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1. Noncommutative Logic - Wikipedia, The Free Encyclopedia The oldest noncommutative logic is the Lambek calculus, which gave rise to the class of logics known as categorial grammars. Since the publication of http://en.wikipedia.org/wiki/Ordered_logic

var wgNotice = ""; var wgNoticeLocal = ""; var wgNoticeLang = "en"; var wgNoticeProject = "wikipedia"; Noncommutative logic From Wikipedia, the free encyclopedia (Redirected from Ordered logic Jump to: navigation search Contents An extension of linear logic Noncommutativity in logic The Lambek calculus Cyclic linear logic ... edit An extension of linear logic Noncommutative logic is an extension of linear logic which combines the commutative connectives of linear logic with the noncommutative multiplicative connectives of the Lambek calculus. Its sequent calculus relies on the structure of order varieties (a family of cyclic orders which may be viewed as a species of structure ), and the correctness criterion for its proof nets is given in terms of partial permutations. It also has a denotational semantics in which formulas are interpreted by modules over some specific Hopf algebras edit Noncommutativity in logic By extension, the term noncommutative logic is also used by a number of authors to refer to a family of substructural logics in which the exchange rule is inadmissible . The remainder of this article is devoted to a presentation of this acceptance of the term. The oldest noncommutative logic is the Lambek calculus , which gave rise to the class of logics known as categorial grammars. Since the publication of

2. A Coq Toolkit For Lambek Calculus Interested users can find more details about this library in the INRIA research report titled Logical Toolkit for Lambek Calculus Files contents http://coq.inria.fr/contribs/Lambek-Calculus.html

A Coq Toolkit for Lambek Calculus Download (archive compatible with Coq V8.1) Authors: Houda Anoun (anoun@labri.fr) (LaBRI, Bordeaux, www.labri.fr) Pierre Castéran (casteran@labri.fr) (LaBRI, Bordeaux, www.labri.fr) Date: March-July 2003 Keywords: Computational linguistic, categorial grammar, Lambek calculus... The README file of the contribution: This page was automatically generated from this description file

3. IngentaConnect The Pentus Theorem For Lambek Calculus With Simple Nonlogical Axi The Lambek calculus introduced in Lambek 6 is a strengthening of the type reduction calculus of Ajdukiewicz 1. We study Associative Lambek Calculus L in http://www.ingentaconnect.com/content/klu/stud/2005/00000081/00000001/00002801

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"); The Pentus Theorem for Lambek Calculus with Simple Nonlogical Axioms Author: Source: Studia Logica , Volume 81, Number 1, October 2005 , pp. 43-59(17) Publisher: Springer view table of contents Key: - Free Content - New Content - Subscribed Content - Free Trial Content CM8ShowAd("Skyscraper"); Abstract: p q , where p and q Keywords: Lambek calculus context-free grammar Document Type: Research article DOI: 10.1007/s11225-005-2801-x

4. Makoto Kanazawa Johan gave me a list of about 10 open problems; the recognizing power of the Lambek calculus with additional connectives was one. http://www.illc.uva.nl/j50/contribs/kanazawa/index.html

Lambek Calculus: Recognizing Power and Complexity Makoto Kanazawa Department of Cognitive and Information Sciences Faculty of Letters Chiba University 133 Yayoi-cho Inage-ku Chiba-shi 2638522 Japan kanazawa@L.chiba-u.ac.jp http://kleene.f.chiba-u.ac.jp/ Abstract I survey known results and results that easily follow from known results about the recognizing power and complexity of various fragments of the Lambek calculus. Dvi-file PS-File PDF-File Bibtex Entry I was a second-year graduate student in Linguistics at Stanford when Johan van Benthem was appointed by the Philosophy Department for the spring quarter of 1991. I understood that John Perry initially asked him to replace Jon Barwise, who had left the Stanford Philosophy Department for Indiana some time earlier, but Johan was not interested in leaving Amsterdam at all, hence the agreement on the part-time appointment. At that time, I was rather disappointed by the program offered by the Stanford Linguistics Department, where formal methods were not appreciated as much as I had hoped, so I was very much looking forward to taking Johan's course, secretly thanking Jon Barwise for making it possible. It was an exciting period for me; Johan's book Language in Action had just appeared, and his course had much to do with formal linguistics. My advisor Stanley Peters suggested I talk with Johan about possible topics for my first Qualifying Paper, which I had decided to write in the area of mathematical linguistics. Johan gave me a list of about 10 open problems; the recognizing power of the Lambek calculus with additional connectives was one. Working on it led to my first publication. (My contribution to this collection summarizes it and other related results.) As Maarten de Rijke once put it, `my life was never the same' after I met Johan.

