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1. Linear Logic - Wikipedia, The Free Encyclopedia
In mathematical logic, Linear logic is a type of substructural logic that denies the structural rules of weakening and contraction. The interpretation is of
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Linear logic
From Wikipedia, the free encyclopedia
Jump to: navigation search In mathematical logic linear logic is a type of substructural logic that denies the structural rules of weakening and contraction . The interpretation is of hypotheses as resources : every hypothesis must be consumed exactly once in a proof . This differs from usual logics such as classical or intuitionistic logic where the governing judgement is of truth , which may be freely used as many times as necessary. To give an example, from propositions A and A B one may conclude A B as follows:
  • Modus ponens (or implication elimination ) on the assumptions A and A B to conclude B Conjunction of the assumption A and (1) to conclude A B
  • This is often symbolically represented as a sequent A A B A B . Both lines in the above proof "consume" the fact that A is true; this " freeness " of truth is usually what is desired in formal mathematics. However, truth is often too abstract or unwieldy when applied to statements about the world. For example, suppose one has a quart of cream from which one can make a pound of butter . If the cream is used to make butter, then it cannot be concluded that one has both cream and butter. Yet, the logical schema outlined above leads one to conclude that

    2. LinearLogic - Home
    Linear logic, Connecting People and Technology. Premier Southeastern Michigan System Integrator Specializing in High Availability Solutions,

    3. Linear Logic (Stanford Encyclopedia Of Philosophy)
    Linear logic is a refinement of classical and intuitionistic logic. Instead of emphasizing truth, as in classical logic, or proof, as in intuitionistic
    Cite this entry Search the SEP Advanced Search Tools ...
    Please Read How You Can Help Keep the Encyclopedia Free
    Linear Logic
    First published Wed 6 Sep, 2006 Linear logic is a refinement of classical and intuitionistic logic. Instead of emphasizing truth , as in classical logic, or proof , as in intuitionistic logic, linear logic emphasizes the role of formulas as resources . To achieve this focus, linear logic does not allow the usual structural rules of contraction and weakening to apply to all formulas but only those formulas marked with certain modals. Linear logic contains a fully involutive negation while maintaining a strong constructive interpretation. Linear logic also provides new insights into the nature of proofs in both classical and intuitionistic logic. Given its focus on resources, linear logic has found many applications in Computer Science.

    4. ScanGaugeII - Trip Computers + Digital Gauges + Scan Tool
    ScanGaugeII By Linear-logic Linear-logic Presents The All New ScanGaugeII With XGauge. New Add-A-Gauge and XGauge Add up to 25 additional gauges to
    ScanGauge 2 Features Photos Support Center ... HOME The Ultra Compact
    3-in-one Automotive Computer
    Trip Computer - Digital Gauges - Scan Tool
    The integrated Trip Computer provides realtime feedback while simultaneously tracking three sets of trip data. The Digital Gauges give you realtime data for your vehicle and the built-in Scan Tool allows you to read trouble codes and diagnose potentially expensive problems before they get out of hand. All this in a small and simple to install unit that can be easily moved from vehicle to vehicle. ScanGaugeII Now Includes
    Click Here to Learn More
    Linear-Logic Presents
    The All New ScanGaugeII With XGauge. New Add-A-Gauge and XGauge
    Add up to 25 additional gauges to your ScanGaugeII Compact size
    Fits almost anywhere! Works On 1996 or Newer Vehicles
    Including: Gas, Diesel, Propane and
    Hybrid Vehicles Detachable Cable
    Allows for easy portability from vehicle to vehicle. Menu Driven Operations No need to memorize complex sequences Works On All OBDII Protocols Including CAN, required on all vehicles

    5. The Linear Logic Pages
    An overview of the current knowledge in Linear logic.
    The Linear Logic Pages An overview of the current knowledge in Linear Logic
    The Linear Logic Corpus
    Sequent calculus
    Phase semantics
    Decision problems
    Proof nets
    Geometry of interaction Denotational semantics
    Game theoretic interpretations
    Variations on Linear Logic
    Non commutative Logic Classical Logic and Unified Logic Bounded complexity systems
    Linear Logic and Computer Science
    Linear logic programming
    Linear Logic and Petri nets Linear functional programming
    Sharing and optimal reduction Interaction nets
    More information
    Open problems Forum Typography Bibliography Index
    Other sites
    Linear Logic in Computer Science: A TMR network
    SRI International Computer Science Laboratory: Linear Logic
    Please contact us if you have any question that is not answered here. We also welcome corrections, suggestions, and of course, new results.

    6. Introduction To Linear Logic
    Abstract and downloadable full text. By Torben Braüner.
    Introduction to Linear Logic
    Torben Braüner December 1996
    Classical and Intuitionistic Logic
    Classical Logic
    Intuitionistic Logic
    The -Calculus
    The Curry-Howard Isomorphism
    Linear Logic
    Classical Linear Logic
    Intuitionistic Linear Logic
    A Digression - Russell's Paradox and Linear Logic
    The Linear -Calculus
    The Curry-Howard Isomorphism
    The Girard Translation
    Concrete Models
    Classical Logic
    Intuitionistic Logic
    Classical Linear Logic
    Intuitionistic Linear Logic
    Cut-Elimination for Classical Linear Logic
    Putting the Proof Together
    Available as PostScript PDF DVI
    BRICS WWW home page

    7. Wadler: Linear Logic
    This paper introduces a new way of attaching proof terms to proof trees for classical Linear logic, which bears a close resemblance to the way that pattern
    Linear Logic
    Philip Wadler
    Down with the bureaucracy of syntax! Pattern matching for classical linear logic
    Philip Wadler. Manuscript, April 2004. This paper introduces a new way of attaching proof terms to proof trees for classical linear logic, which bears a close resemblance to the way that pattern matching is used in programming languages. It equates the same proofs that are equated by proof nets, in the formulation of proof nets introduced by Dominic Hughes and Rob van Glabbeek; and goes beyond that formulation in handling exponentials and units. It provides a symmetric treatment of all the connectives, and may provide programmers with improved insight into connectives such as "par" and "why not" that are difficult to treat in programming languages based on an intuitionistic formulation of linear logic.
    Available in: pdf
    Operational Interpretations of Linear Logic
    David N. Turner and Philip Wadler. Special issue on linear logic, Theoretical Computer Science , to appear. Two different operational interpretations of intuitionistic linear logic have been proposed in the literature. The simplest interpretation recomputes non-linear values every time they are required. It has good memory-management properties, but is often dismissed as being too inefficient. Alternatively, one can memoize the results of evaluating non-linear values. This avoids any recomputation, but has weaker memory-management properties. Using a novel combination of type-theoretic and operational techniques we give a concise formal comparison of the two interpretations. Moreover, we show that there is a subset of linear logic where the two operational interpretations coincide. In this subset, which is sufficiently expressive to encode call-by-value lambda-calculus, we can have the best of both worlds: a simple and efficient implementation, and good memory-management properties.

