Rings / Nearrings / Banach Algebras Teaching Experience: 23 Years Field of Research Interest/ Area of Interest Ring Theory; in particular: - Commutativity theorems for certain rings - Structure of rings Near rings; in particular: - Structure of near rings and its Applications Semi rings, in particular: - Anti-commutative semi rings and its characterizations Field theory; in particular:

45. BCK Page C. A. Meredith introduced the Cpure substitution/detachment system BCK in 1956 of the propositional calculus, Notre Dame Journal of Formal Logic, vol. http://web.ics.purdue.edu/~dulrich/BCK-page.htm

C. A. Meredith introduced the C -pure substitution/detachment system BCK in 1956 [cf. C. A. Meredith and A. N. Prior, Notes on the axiomatics of the propositional calculus Notre Dame Journal of Formal Logic , vol. IV (1963), pp. 171-187]. With substitution and detachment/ modus ponens as rules, Meredith's origjnal axioms were: B CCpqCCrpCrq p q r p r q B Syl* Weak Syl C CCpCqrCqCpr p q r q p r C Com K CpCqp p (q p K Simp Shorter bases exist. For an example that I have not found in the literature, one can replace B with B' CCpqCCqrCpr p q q r p r [also known as Syl ] and then collapse C and K into CCCpCqprr p q p r r giving us a 2-base containing in all just 20 occurrences of letters and the connective [ proofs here Meredith also found [see section 8 of the paper cited above] the first single axiom for BCK BCK-1 CCCpqrCCsCrtCqCst p q r s r t q s t NEW RESULTS I've completed a proof that no shorter single axioms for BCK exist. However, single axioms for it: BCK-2 CCpCqrCCCstqCtCpr / p q r p t q] t s r BCK-3 CpCCqCrsCCCtprCqs / p q r s t p r q s BCK- CCCpqrCqCCsCrtCst / p q r q s r t s t BCK- CpCCCqprCCsCrtCst / p q p r s r t s t BCK- CpCCCqrsCCsCptCrt / p q r s s p t r t BCK- CCCpqrCsCCrCstCqt / p q r s r s t q t BCK- CCpCqrCsCCCtsqCpr / p q r s t s q p r OPEN QUESTIONS There are thirty-three theorems of BCK of length 17 whose status I have not yet been able to determine. Please

46. DBLP: Yang Xu Yong Lin Liu, Yang Xu, Jie Meng BCIimplicative ideals of BCI-algebras. . Yang Xu Uncertainty reasoning based on lattice-valued first-order logic http://www.sigmod.org/dblp/db/indices/a-tree/x/Xu:Yang.html

Yang Xu List of publications from the DBLP Bibliography Server FAQ Coauthor Index - Ask others: ACM DL ACM Guide CiteSeer CSB ... EE Yang Xu, Shengqun Tang Youwei Xu Zukai Tang : Towards Aspect Oriented Web Service Composition with UML. ACIS-ICIS 2007 EE Lina Fang Shengqun Tang ... Xinguo Deng , Yang Xu, Youwei Xu : An User-Driven Slight Ontology Framework Based on Meta-Ontology for Change Management. AINA Workshops (1) 2007 EE Ling Li Shengqun Tang ... Youwei Xu , Yang Xu: VOEditor: a Visual Environment for Ontology Construction and Collaborative Querying of Semantic Web Resources. COMPSAC (1) 2007 EE Jianjun Pu Zhuopeng Zhang ... Jian Kang , Yang Xu, Hongji Yang : Using Aspect Orientation in Understanding Legacy COBOL Code. COMPSAC (2) 2007 EE Yang Xu, Bin Liu Gao Xia Dong Lin : Iteration-Shared Scheduling Algorithms Abolishing the Departure-Time-Compatible Graph in Switch-Memory-Switch Switches. INFOCOM 2007 EE Liping Fan , Yang Xu: A PCA-Combined Neural Network Software Sensor for SBR Processes. ISNN (2) 2007 EE Lina Fang Shengqun Tang ... Youwei Xu , Yang Xu, Xinguo Deng Weiqing Chen : SOF: A Slight Ontology Framework Based on Meta-modeling for Change Management.