5. Preliminaries To The Lambek Calculus Some preliminary facts concerning completeness and decidability problems for the Lambek Calculus 14 are proved as well as some theses and derived rules of http://www.cs.ualberta.ca/~piotr/Mizar/mirror/htdocs/JFM/Vol3/prelamb.html

Journal of Formalized Mathematics Volume 3, 1991 University of Bialystok Association of Mizar Users Preliminaries to the Lambek Calculus Wojciech Zielonka Adam Mickiewicz University, Poznan Summary. Some preliminary facts concerning completeness and decidability problems for the Lambek Calculus [ ] are proved as well as some theses and derived rules of the calculus itself. MML Identifier: PRELAMB The terminology and notation used in this paper have been introduced in the following articles [ Contents (PDF format) Bibliography 1] Grzegorz Bancerek. Cardinal numbers Journal of Formalized Mathematics 2] Grzegorz Bancerek. The fundamental properties of natural numbers Journal of Formalized Mathematics 3] Grzegorz Bancerek. Introduction to trees Journal of Formalized Mathematics 4] Grzegorz Bancerek. Journal of Formalized Mathematics 5] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences Journal of Formalized Mathematics 6] Czeslaw Bylinski. Binary operations Journal of Formalized Mathematics 7] Czeslaw Bylinski. Functions and their basic properties Journal of Formalized Mathematics 8] Czeslaw Bylinski.

6. On Reduction Systems Equivalent To The Non-associative Lambek Calculus With The The article concludes a series of results on cutrule axiomatizability of the Lambek calculus. It is proved that the non-associative product-free Lambek http://logcom.oxfordjournals.org/cgi/content/abstract/exl041v1

@import "/resource/css/hw.css"; @import "/resource/css/logcom.css"; Skip Navigation Oxford Journals Contact Us My Basket ... Journal of Logic and Computation Advance Access 10.1093/logcom/exl041 Journal of Logic and Computation Advance Access published online on December 27, 2006 Journal of Logic and Computation, doi:10.1093/logcom/exl041 This Article Full Text Full Text (PDF) All Versions of this Article: most recent Alert me when this article is cited Alert me if a correction is posted Services Email this article to a friend Similar articles in this journal Alert me to new issues of the journal Add to My Personal Archive ... Request Permissions Google Scholar Articles by Zielonka, W. Search for Related Content Original papers On Reduction Systems Equivalent to the Non-associative Lambek Calculus with the Empty String Wojciech Zielonka University of Warmia and Mazury, Faculty of Mathematics and Computer Science, o nierska 14a, 10-561 Olsztyn, Poland. E-mail: Abstract The article concludes a series of results on cut-rule axiomatizability of the Lambek calculus. It is proved that the non-associative

7. JSTOR Product-Free Lambek Calculus And Context-Free Grammars In this paper we prove the Chomsky Conjecture (all languages recognized by the Lambek calculus are contextfree) for both the full Lambek calculus and its http://links.jstor.org/sici?sici=0022-4812(199706)62:2<648:PLCACG>2.0.CO;2-F

8. Lambek Calculus@Everything2.com lambada calculus, lojban, Knights of the Lambda Calculus, Forum 2000. Joachim Lambek, Timerio, Reversal of Man, symbolic logic http://everything2.com/index.pl?node_id=745938

9. Relational Semantics And A Relational Proof System For Full Lambek In this paper we give relational semantics and an accompanying relational proof theory for full Lambek calculus (a sequent calculus which we denote by FL). http://projecteuclid.org/handle/euclid.jsl/1183745525

Log in RSS Title Author(s) Abstract Subject Keyword All Fields FullText more options Home Browse Search ... next Relational Semantics and a Relational Proof System for Full Lambek Calculus Wendy MacCaull Source: J. Symbolic Logic Volume 63, Issue 2 (1998), 623-637. Abstract Keywords: Full Lambek Calculus; Relational Semantics; Proof Theory; Ternary Relations; Kripke Semantics; Tableaux; Countermodel Full-text: Remote access If you are a member of the ASL, log in to Euclid for access. Full-text is available via JSTOR, for JSTOR subscribers. Go to this article in JSTOR. Links and Identifiers Permanent link to this document: http://projecteuclid.org/euclid.jsl/1183745525 JSTOR: links.jstor.org back to Table of Contents previous next Journal of Symbolic Logic Current Issue All Issues Search this Journal Editorial Board ... Log in RSS Cornell University Library

10. A Sign-Based Extension To The Lambek Calculus For Discontinuous This paper takes as its starting point the work of Moortgat (1991) and aims to provide a linguisticallymotivated extension to the basic Lambek calculus http://jigpal.oxfordjournals.org/cgi/content/abstract/3/4/555

11. Title_page Linear Logic; Proof Nets; Categorial Grammar; Lambek linear logic; proof nets; categorial grammar; Lambek calculus; substructural logic; multiplicative linear; intuitionistic linear; linear type; linear lambda http://cs-structure.inr.ac.ru/cs-bin/theme2.py?level=1&index1=6132

12. Lambek Calculus A pure Lambek Calculus for categorial grammar is a Gentzen sequent system lacking all of the structural rules like associativity, interchange, contraction, http://philo.at/phlo/199704/msg00152.html

Date Prev Date Next Thread Prev Thread Next ... Thread Index Lambek Calculus To phil-logic@bucknell.edu Subject : Lambek Calculus From g.solomon@phil.canterbury.ac.nz (Graham Solomon) Date : Mon, 14 Apr 1997 10:38:46 +0000 Follow-Ups Re: Lambek Calculus From: Prev by Date: Self-introduction Next by Date: Re: Lambek Calculus Prev by thread: Self-introduction Next by thread: Re: Lambek Calculus Index(es): Date Thread