    8. Game Semantics Or Linear Logic?
    The page is about an alternative to Linear logic called computability logic. It is semanticsbased unlike the syntax-based Linear logic.
    Game semantics
    linear logic?*
    * This material is based upon work supported by the National Science Foundation under Grant No. 0208816. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation. So far games in logic have been mostly used to find models and semantical justifications for syntactically introduced popular systems such as intuitionistic logic or linear logic. The recently initiated line called computability logic changes the approach and views games as foundational entities in their own right. This approach is in the taste of those who believe that syntax (axiomatizations or other deductive constructions) should serve a meaningful and motivated semantics rather than vice versa. The (new) concept of games that computability logic is based on appears to be an adequate formal counterpart of our broadest intuition of interactive computational tasks - tasks performed by a machine for a user/environment. What is a task for a machine is a resource for the environment and vice versa, so computability-logic games, at the same time, formalize our intuition of

    9. Bibliography On Linear Logic - Home Page
    This bibliography is a byproduct of the seminar on Linear logic and Applications held at Carnegie Mellon University in Spring 1995.
    Bibliography on Linear Logic
    This bibliography is a by-product of the seminar on Linear Logic and Applications held at Carnegie Mellon University in Spring 1995. This seminar was organized by Frank Pfenning Iliano Cervesato and . Although it concentrates on linear logic and its applications in computer science, a few related references concerning other substructural logics (such as relevance logic or the Lambek calculus) are included. Whenever available, URL links to the referenced papers are provided. We encourage the individual authors to make their contributions available on the World Wide Web. This, like any bibliography, is incomplete. Any corrections, updates, comments, suggestion, new entries, new URLs for papers, etc. are very much appreciated. In particular, help us filling the "". Please, send feedback to the maintainers. When compiling this bibliography, we had the chance to access material from Andre Scedrov's and Harold Schellinx and Anne S. Troelstra's linear logic bibliographies, as well as from the DIKU types bibliography. We would like to acknowledge their authors. This document is also available in the following formats:
  • HTML (including hypertext links)
  • BibTeX (the source)
  • Postscript with citation labels
  • DVI with citation labels ... What's new in linear logic This bibliography is currently maintained by
  • 10. Anti-Linear Logic - 6k - Cached - Similar pages A Survey of Linear logic ProgrammingSuch rippling has been observed during the past eight years since the first introduction of Linear logic Girard 1987. This exciting advance in logic
    Your browser does not support inline frames or is currently configured not to display inline frames. For contact information see Resume.

    11. Roberto Di Cosmo: Linear Logic Course Notes
    Introductory course by Vincent Danos and Roberto Di Cosmo.
    The Linear Logic Primer
    This is an ever evolving support, elaborated together with one of the very experts of Linear Logic, Vincent Danos , for an introductory course on Linear Logic that has been taught on several occasions in different continents over the past years.
    The original Pisa Version (1992)
    This is a short rugged presentation of the material, but already has a fan club, so we still keep it here ( dvi
    The current major revision
    This is the support being used since 1996/1997 in a course in the Currently available chapters:
    Selected exam texts
    General surveys
    Some useful web pointers: (no claim to give a complete list implied)

    12. Linear Logic In Computer Science - Cambridge University Press
    Linear logic is a branch of proof theory which provides refined tools for the study of the computational aspects of proofs.
    Home Catalogue Google Book Search Search this book var addthis_pub = 'rdacosta';
    • 75 exercises 150 figures 50 worked examples Page extent: 392 pages Size: 228 x 152 mm Weight: 0.545 kg
    Library of Congress
    • Dewey number: 511.3'6 Dewey version: 22 LC Classification: QA9.54 .L56 2004 LC Subject headings:
      • Proof theory Logic, Symbolic and mathematical Computer scienceMathematics
      Library of Congress Record
      Linear Logic in Computer Science
      Series: London Mathematical Society Lecture Note Series (No. 316)
      Edited by Thomas Ehrhard
      Institut de Math©matiques de Luminy, Marseille
      Jean-Yves Girard
      Institut de Math©matiques de Luminy, Marseille
      Paul Ruet
      Institut de Math©matiques de Luminy, Marseille
      Philip Scott
      University of Ottawa View list of contributors...
        For price and ordering options, inspection copy requests, and reading lists please select:
        Europe, Middle East and Africa Americas Asia Australia and New Zealand Linear Logic is a branch of proof theory which provides refined tools for the study of the computational aspects of proofs. These tools include a duality-based categorical semantics, an intrinsic graphical representation of proofs, the introduction of well-behaved non-commutative logical connectives, and the concepts of polarity and focalisation. These various aspects are illustrated here through introductory tutorials as well as more specialised contributions, with a particular emphasis on applications to computer science: denotational semantics, lambda-calculus, logic programming and concurrency theory. The volume is rounded-off by two invited contributions on new topics rooted in recent developments of linear logic. The book derives from a summer school that was the climax of the EU Training and Mobility of Researchers project ‘Linear Logic in Computer Science’. It is an excellent introduction to some of the most active research topics in the area.

    13. [quant-ph/0312174] Quantum Computation, Categorical Semantics And Linear Logic
    Quantum Computation, Categorical Semantics and Linear logic. Authors André van Tonder Comments 21 pages, Latex2e.. Minor corrections and improvements quant-ph
    Search or Article-id Help Advanced search All papers Titles Authors Abstracts Full text Help pages
    Full-text links: Download:
    Citations p revious n ... ext
    Quantum Physics
    Title: Quantum Computation, Categorical Semantics and Linear Logic
    Authors: Miquel Dorca (Submitted on 20 Dec 2003 ( ), last revised 17 Feb 2007 (this version, v4)) Abstract: We develop a type theory and provide a denotational semantics for a simple fragment of the quantum lambda calculus, a formal language for quantum computation based on linear logic.
    In our semantics, terms inhabit certain Hilbert bundles, and computations are interpreted as the appropriate inner product preserving maps between Hilbert bundles. These bundles and maps form a symmetric monoidal closed category, as expected for a calculus based on linear logic. Comments: 21 pages, Latex2e. Expanded note added Subjects: Quantum Physics (quant-ph) ; Logic in Computer Science (cs.LO); High Energy Physics - Theory (hep-th) Cite as: arXiv:quant-ph/0312174v4
    Submission history
    view email
    Sat, 20 Dec 2003 17:47:33 GMT (16kb)

    14. A Linear Logic Prover (llprover)
    A Linear logic prover that searches a cutfree proof for the given two-sided sequent of first-order Linear logic.
    A Linear Logic Prover (llprover)
    Last modified: Wed May 9 16:39:04 2007 JST
  • Please select a system: CLL (Classical) ILZ (ILL+Negation+0) ILL (Intuitionistic) Full fragment q (Quantifiers) e (Exponentials) (Propositional)
  • Please input a sequent:
    or select an example: Not selected
    See more examples . See documents for the syntax.
  • Please select a output style: Two-sided One-sided Proof-net (not perfect) Pretty form LaTeX form Indented form Internal form
  • Please limit the number of contraction rules for each path:
  • Now, press or
    Process spending more than 120 seconds will be killed.
  • Example outputs are here and here and here and here If you are looking for a sequent calculus prover for first-order classical logic (LK), please check Sequent Prover (seqprover)
    What's New
    • Feb 26, 2000: LaTeX output is modified for SICStus Prolog 3.7.1.
    • May 29, 1998: English documents are now available.
    • May 25, 1998: Input form is re-displayed in the result page.
    • May 6, 1998: Now, you can use formulas for input (e.g. a+b->b+a ). In the previous version, you need to write it as
  • 15. Linear Logic Papers Of Andreas R. Blass
    Collection of downloadable papers on Linear logic and game semantics by A. Blass.
    Linear Logic Papers
    Andreas R. Blass
    Propositional Connectives and the Set Theory of the Continuum (CWI Quarterly (Special issue for SMC 50 jubilee) 9 (1996) 25-30) PostScript or PDF This talk is a survey of two topics of recent interest in mathematical logic, namely linear logic and cardinal characteristics of the continuum. I shall try to explain enough about each of them to be able to point out how they are connected. Since the underlying ideas of the two topics are quite different, I regard the existence of a connection as surprising. Is Game Semantics Necessary? (Computer Science Logic: 7th Workshop, CSL '93, Springer Lecture Notes in Computer Science 832 (ed. E. Boerger, Y. Gurevich, and K. Meinke) (1994) 66-77) PostScript or PDF We discuss the extent to which game semantics is implicit in the formalism of linear logic and in the intuitions underlying linear logic. Resource Consciousness in Classical Logic (in Games, Logic, and Constructive Sets, Proceedings of LLC9, the 9th conference on Logic, Language, and Computation, held at CSLI (ed. G. Mints and R. Muskens) (2003) 61-74) PostScript or PDF Using Herbrand's Theorem, we define simple Herbrand validity, a sort of resource consciousness that makes sense in classical predicate logic. We characterize the propositional formulas all of whose first-order instances are simply Herbrand valid. The characterization turns out to coincide with a known characterization of game semantical validity for multiplicative formulas.

    16. Summer School On Linear Logic
    The thematic sessions will cover stateof-the-art research in Linear logic. Each session has an organizer responsible for inviting speakers who will talk
    The LINEAR International Summer School (Linear Logic and Applications) August 30 to September 7, 2000 Hotel Terra Nostra, S.Miguel, Azores, Portugal
    Check the updated LOCAL INFORMATION
    To check the PRELIMINARY TIMETABLE click here
    Check also the contents of the lectures and of the thematic sessions

    (last update: 17/8/2000)
    The LINEAR TMR research network "Linear Logic in Computer Science" ( ) is proud to announce its first International Summer School on Linear Logic and Applications. The school will be held in the island of S.Miguel, Azores , amid luxurious vegetation and hot water springs. The entrance to the mythic kingdom of the Atlantis is believed to be located near Hotel Terra Nostra, some say at the bottom of its famous red and hot water swimming pool... The school is directed to everyone doing postgraduate work in Computer Science or Mathematics with an interest in the field of Formal Logic and its applications. A background in Discrete Mathematics will be helpful. Applicants and participants who wish to get an overview of the subject may consult the following documents: Jean-Yves Girard, Linear Logic, it syntax and semantics, in Advances in Linear Logic, CUP, LMSLNS 222

    17. ACM Sigplan Notices 29, 2 (Feb. 1994), 13-18.
    The Linear style of programming inspired by Linear logic has been Linear logic Girard87 Lafont88 Abramsky93 has been proposed as the basis for a
    A "Linear Logic" Quicksort
    Henry G. Baker Nimble Computer Corporation, 16231 Meadow Ridge Way, Encino, CA 91436
    (818) 986-1436 (818) 986-1360 (FAX) This material is based upon work supported by the National Science Foundation under Grant No. III-9261682.
    The linear style of programming inspired by linear logic has been proposed to reduce garbage collection and synchronization costs in serial and parallel systems. We programmed Quicksort for both lists and arrays in a "linear" fragment of Lisp to estimate the performance impact of linearity on a serial machine. Even though Quicksort is well-tuned for current non-linear architectures, we find that linearity extracts no real penalty. Our "linear" list Quicksort is as fast as any non-linear list Quicksort, and our "linear" vector Quicksort is only 3.5% slower than a non-linear vector Quicksort. The linear style is moderately pleasant, and the redundancy of linearity checking can aid in finding bugs.
    Linear logic [Girard87] [Lafont88] [Abramsky93] has been proposed as the basis for a "linear" computer language which preserves the cleanliness of functional programming, yet allows efficient "update-in-place" array operations, no tracing garbage collection and no synchronization. However, some early measurements on linear languages have been disappointing. Wakeling [Wakeling91] complains about the inefficiencies of his version of "linear" ML, especially on list and array variants of Quicksort sorting algorithm, as well as about the stilted linear programming style.