13. Proof Nets And Roorda's Thesis This shows an embedding of Lambek Calculus into Modal Logic. My personal research interest is on Lambek Calculus and Modal Logic. http://www.cis.upenn.edu/~bcpierce/types/archives/1991/msg00099.html

[Prev] [Next] [Index] [Thread] Proof Nets and Roorda's Thesis To : types Subject : Proof Nets and Roorda's Thesis From hfc@doc.imperial.ac.uk Date : Sat, 23 Nov 91 14:22:57 EST Sender meyer@theory.lcs.mit.edu Prev: more about kingdoms; with reply by de Groote; comments by Lincoln Next: Re: Why Prolog and CBV? Index(es): Main Thread

14. Abstract Of A `Natural Logic Inference System Using The Lambek The system that we propose is based on the Lambek calculus and works directly on the CurryHoward counterparts for syntactic representations of natural http://www.cs.technion.ac.il/~winter/papers/natlog-lambek.abs.html

Abstract of "A `Natural Logic' inference system using the Lambek calculus" Anna Zamansky Nissim Francez and Yoad Winter Journal of Logic, Language and Information pdf This paper develops an inference system for natural language within the `Natural Logic' paradigm as advocated by Van Benthem, Sanchez and others. The system that we propose is based on the Lambek calculus and works directly on the Curry-Howard counterparts for syntactic representations of natural language, with no intermediate translation to logical formulae. The Lambek-based system we propose extends the system by Fyodorov et al., which is based on the Ajdukiewicz/Bar-Hillel (AB) calculus. This enables the system to deal with new kinds of inferences, involving relative clauses, non-constituent coordination, and meaning postulates that involve complex expressions. Basing the system on the Lambek calculus leads to problems with non-normalized proof terms, which are treated using normalization axioms. Back to Yoad Winter's homepage

15. Studia Logica, Volume 71 355388 BibTeX Wendy MacCaull, Ewa Orlowska Correspondence Results for Relational Proof Systems with Application to the Lambek Calculus. http://www.informatik.uni-trier.de/~ley/db/journals/sLogica/sLogica71.html

Studia Logica , Volume 71 Volume 71, Number 1, June 2002 M. Angela Weiss Rohit Parikh : Completeness of Certain Bimodal Logics for Subset Spaces. 1-30 BibTeX Wojciech Zielonka : On Reduction Systems Equivalent to The Lambek Calculus with the Empty String. 31-46 BibTeX Bob Coecke : Disjunctive Quantum Logic in Dynamic Perspective. 47-56 BibTeX Mamoru Kaneko Takashi Nagashima Nobu-Yuki Suzuki ... Yoshihito Tanaka : A Map of Common Knowledge Logics. 57-86 BibTeX : First Order Extensions of Classical Systems of Modal Logic; The role of the Barcan schemas. 87-118 BibTeX Susan Rogerson Sam Butchart BibTeX Volume 71, Number 2, July 2002 BibTeX : Monadic Fuzzy Predicate Logics. 165-175 BibTeX William C. Purdy : Complexity and Nicety of Fluted Logic. 177-198 BibTeX Francesc Esteva Joan Gispert Lluis Godo ... Franco Montagna : On the Standard and Rational Completeness of some Axiomatic Extensions of the Monoidal T-norm Logic. 199-226 BibTeX Franco Montagna Hiroakira Ono : Kripke Semantics, Undecidability and Standard Completeness for Esteva and Godo's Logic MTL[forall]. 227-245 BibTeX Anthony Hunter BibTeX Volume 71, Number 3, August 2002

16. Small Workshop On DEEP INFERENCE Grishin interactions for the symmetric Lambek calculus; 12.30 14 Lunch; 14 - 14.45 Richard McKinley (Bern University, Switzerland) http://www.pps.jussieu.fr/~eleph/swodi.html

Small Workshop on DEEP INFERENCE Paris, December 1 and 2, 2006. Small workshop of the European TYPES IST Action and of the Bath - Paris Alliance action on Deep Inference. This workshop is intended to gather people interested in structural proof theory and especially in Deep Inference. Deep inference systems are deduction systems with rules applying directly to connectives arbitrary deep inside formulas, unlike sequent calculus or natural deduction where rules apply only to the main connectives. The workshop is open to everyone interested. All talks about questions of structural proof theory are welcome, including works in progress and discussions of open problems. If you intend to give a talk or simply participate, please let us know before wednesday 29 november by sending an email to parigot@pps.jussieu.fr and manoury@pps.jussieu.fr See below the programme and the location of the workshop. Planned talks (preliminary list): Kai Bruennler (Bern University, Switzerland): What's a cut-free derivation in the calculus of structures? (abstract) Paola Bruscoli (Bath University, UK):

17. CiteULike: A Dialectica Model Of The Lambek Calculus A Dialectica Model of the Lambek Calculus lextensiva linear logic logical modal nets own petri predicates proofnets review semantics sequentcalculus http://www.citeulike.org/user/vcvpaiva/article/401382

Register Log in FAQ CiteULike News Discussion Journals Browse current issues Groups Search groups Profile Library Watchlist ... Import A Dialectica Model of the Lambek Calculus Authors de Paiva V Online Article CiteSeer (at Penn State): View article online CiteSeer (at MIT): View article online CiteSeer (at Zurich): View article online CiteSeer (at Singapore): View article online Note: You or your institution must have access rights to this article. CiteULike will not help you view an online article which you aren't authorized to view. Copy-and-Pasteable Citation Citation format: Plain APA Chicago Elsevier Harvard MLA Nature Oxford Science Turabian Vancouver vcvpaiva's tags for this article linear logic Everyone's tags for this article linear logic Abstract this paper. But it must be said from the start that the `answer' is only provided in semantical terms. The Proof Theory of the systems considered should be investigated in future work. Another warning is that the perspective of this note is basically from Category Theory as a branch of Mathematics, so words like categories and functors are always meant in their mathematical, rather than linguistical or philosophical sense. We first recall Linear Logic and provide the transformations to show... BibTeX Note: You may cite this page as: http://www.citeulike.org/user/vcvpaiva/article/401382

18. Jørgen Villadsen - Course Proposal: Semantics and Grammar Fragments Exercises 2 Simply Typed lCalculus 2.1 Simple Types Postulates Exercises 5 The Lambek Calculus 5.1 Lambek s Sequent Calculus http://www2.imm.dtu.dk/~jv/semantics/

Course Proposal: Semantics Jørgen Villadsen I am interested in giving a course on natural language processing and semantics. It should provide an introduction to modern logical and grammatical theories for natural language. A possible textbook is Bob Carpenter: Type-Logical Semantics, MIT Press 1997. Based on an introductory course on natural-language semantics, this book provides an introduction to type-logical grammar and the range of linguistic phenomena that can be handled in categorial grammar. It also contains a great deal of original work on categorial grammar and its application to natural-language semantics. The author chose the type-logical categorial grammar as his grammatical basis because of its broad syntactic coverage and its strong linkage of syntax and semantics. Although its basic orientation is linguistic, the book should also be of interest to logicians and computer scientists seeking connections between logical systems and natural language. Table of Contents Jørgen Villadsen imm.dtu.dk/~jv/semantics

19. British Library Direct: Order Details Order from the British Library From Proof Trees in Lambek Calculus to Ajdukiewicz BarHillel Elimination Binary Trees. http://direct.bl.uk/research/5F/46/RN134177185.html

This is an article from British Library Direct, a new service that allows you to search across 20,000 journals for free and order full text using your credit card. Buy this article Search British Library Direct Return to Search Engine Article details Article title From Proof Trees in Lambek Calculus to Ajdukiewicz Bar-Hillel Elimination Binary Trees Author Le Nir, Y. Journal title RESEARCH ON LANGUAGE AND COMPUTATION Bibliographic details 2003, VOL 1; NUMBER 3/4, pages 181-201 Publisher Kluwer Academic Publishers Country of publication Not Quoted ISBN ISSN Language English Pricing To buy the full text of this article you pay: service charge To see the abstract, point to the 'A' icon Abstract: In this article, we will use the formalism allowing to consider the natural deductions of Lambek calculus as proof tree structures (Tiede, 1999), so as to define transformations on these deductions. With these transformations, we obtain singular deductions close to AB calculus deductions (Ajdukewicz, 1935; Bar-Hillel, 1953) from deductions: t1, , tnS in Lambek calculus (Lambek, 1958). These deductions consist of two parts; firstly deductions in Lambek calculus of terms: titi for all i[1,n]; secondly a deduction t1, , tnS in AB calculus. After the transformation, the implicit bracketing of the proof trees is kept and this characteristic is a significant property of our transformation. Thus, we hope to extend the results from the learning of AB grammars (from positive examples) (Kanazawa, 19...

20. Logic And Linguistics Title Parsing Lambek calculus using partial composition Abstract A proof of Lambek calculus or cyclic linear logic can be seen as a planar proofnet. http://iml.univ-mrs.fr/~mrd/linguistic_page.html

Logic and Linguistics Marseille - Luminy February 13 - 17, 2006 Organizer M. Quatrini J. Vauzeilles Context : This workshop is part of Geometry of Computation 2006 ( ), a special series of events in theoretical computer science organized by the GEOCAL group and taking place at the CIRM from January 30 to March 3. Geocal06 is supported by the following institutions: CIRM CNRS Luminy Paris Nord ... Preliminary programme Thematic presentation : Linguists and logicians tried to specify grammars as logical theories. As compared to the traditional linguistics, the advantage of the logical approach is that we can prove important properties of the grammars such as soundness, completeness and consistency. J-Y. Girard has introduced linear logic (1987) as a resource sensitive logic. The basic feature of linear logic that is essential for linguistic applications, is the absence of contraction and weakening rules. But the unlimited commutativity rule is not acceptable from the linguistic point of view either. If we do not allow the commutativity rule at all, we strongly restrict the expressive power to generate natural languages. One of the promising approaches among various extensions and generalizations of Lambek grammars is to use proof nets technics inspired by linear logic.