    18. That Logic Blog: Linear Logic - Naturally!
    Linear logic has enjoyed enormous popularity over the last couple of decades or so. For those without some training in structural proof theory,
    @import url(""); @import url("");
    That Logic Blog
    February 13, 2006
    Linear Logic - Naturally!
    Linear logic has enjoyed enormous popularity over the last couple of decades or so. For those without some training in structural proof theory, understanding the system can be quite intimidating, especially because of the funny notation and weird jargon. In this post, I am going to show you that, in fact, you could have invented linear logic
    fragment of linear logic, whereas the x and + connectives come from the additive fragment of linear logic. To complete the scenario, all you need to do is add in some units for the additive fragment.

    Apart from some added formalism and some theorems to say that everything works out as it should, you have now created linear logic! posted by Jon @ 12:00 PM
    Anonymous said...
    Peter, have you seen computability logic ? It claims to be the logic of resources just as linear logic does. And the two appear to disagree on many principles. I wonder which one is "right". 7:53 AM
    Ole Thomassen Hjortland said...

    19. CTO : Linear Logic Comments
    I m sure that when I did, the current Linear logic article wasn t in the Wiki, and I think it very authoritatively replaces this essay at least in the
    CTO CLiki for the TUNES project Home Recent Changes About CLiki Text Formatting ... Create New Page
    Linear Logic Comments
    An essay by Jack Waugh [I don't remember how long ago I wrote this. I'm sure that when I did, the current linear logic article wasn't in the Wiki, and I think it very authoritatively replaces this "essay" at least in the latter's original intent. I could see deleting this page, except perhaps for the value of the comments and discussion that people have added after my original text. Jack Waugh 2003-08-11] I have seen the term " linear logic " used in a way that has nothing to do with linear equations (described at Linear Programming ). In the literature I saw, Linear Logic means a discipline where an object reference can only be used once. In fact, you must use it exactly once. To destroy a reference requires an explicit mention. To "split" a reference also does. So for example, where in, say Smalltalk (an imperative OO language), you could write: foo: anArgument ^ anArgument + anArgument a linear-logic language would (assuming a similar syntax) require something like (This assumes an extention to Smalltalk syntax where you could have a call return multiple results, which you could then store in variables.)

    20. FLoC '02 - LL
    Linear logic was invented by Girard in 1986, and first appeared as a finer analysis of his denotational semantics of system F. It provides a decomposition
    LL 2002 Linear Logic
    Copenhagen, Denmark, July 25-26, 2002
    Affiliated with LICS 2002
    Linear logic, by the decomposition of intuitionistic connectives it provides, but also by the new concepts it carries (geometrization of computations, management of resources at the logical level) opens a new dimension, already intensely exploited, in applications of logic to computer science, and at the same time is the locus of a mathematical theory within which mathematics is comfortably done. It is both a theory and a tool. New possibilities show up at two levels, which we might call the language level and the operational level. Let us say that linear logic acts as a kind of a looking-glass through which phenomena of the field are better understood. Though only fifteen years old, linear logic has spread in the computer science and logic communities, so that it can now be considered as a mature topic. The success of various international initiatives, such as the linear logic workshops at Cornell in 1993, in Tokyo in 1996, in Marseille in 1998; the linear logic bibliography web page at Carnegie Mellon University in the U.S. and also the various European projects where linear logic played or currently plays an important role, such as the Esprit projects CLiCS and CONFER, the HCM Network "Typed Lambda-Calculus" and the current LINEAR TMR network are concrete signs of vitality of the subject. In 2002 four years will have passed since the most recent international workshop on linear logic held in 1998 in Marseille, and the time will be ripe for a successful meeting.

    21. TBogg - “…a Somewhat Popular Blogger” » The Linear Logic Of J
    22 Responses to “The Linear logic of J. Assrocket Squirrel Esq.” humboldtblue December 10th, 2007 at 714 pm. 1. Oh. My God. TBogg.
    var blogHome=""; document.write(''); Login Here Username: Password Remember Me
    The Linear Logic of J. Assrocket Squirrel Esq.
    By: TBogg Monday December 10, 2007 7:07 pm It must be very strange to be President Bush John Hinderaker. A man of extraordinary vision lunacy and brilliance wrongheadedness approaching to genius dementia, he can't get anyone to notice stage an intervention. He is like a great painter or musician street person shouting at a bus stop who is ahead of his time off his meds, and who unveils one masterpiece absurd leap of logic after another to a reception that, when not bored amused, is hostile appalled. The Rocketman provides a ....unique observation about the shootings in Colorado The man who shot up a training center for missionaries and a church in suburban Denver, killing four people and wounding a number of others, has been identified: A law enforcement official says the deadly rampages at a megachurch and a missionary training school were believed to have been carried out by the same person—Matthew Murray, a 24-yeare-old suburban Denver man who "hated Christians." It is perhaps worth noting that the toll in Sunday's shootings exceeded the combined total in all "hate crimes" against Muslims in the six years since September 11.

    22. The Apple Core Vs. Linear Logic: Design Education: Education: AIGA
    Linear logic. by Maggie MacnabMarch 05, 2006. Humans have often been called the symbolizing animal. We use words, number, shape, and other deduced imagery
    From AIGA Design Archives
    Flight 001 (New York, New York) , Package design, 2004 F1 Spacepak System
    From The Archives
    The Apple Core vs. Linear Logic
    by Maggie Macnab March 05, 2006 H
    I've explored and implemented what I call the philosophy of visual communication for almost 25 years, and have presented it to design students and at conferences for the last eight. In my work as an identity designer I've rediscovered the obvious: It has to land in the gut before it has any chance of making it to the head. If you want your logo to be remembered and recognized, you've got to provide a pay-off to the viewer. The more immediate story you can tell, the more likely you are to create a relationship with the viewer. And when I say story I don't mean fable. I mean the more appropriate visual content you use to represent your client's message, the much higher the likelihood you have of creating a direct line from invitation to connection. This is information value, as opposed to information junk, which is in abundance today.
    Humans love myth and symbol, as we should. There's a lot of knowledge in this synaptic bit, distilled through the experiences of millions of predecessors. A symbol is certainly a collapsed piece of information, but symbols also expand to fit any human in any culture in any time by linking to universal principles. These cues in turn inform choice: Is it good? What is in it for me? Do I trust? Taking the wrong turn into the jaws of a famished predator is as relevant a concern today in the concrete jungle as it was in the Savannah millennia ago. A composite that immediately tells the story without excessive processing is invaluable. It enhances our edge of appropriate response by giving us access to response-ability.

    23. LINEAR LOGIC, LUDICS, IMPLICIT COMPLEXITY, OPERATOR ALGEBRAS - SIENA, May 17-20 4k - Cached - Similar pages Linear logic LinksLectures on Linear logic, CSLI Lecture Notes No.29, 1992. by Anne Troelstra. LLP A Linear logic Programming Language and its Compiler System,

    24. Lectures On Linear Logic
    Linear logic is an example of a resourcesensitive logic, keeping track of the number of times data of given types are used. Formulas in Linear logic
    Lectures on Linear Logic A.S. Troelstra Linear logic is an example of a "resource-sensitive" logic, keeping track of the number of times data of given types are used. Formulas in linear logic represent either the data themselves or data types, whereas in ordinary logic a formula is a proposition. If ordinary logic is a logic of truth, linear logic is a logic of actions. Linear logic and its implications are explored in depth in this volume. Particular attention has been given to the various formalisms for linear logic, embeddings of classical and intuitionistic logic into linear logic, the connection with certain types of categories, the "formulas-as-types" paradigm for linear logic and associated computational interpretations, and Girard's proof nets for classical linear logic as an analogue of natural deduction. It is also shown that linear logic is undecidable. A final section, contributed by D. Roorda, presents a proof of strong normalization for cut elimination in linear logic. Linear logic is of interest to logicians and computer scientists, and shows links with many other topics, such as coherence theorems in category theory, the theory of Petri nets, and abstract computing machines without garbage collection. ISBN (Paperback): 0937073776 ISBN (Cloth): 0937073784 Subject: Mathematics; Logic

    25. The Linear-Logic ScanGauge II Review At Gear Diary
    When I discovered a new device called the ScanGauge II Automotive Computer made by Linearlogic, I was in awe that a little black box with a LCD could offer
    • Main About Judie Gear Diary Team Gear Diary Contributors ...
      The Linear-Logic ScanGauge II Review
      Published by Allen Hong June 10th, 2007 in Automotive Gear Allen Hong and Reviews Tags: Allen Hong Automotive Gear Reviews
      ScanGauge II package contents
      picture of my boring 2003 Honda CR-V dashboard
      the back of the ScanGauge II, with ports on side and back Initially, I could not decide where to mount the ScanGauge II, so I just placed it on the rubber lined shelf on the passenger side of the dash.
      the ScanGauge II on CR-V dashboard shelf Later on, I wanted to mount it to the center storage compartment door under the air conditioner vents. But I decided against doing so, because when I temporary placed the ScanGauge II on the opened door using it like a small shelf - I found myself playing with the buttons and changing gauge views while I was driving. Not a safe thing to do at all! So back to the shelf on the passenger side it goes!
      the center dash and shelf on the right with the ScanGauge II On first use, a bit of setup configuration is required, you just have to set units of measure, engine size, fuel type, and fuel tank size; after that the ScanGauge II to be ready for use. The unit turns on when the ignition key is turned to the ON position and it will connect to the car computer to pull information for which the user configured for display. In the sample display picture below, it is set to display RPM, MPH, MPG, and battery voltage.
      sample display, showing RPM, MPG, MPH, and Voltage

    26. Linear Logic LLC Semiconductor IP Listing -- D&R Silicon IP Catalog
    Linear logic LLC Semiconductor IP Listing D R Silicon IP Catalog.

    27. Re: Linear Logic Semantics (Barwise) (353 Lines)
    A long reply by Vaughan Pratt to a question on the Linear logic mailing list by the late Jon Barwise.
    [Prev] [Next] [Index] [Thread]
    Re: Linear logic semantics (Barwise) (353 lines)

    28. LLPN - Linear Logic Petri Nets
    Linear logic Petri Nets combine the benefits of Petri nets with the elegance of Linear logic. They have been successfully used for the specification of
    image1 = new Image(110,49) image1.src = "images/News_ovr_c.jpg" image2 = new Image(110,49) image2.src = "images/References_ovr_c.jpg" image3 = new Image(110,49) image3.src = "images/Chronology_ovr_c.jpg" image4 = new Image(110,49) image4.src = "images/Presentations_ovr_c.jpg" image5 = new Image(110,49) image5.src = "images/OPNs_ovr_c.jpg"
    LLPN - Linear Logic Petri Nets
    Linear Logic Petri Nets combine the benefits of Petri nets with the elegance of Linear Logic. They have been successfully used for the specification of dynamic Petri net structures, i.e. Petri nets that can have their underlying net structure modified at run-time.
    LLPNs can be viewed as one of the siblings of the family of Object Petri Nets (OPNs). They were introduced in 1998 as a means to study the semantics of Object Systems, the OPN formalism introduced by Valk.