21. Proof Nets For The Multimodal Lambek Calculus - Puite, Moot (ResearchIndex) Ion of Gamma Let us call a sequence Sigma of formulas aC sequence if, for some Gamma such that hh Gammaii Sigma, the sequent Gamma C is derivable. http://citeseer.comp.nus.edu.sg/195896.html

Proof Nets for the Multimodal Lambek Calculus (1999) (Make Corrections) (4 citations) Quintijn Puite, Richard Moot @ NUS Home/Search Context Related View or download: math.uu.nl/publications/pr 1096.ps.gz Cached: PS.gz PS PDF Image ... Help From: math.uu.nl/publications/Prepri (more) (Enter author homepages) Rate this article: (best) Comment on this article (Enter summary) Abstract: (Update) Context of citations to this paper: More the use of the modal enrichments is provided by the treatment of English wh extraction to form a relative clause, taken from Puite and Moot Take the following lexicon, in which the wh word has unary modal operators in its assigned type. 1) I G (agent) n I G (Mulder)... Cited by: More Learnability of Type-Logical Grammars - Fulop (2001) (Correct) A Graph-Theoretic Approach to Sequent Derivability in the Lambek.. - Penn (2001) ... (Correct) Similar documents (at the sentence level): Proof Nets for the Multimodal Lambek Calculus - Moot, Puite (2001) (Correct) Active bibliography (related documents): More All Non-commutative logic I: the multiplicative fragment - Abrusci, Ruet (1998)

22. Re Lambek Calculus To phillogic@bucknell.edu; Subject Re Lambek Calculus Prev by Date Lambek Calculus; Next by Date Re Hierarchical Multimodal Logics http://hhobel.phl.univie.ac.at/phlo/199704/msg00171.html

23. Lambek-Calculus With General Elimination Rules And Continuation Semantics LambekCalculus with General Elimination Rules and Continuation Semantics. Nissim Francez. Journal Title Logical Aspects of Computational Linguistics http://wotan.liu.edu/docis/dbl/laclla/2005__101_LWGERA.htm

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 Logical Aspects of Computational Linguistics Lambek-Calculus with General Elimination Rules and Continuation Semantics Nissim Francez Journal Title: Logical Aspects of Computational Linguistics Date: 2005 For further information about this item go to: http://dx.doi.org/10.1007/11422532_7 This data comes from DBLP This page is maintained by Angela Cornwell and Thomas Krichel It was last updated on 2006-04-12

24. Re: Introduction To Lambek-Calculus Oops, sorry for my ignorance of Lambek calculus. Prev by Date Re Introduction to LambekCalculus; Next by Date Re Introduction to Lambek-Calculus http://sci.tech-archive.net/Archive/sci.logic/2005-08/msg00227.html

Re: Introduction to Lambek-Calculus From : "r.e.s." < Date : Mon, 08 Aug 2005 20:05:28 GMT Oops, sorry for my ignorance of Lambek calculus. Obviously I thought you meant "lambda calculus". -r.e.s. . Follow-Ups Re: Introduction to Lambek-Calculus From: emjotde References Re: Introduction to Lambek-Calculus From: r.e.s. Prev by Date: Re: Introduction to Lambek-Calculus Next by Date: Re: Introduction to Lambek-Calculus Previous by thread: Re: Introduction to Lambek-Calculus Next by thread: Re: Introduction to Lambek-Calculus Index(es): Date Thread We are proud to have Web Hosting and Rack Housing from 9 Net Avenue Deutschland. Windows Science Usenet Archive ... sci.logic

25. How Forum 2000 Works Lambek Calculus is a structurally free logic which gives a good formal model for natural language understanding and generation. In a Lambek system, http://www.cs.cmu.edu/afs/cs.cmu.edu/user/andrej/www/quadratic.html

How Forum 2000 Works Due to increasing interest in quadratic search algorithms (QSA), I was asked to give a short overview. QSA is an algorithm used by Forum 2000 , developed at Carnegie Mellon University 1. The theory behind the Quadratic Search Algorithm QSA is an algorithm that combines ideas from computational linguistics and natural language processing , connectionist methods and category theory. It consists of two parts: a) the inference phase, which derives the meaning of a sentence, b) the executive agent, which composes a reply to the sentence. Each phase is described briefly in the following sections. 1.1. Connectionist Approach to Derivations in Lambek Calculi Lambek Calculus is a structurally free logic which gives a good formal model for natural language understanding and generation. In a Lambek system, a sentence corresponds to a proposition, and its meaning corresponds to a proof of the proposition. Different proofs of the same proposition give alternate meanings of ambiguous sentences. The propositions as types paradigm (for a nice introduction, see

26. That Logic Blog: Linguistic Tricks Forever more, the system presented in that paper has been known as Lambek Calculus. If you are familiar with Sequent calculus, Lambek Calculus fits in like http://thatlogicblog.blogspot.com/2005/05/linguistic-tricks.html