    29. Linear Logic ScanGuage - Erik
    Linear logic ScanGuage Linear logic ScanGuage 2005/02/15, Viewed 165 times this month, last update 2005/02/15 Logic ScanGuage

    30. Week40
    When I first heard about Linear logic, it made utterly no sense. The point is that in Linear logic one should not think of S as a *set* of premisses,
    This Week's Finds in Mathematical Physics (Week 40)
    John Baez
    Anyway, it's now quite clear to me that I just hadn't been reading the right stuff. I think Rota has said that the really interesting work in logic now goes under the name of "computer science', but for whatever reason, I didn't dig into the Journal of Philosophical Logic, other logic journals, or proceedings of conferences on category theory, computer science and the like and find the stuff that would have excited me. It goes to show that one really needs to keep digging! Anyway, I just went to a conference called the Lambda Calculus Jumelage up in Ottawa, thanks to a kind invitation by Prakash Panangaden and Phil Scott, who thought my ideas on category theory and physics might interest (or at least amuse) the folks who attend this annual bash. It became clear to me while up there that logic is alive and well! Of course, I don't actually understand most of what these people are up to, so take what I say with a large grain of salt. My goal here is more to draw attention to some interesting-sounding ideas than to explain them. One interesting subject, which I think I'm finally beginning to get an inkling of, is "linear logic". This was introduced in the following paper (which I haven't gotten around to looking at):

    31. Multiplicative Intuitionistic Linear Logic | The N-Category Café
    I’ve been trying to understand multiplicative intuitionistic Linear logic (MILL) from a categorytheoretic perspective, and I think I’ve figured out what’s
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    A group blog on math, physics and philosophy
    Skip to the Main Content
    Enough, already! Skip to the content. Note: These pages make extensive use of the latest XHTML and CSS Standards only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser. Main
    January 18, 2007
    Multiplicative Intuitionistic Linear Logic
    Posted by John Baez
    The following guest post is by Mike Stay Richard Guy Here are the inference rules, copied from this paper by Benton, Bierman, de Paiva, and Hyland: A A Identity A B C B A C Exchange B B C C Cut A I A I L I I R A B C A B C L A B A B R A B C A B C L A B A B R A MILL M is, apparently, a poset. (Recall that a poset is a category with at most one morphism, called , between any pair of objects). The objects are propositions or resources. A B if B is deducible from A There are three functors of particular note:
    • the tensor product, denoted either by the crossed circle (

    32. The Completeness Of Linear Logic For Petri Net Models -- Ishihara And Hiraishi 9
    The completeness between Linear logic and Petri nets has been shown for several versions of Linear logic. For example, Engberg and Winskel considered the
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    Oxford University Press

    This Article Full Text (PDF) 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 Ishihara, K Articles by Hiraishi, K Search for Related Content
    The completeness of linear logic for Petri net models
    K Ishihara and K Hiraishi School of Information Science, Japan Advanced Institute of Science and Technology, 1-1 Asahidai Tatsunokuchi Ishikawa, 923-1292, Japan E-mail:; The completeness between linear logic and Petri nets has been shown for several versions of linear logic. For example, Engberg and Winskel considered the [sqcup ]-free fragment of propositional showed soundness and completeness of the logic for a Petri net model. One of difficulties in proving completeness for full

    33. SSRN-Modelling Product Innovation Processes, From Linear Logic To Circular Chaos
    SSRNModelling Product Innovation Processes, from Linear logic to Circular Chaos by Jan Buijs.

    34. Logic Programming With Focusing Proofs In Linear Logic ANDREOLI
    The deep symmetry of Linear logic 18 makes it suitable for providing abstract models of computation, free from implementation details which are,

    35. First Order Linear Logic In Symmetric Monoidal Closed Categories
    The aim of the thesis is to test the strength and flexibility of this paradigm by studying the specific case of Girard s Linear logic .
    First Order Linear Logic in Symmetric Monoidal Closed Categories
    Simon Ambler Abstract: There has recently been considerable interest in the development of `logical frameworks' which can represent many of the logics arising in computer science in a uniform way. Within the Edinburgh LF project, this concept is split into two components; the first being a general proof theoretic encoding of logics, and the second a uniform treatment of their model theory. This thesis forms a case study for the work on model theory. The models of many first and higher order logics can be represented as fibred or indexed categories with certain extra structure, and this has been suggested as a general paradigm. The aim of the thesis is to test the strength and flexibility of this paradigm by studying the specific case of Girard's linear logic . It should be noted that the exact form of this logic in the first order case is not entirely certain, and the system treated here is significantly different to that considered by Girard. To secure a good class of models, we develop a carefully restricted form of first order intuitionistic linear logic, called

    36. Asudeh :: ESSLLI06 :: Linear Logic, Linguistic Resource Sensitivity And Resumpti
    This course investigates linguistic applications of resource logics, particularly Linear logic. The resource accounting of Linear logic forms the basis for
    Linear Logic, Linguistic Resource Sensitivity and Resumption
    ESSLLI 2006
    :: Malaga, Spain :: 31.07.2006 - 04.08.2006
    Ash Asudeh
    Institute of Cognitive Science

    School of Linguistics and Applied Language Studies

    Carleton University
    ... Slides
    Course Description:
    This course investigates linguistic applications of resource logics, particularly linear logic. The resource accounting of linear logic forms the basis for a linguistic hypothesis/research program, Resource Sensitivity (RS). The linguistic utility of logical resource accounting is shown to have a tight connection to choice of logical connectives. A number of theoretical proposals in linguistics are shown to be reducible to resource-sensitive semantic composition. Resource Sensitivity is challenged by pronominal resumption, which seems to constitute a kind of resource surplus. A theory of resumption is presented that maintains RS and sheds new light on resumption as a problem of semantic composition, rather than a principally syntactic problem, as in other treatments. The theory also forms the basis for a processing model (production and parsing) of non-grammaticized resumptive pronouns. The theory is further applied to copy raising, a phenomenon whose syntax and semantics is both underinvestigated and challenging and whose intuitive similarity to resumptive pronouns has previously not been captured. Prerequisites : Students should have some background in logic or linguistic syntax and semantics

    37. JSTOR Quantales And (Noncommutative) Linear Logic
    THE JOURNAL OF SYMBOLIC logic Volume 55, Number 1, March 1990 QUANTALES AND (NONCOMMUTATIVE) Linear logic DAVID N. YETTER It is the purpose of this paper to<41:QA(LL>2.0.CO;2-A

    38. Lolli: A Linear Logic Programming Language
    Lolli is a logic programming language based on a fragment of Linear logic. As such it allows the programmer to exercise a significant degree of control over
    Lolli: A Linear Logic Programming Language
    Based on the web page by Josh Hodas, written 26 July 1995. Recent updates by Dale Miller.
    Table of Contents
    Lolli is a logic programming language based on a fragment of linear logic. As such it allows the programmer to exercise a significant degree of control over the pattern of use of certain program clauses (or resources ) during proof search. The language was designed by Josh Hodas and Dale Miller and is described in the papers listed below.
    • Logic Programming in a Fragment of Intuitionistic Linear Logic , by Joshua S. Hodas and Dale Miller, Information and Computation , Vol. 110, No. 2, May 1, 1994, pp. 327-365. (DVI Postscript) Logic Programming in Intuitionistic Linear Logic: Theory, Design, and Implementation , by Joshua S. Hodas, Ph.D. Dissertation from University of Pennsylvania, Department of Computer and Information Science, May 1994. Available as University of Pennsylvania Technical Reports MS-CIS-92-28 or LINC LAB 269 (Postscript) Lolli: An Extension of Prolog with Linear Logic Context Management , by Joshua S. Hodas

    39. Linear Logic - The Twelf Project
    The fact that the LF type theory uses only unrestricted assumptions has led some to assume that it cannot encode substructural logics such as Linear logic.
    var skin = 'monobook';var stylepath = '/w/skins';
    Linear logic
    From The Twelf Project
    Jump to: navigation search The fact that the LF type theory uses only unrestricted assumptions has led some to assume that it cannot encode substructural logics such as linear logic. In fact, LF admits a very elegant encoding of linear logic, given below. The key idea is a judgement used to enforce the linear use of each linear assumption.
    • The Encoding
      edit The Encoding
      edit Syntax
      %%% Types %%% tp : type. %name tp T -o %infix right 7 -o. %infix right 10 *. %infix %infix right 8 +. zero : tp. top : tp. %% 1 need not be primitive. one : tp = ! top. %%% Terms %%% term : type. %name term M lam app tensor lett pair case bang letb any unit : term. %% derived syntax for 1 star : term = bang unit. leto
      edit Linearity
      Linearity is enforced by employing a linearity judgement for each linear assumption. The linearity judgement ensures that a variable is used linearly (roughly speaking, exactly once) within its scope. (For an example of its use, see the rule of/lam for linear lambda abstraction below, which ensures that its argument is used linearly within its body.)

    40. The Focused Inverse Method For Linear Logic
    Linear logic presents a unified framework for describing and reasoning about stateful systems. Because of its view of hypotheses as resources,

    41. DI & CoS - Commutative/Non-commutative Linear Logic
    We conservatively extend mixed multiplicative and multiplicative exponential Linear logic with a selfdual non-commutative operator.
    This page is no longer updated, please refer to this page Alessio Guglielmi's Research Deep Inference and the Calculus of Structures / Commutative/Non-commutative Linear Logic
    Deep Inference and the Calculus of Structures
    Commutative/Non-commutative Linear Logic
    We conservatively extend mixed multiplicative and multiplicative exponential linear logic with a self-dual non-commutative operator. The systems so obtained cannot be presented in the sequent calculus, but they enjoy the usual properties of locality, decomposition and cut elimination available in the calculus of structures. We can present Yetter's cyclic linear logic in the calculus of structures and prove cut elimination; interestingly, cyclicity is naturally subsumed by deep inference. New, purely proof-theoretical, techniques are developed for reducing the non-determinism in the calculus of structures. We introduce the calculus of structures: it is more general than the sequent calculus and it allows for cut elimination and the subformula property. We show a simple extension of multiplicative linear logic, by a self-dual non-commutative operator inspired by CCS, that seems not to be expressible in the sequent calculus. Then we show that multiplicative exponential linear logic benefits from its presentation in the calculus of structures, especially because we can replace the ordinary, global promotion rule by a local version. These formal systems, for which we prove cut elimination, outline a range of techniques and properties that were not previously available. Contrarily to what happens in the sequent calculus, the cut elimination proof is modular.

    42. My Secret Spiritual Dance: On Linear And Mystic Logic
    Of course, Law of Attraction makes absolutely no sense at all when looked at from Linear logic. Do the critics really think that all who see this Law think
    Let's Play An Abundance Game!!!
    I was introduced to Abraham in 1995 and started the weekly tape program a few years later. Since that time, I have been on and off the program, mostly on. I am on it again. I have a gabillion Abraham tapes. That means lots. It feels yummy when I think about giving them away. Sofor the next whatever amount of time, I'll keep track of the direct referrals to my blog from other sites. I will send the person with the most referral clicks, three random Abraham tapes as my gift. No postage fee required. All you have to do is be your lovely self and receive. I'll post the weekly winners. No real efforting involved here....let's just play. Could be fun!! So Far: Karen Grace Lyman Phylameana ... Kara-Leah
    I offer here insights and places I have been and come to in my Spiritual Journey. I'm not right. I love respectful and honoring discussion of Paths that are both similar and different than mine. My journey continues. Thanks for stopping by. -Pamm
    Blog Archive

    43. Minimal Linear Logic
    the formulas of minimal Linear logic; the system ND of natural deduction; the system AND of annotated natural deduction in which the classification of
    Minimal Linear Logic
    This example investigates minimal linear logic (often called intuitionistic linear logic). It specifies
  • the formulas of minimal linear logic
  • the system ND of natural deduction
  • the system AND of annotated natural deduction in which the classification of introduction and elimination rules is explicit in the judgments
  • the sequent calculus LV with linear and modal zone on the left and single conclusion on the right
  • translations which are inductive and therefore demonstrate the equivalence of these systems from the point of view derivability:
  • natural deductions to annotated natural deductions ND -> AND
  • annotated natural deductions to sequent derivations AND -> LV
  • sequent derivations to annotated natural deductions LV -> AND
  • annotated natural deduction to natural deductions AND -> ND
  • some reduction rules for annotated natural deduction formulas nd and lv ... Frank Pfenning DEL("Frank Pfenning", fp)
  • 44. Efficient Linear Logic Meaning Assembly
    The glue approach to semantic composition in LexicalFunctional Grammar uses Linear logic to assemble meanings from syntactic analyses Dalrymple et al,
    Efficient Linear Logic Meaning Assembly
    Vineet Gupta, John Lamping
    The "glue" approach to semantic composition in Lexical-Functional Grammar uses linear logic to assemble meanings from syntactic analyses [Dalrymple et al, 1993]. It has been computationally feasible in practice [Dalrymple et al, 1997b]. Yet deduction in linear logic is known to be intractable. Even the propositional tensor fragment is NP complete. In this paper, we investigate what has made the glue approach computationally feasible and show how to exploit that to efficiently deduce underspecified representations. In the next section, we identify a restricted pattern of use of linear logic in the glue analyses we are aware of, including those in [Crouch and van Genabith 1997, Dalrymple et al 1996,Dalrymple et al 1995] . And we show why that fragment is computationally feasible. In other words, while the glue approach could be used to express computationally intractable analyses, actual analyses have adhered to a pattern of use of linear logic that is tractable. The rest of the paper shows how this pattern of use can be exploited to efficiently capture all possible deductions. We present a conservative extension of linear logic that allows a reformulation of the semantic contributions to better exploit this pattern, almost turning them into Horn clauses. We present a deduction algorithm for this formulation that yields a compact description of the possible deductions. And finally, we show how that description of deductions can be turned into a compact underspecified description of the possible meanings.