@import url("http://www.blogger.com/css/blog_controls.css"); @import url("http://www.blogger.com/dyn-css/authorization.css?targetBlogID=9821078"); That Logic Blog May 11, 2005 Linguistic Tricks So, I have been doing a little linguistics. A shocking revelation, I know, but I may as well jabber on about it a bit. Way back in the days before colour television, Jim Lambek turned the linguistic notion of categorial grammar into logic. We've all seen something of the sort in school - nouns, adjectives, subjects and silly old intransitive verbs - to name but a few of the categories you may very well have been subjected to. The question then is how members of these categories can be glued together to form a valid sentence. Or, to turn the problem on its head, how we can get a computer to decide whether or not a sentence is grammatical. Lambek addresses these in his paper: The Mathematics of Sentence Structure , J. Lambek Linguistically, however, getting rid of all of the structural rules is a bit on the dodgy side. For instance, it has been shown that the Lambek Calculus only really generates context free grammars. Most unsatisfactory! What we can do, however, is to let the structural rules back in a controlled and regimented way (much like linear logic...). So, for instance, we may have a unary operator that says that everything in its scope has access to weakening. Here is an example where this all makes sense: (1) Take the book to John Let us now postulate that the noun phrase "John" has access to weakening. In which case, we derive that the following is grammatically valid:

27. How Forum 2010 Works Connectionist Approach to Derivations in Lambek Calculi. Lambek Calculus is a structurally free logic which gives a good formal model for natural language http://forum2010.org/f2010overview.html

How Forum 2010 Works Due to increasing interest in quadratic search algorithms (QSA), I felt it was necessary to provide a short overview, with special thanks to Andrej Bauer for orignally writing this document. QSA is an algorithm used by Forum 2010 , developed at Carnegie Mellon University 1. The theory behind the Quadratic Search Algorithm QSA is an algorithm that combines ideas from computational linguistics and natural language processing , connectionist methods and category theory. It consists of two parts: a) the inference phase, which derives the meaning of a sentence, b) the executive agent, which composes a reply to the sentence. Each phase is described briefly in the following sections. 1.1. Connectionist Approach to Derivations in Lambek Calculi Lambek Calculus is a structurally free logic which gives a good formal model for natural language understanding and generation. In a Lambek system, a sentence corresponds to a proposition, and its meaning corresponds to a proof of the proposition. Different proofs of the same proposition give alternate meanings of ambiguous sentences. The propositions as types paradigm yields a correspondence between a propositional system and a lambda calculus variants of this are also known as the "Curry-Howard Isomorphism" . Lambek calculus corresponds to a typed lambda calculus with subtypes. The more specific the type, the better the meaning of the sentence is preserved. For example, consider the sentence

28. LISTSERV 14.4 and Construction of Arabic Nissim Francez `LambekCalculus with General Hermann Haeusler `On the Selective Lambek Calculus Areski Nait Abdallah, http://listserv.linguistlist.org/cgi-bin/wa?A2=ind0504&L=ln&P=9155

29. Substructural Logics (Stanford Encyclopedia Of Philosophy) R differs from linear logic and the Lambek calculus in having a For logics like as the Lambek calculus, the interpretation of the semantics is http://plato.stanford.edu/entries/logic-substructural/

Cite this entry Search the SEP Advanced Search Tools ... Please Read How You Can Help Keep the Encyclopedia Free Substructural Logics First published Tue Jul 4, 2000; substantive revision Fri May 17, 2002 Substructural logics are non-classical logics weaker than classical logic, notable for the absence of structural rules present in classical logic. These logics are motivated by considerations from philosophy (relevant logics), linguistics (the Lambek calculus) and computing (linear logic). In addition, techniques from substructural logics are useful in the study of traditional logics such as classical and intuitionistic logic. This article provides a brief overview of the field of substructural logic. For a more detailed introduction, complete with theorems, proofs and examples, the reader can consult the books and articles in the Bibliography. Residuation Logics in the Family Proof Systems Semantics ... Related Entries Residuation Logic is about logical consequence . As a result, the conditional is a central notion in logic because of its intimate connection with logical consequence. This connection is neatly expressed in residuation condition p q r if and only if p q r It says that r follows from p together with q just when q r follows from p alone. The validity of the transition from

30. Formal Grammar Buszkowski (2005) showed that all Nonassociative Lambek Calculus with finitely many nonlogical axioms are decidable in polynomial time and generate http://cs.haifa.ac.il/~shuly/fg06/program.html

FG-2006 FG-2006: The 11th conference on Formal Grammar Collocated with the European Summer School in Logic, Language and Information Malaga, Spain, July 29-30, 2006 Background FG is a series of conferences on Formal Grammar, held in conjunction with the European Summer School in Logic, Language and Information , which takes place yearly in Europe. FG provides a forum for the presentation of new and original research on formal grammar, with particular regard to the application of formal methods to natural language analysis. Themes of interest include, but are not limited to, formal and computational phonology, morphology, syntax, semantics and pragmatics; model-theoretic and proof-theoretic methods in linguistics; constraint-based and resource-sensitive approaches to grammar; learnability of formal grammar; integration of stochastic and symbolic models of grammar; foundational, methodological and architectural issues in grammar. Previous conferences in this series have welcomed papers from a wide variety of frameworks. Program Saturday, July 29th