    45. Proof Theory For Full Intuitionistic Linear Logic, Bilinear Logic, And MIX Categ
    Proof theory for full intuitionistic Linear logic, biLinear logic, and MIX categories. J.R.B. Cockett and R.A.G. Seely. This note applies techniques we have
    Proof theory for full intuitionistic linear logic, bilinear logic, and MIX categories
    J.R.B. Cockett and R.A.G. Seely
    Keywords: monoidal closed categories, tensorial strength, coherence, categorical proof theory. 1991 MSC: 03B70, 03F07, 03G30, 18D10, 18D15, 19D23. Theory and Applications of Categories , Vol. 3, 1997, No. 5, pp 85-131.
    TAC Home

    46. A Mixed Linear And Non-Linear Logic: Proofs, Terms And Models
    Intuitionistic Linear logic regains the expressive power of intuitionistic logic through the

    47. A Parigot-style Linear Lambda-calculus For Full Intuitionistic Linear Logic
    This paper describes a natural deduction formulation for Full Intuitionistic Linear logic ($sf FILL$), an intriguing variation of multiplicative Linear

    48. Linear Logic Consulting Home Page
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    49. Texts That Run Rings Around Everyday Linear Logic - New York Times
    Perhaps the order behind the sounds is simply not being heard; perhaps the logic of the argument is not being understood. Paying attention to anything alien
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    Texts That Run Rings Around Everyday Linear Logic

    By EDWARD ROTHSTEIN Published: March 26, 2007 The feeling is familiar. You are listening to a piece of music, and nothing links one moment with the next. Sounds seem to emerge without purpose from some unmapped realm, neither connecting to what came before nor anticipating anything after. The same thing can happen while reading. Passages accumulate like tedious entries in an exercise book. Chaos, disorder, clumsiness, disarray: these must be the marks of poor construction or, perhaps, of deliberate provocation. Skip to next paragraph Jonathan Player for The New York Times The anthropologist and author Mary Douglas at her home in London. In a strange way, though, the very same sensations might also be marks of our own perceptual failures. Perhaps the order behind the sounds is simply not being heard; perhaps the logic of the argument is not being understood. Paying attention to anything alien can be like listening to a foreign language. There may be logic latent in the sounds, but it is not evident to untrained ears.

    50. Linear Logic As A Good Logical Foundation For Computer Science.
    A brief analysis of the computational aspects of Linear logic by Max Kanovich.
    Linear Logic as a good logical foundation for computer
    A specification language for a computational system is supposed to be capable of handling both resource-sensitive and time-dependent properties of computational processes. Covering both of the two aspects is necessitated by the fact that computational processes (in particular, concurrent processes) consume their resources and develop in time, so that the state of the resources consumed by the processes and the order of their actions are time-dependent. In consequence, such formal systems should be based on the concept of dynamical state and provide updating, inheritance,and parallelism. Traditional logics like intuitionistic logic (whose proofs are represented with the means of the lambda calculus), have not been very successful at the task of modelling states and changes of states; researchers have had to invent formal systems, like Milner's CCS and pi-calculus, for which it is hard to establish a connection with logic in the traditional sense. Linear logic has been introduced by Girard as resource-sensitive refinement of classical logic. Research in linear logic has been the thrust behind the development of a closer relationship between logic and computer science, as evid enced for example by natural characterizations of major complexity classes in terms of natural fragments of linear logic (Lincoln, Mitchell, Scedrov, Shankar, Kanovich, Lafont). Direct connections have been established between linear logic and various models of concurrency like Petri nets (Asperti, Brown, Meseguer, Gunter), games (Blass, Abramsky, Lamarche), stochastic Petri nets, Dijkstra's guarded commands (Kanovich); all this shows linear logic to be a good logical foundation for the theory of computational processes that may consume resources.

    51. Welcome Linear Logic - Forums
    Home of the Scan Gauge II, Linear logic is based in Mesa, AZ. Joey Snyder, Linear logic , will be glad to answer your questions about Scan Gauge II.

    52. Deep Blue At The University Of Michigan: A Game Semantics For Linear Logic
    We propose that the connectives of Linear logic can be naturally interpreted as the operations on games introduced for entirely different purposes by Blass
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    Edit Your Profile Get more info About Deep Blue Help Contact Us Deep Blue at the University of Michigan ... Interdisciplinary and Peer-Reviewed Please use this persistent URL to cite or link to this item:
    Title: A game semantics for linear logic Author(s): Blass, Andreas Issue Date: 29-Apr-1992 Publisher: Elsevier Citation: Blass, Andreas (1992/04/29)."A game semantics for linear logic." Annals of Pure and Applied Logic 56(1-3): 183-220. <> Abstract: Persistent URL (URI):

    Other Identifiers: Appears in Collections: Interdisciplinary and Peer-Reviewed Files in This Item: File Description Size Format 0000469.pdf Adobe PDF View/Open

    53. » Linear Logic And Permutation Stacks: The Forth Shall Be First
    Recently on the Concatenative mailing list the paper Linear logic and Permutation Stacks The Forth Shall be First by Henry Baker was referenced when
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    54. Untitled Document
    Our new website will be posted soon. New Life Digital Media 1233 Northgate Business Parkway Madison, TN 371152475. 615-868-1179.

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