31. Conference (November, 2004) The talk by Mati Pentus (Moscow State University) Algorithmic complexity of the Lambek syntactic calculus gave an overview of modern state of art in http://www.mccme.ru/lifr/cfs/cfnov04.htm

Moscow Russian-French conference "Foundations of Internet" Moscow, November 15-19, 2004 The conference program included eleven one-hour talks and four free discussions. The talk by Senya Shlosman (CPT, Marseille) "Phase transitions in Internet models and eternal transient dynamical systems" gave a model of Internet as a dynamical system using methods of statistical physics. The talk by Alexander Rybko (IITP, Moscow) "Fluid Limit for Queueing Networks" described his recent joint work with Senya Shlosman on limiting behavior of the large queueing systems in some simple ("mean-field") cases. It turns out that the Poisson Hypothesis holds in this case, and the corresponding dynamical systems, defined by the non-linear Markov processes, have a line of fixed points which are global attractors. The talk by Matthieu Latapy (LIAFA, Paris 7) "Complex Networks: State of the Art and Perspectives" described his recent joint work with Jean-Loup Guillaume on a new approach to representation of Internet topology using exploration process. The talk by Kave Salamatian (LIP 6, Paris) "Internet measurements: state of the art and challenges" gave an overview of user and application traffic characteristics of Internet traffic, including different aspects of user, application and backbone traffic behaviour. The talk also discussed different models for Internet traffic and quality of Service measures.

32. Studia Logica An International Journal For Symbolic Logic, 2002; 71 (3) Classical Conservative Extensions of Lambek Calculus / Abrusci, V Michele, 277314 Classical Non-Associative Lambek Calculus / de Groote, http://www.ucm.es/BUCM/compludoc/W/10210/00393215_1.htm

Sumario Título / Autor(es) Página(s) Editorial Introduction / Buszkowski, Wojciech Moortgat, Michael Classical Conservative Extensions of Lambek Calculus / Abrusci, V Michele A Tale of Four Grammars / Casadio, Claudia Lambek, Joachim Bicartesian Coherence / Dosen, Kosta Petric, Zoran Classical Non-Associative Lambek Calculus / de Groote, Philippe Lamarche, François Correspondence Results for Relational Proof Systems with Application to the Lambek Calculus / Maccaull, Wendy Orllowska, Ewa Proof Nets for the Multimodal Lambek Calculus / Moot, Richard Puite, Quintijn A Rule-Extension of the Non-Associative Lambek Calculus / Wansing, Heinrich

33. Student Publications Huang, D. Stott Parker, ``A Study of Variants of the Lambek Calculus and an The Lambek Calculus was proposed as a syntactic calculus of expressions of http://www.cs.ucla.edu/~stott/spub/

Ph.D. Dissertations Prasad Ram, ``A New Understanding of Greed'', Ph.D. Dissertation, UCLA Computer Science Department, June 1993. Ram.PhDdissertation.ps.Z Anyone who has read traditional presentations of what greedy solvable problems are, comes away with a sense of confusion or disbelief or both. The pervasiveness of greedy algorithms is due to their simplicity. It is just not comprehensible that a complex theory (e.g. matroids, greedoids) underlies the class of optimization problems that can be solved using greedy algorithms. It seems impossible to reconcile one's intuitive understanding with matroids, polymatroids, greedoids, submodularity, etc. Our chief contribution is to provide a new and simple characterization of greed. This is a very important problem with several applications in all the areas of computer science. Befitting the prominence of the problem, it has been worked on by many reasearchers. Our characterization of greed is a radical departure from the other characterizations based on matroids, polymatroids, greedoids, and submodular functions, and subsumes all these existing formalisms. In this dissertation we have proposed a novel solution to an important problem. Greedy algorithms to optimization problems are typically iterative and generate a sequence as an answer. Each step of the iteration uses the prefix sequence generated up to the previous iteration to compute the next element of the sequence. We define a very useful partial order relation on sequences called

34. Abstracts Of Talks The approach of categorial grammar is firstly introduced, as well as a specific categorial system, the (productfree) Lambek calculus. http://www.cs.bham.ac.uk/research/projects/xslam/dagstuhl/abstracts.html

Abstract of Talks Gianluigi Bellin (Univ. Verona) and Carlo Dalla Pozza (Univ. Lecce) A pragmatic interpretation of Substructural logics Alessandra Carbone, University of Paris XII On the Complexity of Proofs We present a survey of results on graphs associated to formal proofs. From graph-theoretical properties we derive consequences that explain the expansion of proofs under cut-elimination, both for the propositional calculus and the predicate calculus. Linear, polynomial, exponential and super-exponential complexities arise from the topology of the graphs. Dick Crouch, Xerox Palo Alto Research Center Linear Logic for Natural Language Semantics: skeletons, modifiers and proofs with identities We start by briefly reviewing the `glue language' approach to the semantic interpretation of natural language. This uses an (implicational) fragment of linear logic to assemble the meanings of words and phrases in a syntactically analysed sentence. This raises an important, but neglected, proof search problem of relevance to computational linguistics. From a given set of premises (representing lexical items), and with given conclusion in mind (representing an analysis of the full sentence), there is typically a multitude of distinct normal form derivations, each assigning a distinct proof/meaning term to the sentence. Many applications in computational linguistics require exhaustive proof search, where all alternative derivations are found, and as efficiently as possible.

35. Baztech Informacja O Publikacji Streszczenie angielskie We introduce the notion of derived tree corresponding with a derivation in the nonassociative Lambek calculus with product, http://baztech.icm.edu.pl/baztech/cgi-bin/btgetdoc.cgi?BUS2-0004-0118

36. INRIA :: [inria-00186911, Version 1] Proof Nets For Display Logic It concludes with an exploration of the generative capacity of the LambekGrishin calculus, presenting an embedding of lexicalized tree adjoining grammars http://hal.inria.fr/inria-00186911/en/

Home Submit Login Register ... Detailed view Domain(s): Computer Science/Computation and Language Title: Proof nets for display logic Author: Richard Moot N° : Organization Laboratory(ies) Research team CNRS INRIA Futurs LaBRI SIGNES Abstract: This paper explores several extensions of proof nets for the Lambek calculus in order to handle the different connectives of display logic in a natural way. The new proof net calculus handles some recent additions to the Lambek vocabulary such as Galois connections and Grishin interactions. It concludes with an exploration of the generative capacity of the Lambek-Grishin calculus, presenting an embedding of lexicalized tree adjoining grammars into the Lambek-Grishin calculus. ACM Classification: F.: Theory of Computation/F.4: MATHEMATICAL LOGIC AND FORMAL LANGUAGES/F.4.1: Mathematical Logic/F.4.1.7: Proof theory F.: Theory of Computation/F.4: MATHEMATICAL LOGIC AND FORMAL LANGUAGES/F.4.1: Mathematical Logic/F.4.1.4: Mechanical theorem proving Full text language: English Year: Description: Draft paper of a proposal to use proof nets for display logic by generalising techniques used for the multimodal Lambek calculus Keywords: url: http://hal.inria.fr/inria-00186911/en/

37. EULER Record Details Undecidable varieties of semilatticeordered semigroups, of Boolean algebras with operators, and logics extending Lambek calculus. http://www.emis.de/projects/EULER/detail?ide=1993kuruundevarisemi&matchno=117&ma

38. Glyn Morrill Morrill, Fadda and Valentín 2007 ` Nondeterministic Discontinuous Lambek Calculus , in Proceedings of the Seventh International Workshop on Computational http://www.lsi.upc.es/~morrill/

Glyn Morrill Jordi Girona Salgado, 13 08028 Barcelona. Tel.: +34 93 413 7843 Fax: +34 93 413 7786 e-mail: morrill_DELETE_THIS_@lsi.upc.edu http: http://www.lsi.upc.edu/~morrill/ Photo circa 1992. Her web page is http://seneca.uab.es/ggt/Gavarro/gavarro.html. Photo of our children Lena and Joan. Me with Lena and Joan. I'm interested in language, logic and computation; particularly the roles of logic in grammar and language processing. Photo: me, Michael Moortgat and Raffaella Bernardi, Ottawa 2003. (For a putative categorial syntactic structure to be correct, a proof net, it has to contain a Girard ``long trip''.) Nondeterministic Discontinuous Lambek Calculus ', in Proceedings of the Seventh International Workshop on Computational Semantics , IWCS7, Tilburg. Morrill 2006: `Categorial Grammars: Deductive Approaches', in Keith Brown (ed.) Encyclopedia of Language and Linguistics , 2nd Edition, Elsevier, Oxford, Volume Two, 242248. Carpenter and Morrill 2005: ` Switch Graphs for Parsing Type Logical Grammars ', in Proceedings of the International Workshop on Parsing Technology , IWPT05, Vancouver.

39. ICCL Workshop - Proof Theory 2004 The aim of this paper is to investigate the effect of passing to GErules in the (associative) Lambek-calculus L, the heart of Type-Logical Grammar http://www.computational-logic.org/iccl/events/WPT-2004/

ICCL Workshop Proof Theory 2004 September 27-28, 2004 This meeting is dedicated to structural proof theory and related matters. The meeting will mostly take place on September 27 and 28. Participants (Bern) Paola Bruscoli (TU Dresden) Ugo Dal Lago (Bologna) Jeremy Dawson (NICTA, Canberra) Roy Dyckhoff (St Andrews) Birgit Elbl (Universidade do Minho) Nissim Francez (Technion Haifa) Alessio Guglielmi (TU Dresden) Ozan Kahramanogullari (Leipzig) (LORIA, Nancy) (St Andrews and Paris VII) Michel Parigot (CNRS, Paris) Elaine Pimentel (Belo Horizonte and Torino) Charles Stewart (TU Dresden) (LORIA, Nancy) Talks I'm collecting the abstracts, here is a partial list: Deep Inference and Cut Elimination for Predicate Logic (Bern) In my thesis I gave a proof of cut elimination for the presentation of predicate logic in the calculus of structures. This proof is indirect: it translates a proof to the sequent calculus by introducing lots of new cuts, makes use of Gentzen's cut elimination result, and translates back the cut-free proof. My talk will be about a new, direct proof without any reference to the sequent calculus. On Light Logics, Uniform Encoding and Polynomial Time