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# Topic: Linear logic

###### In the News (Sun 21 Apr 19)

 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. Linear logic was proposed by the French mathematician Jean-Yves Girard in 1987. Linear logic uses an idea from modal logic to embed the usual logic by means of a pair of exponential operators. en.wikipedia.org /wiki/Linear_logic   (1462 words)

 Lectures on Linear Logic 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. 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. csli-publications.stanford.edu /site/0937073776.html   (186 words)

 COMPUTABILITY LOGIC: a theory of interactive computation HOMEPAGE Technically CL is a game logic: it understands interactive computational problems as games played by a machine against the environment, their computability as existence of a machine that always wins the game, logical operators as operations on computational problems, and validity of a logical formula as being a scheme of "always computable" problems. The name "universal" is related to the potential of this logic to integrate, on the basis of one semantics, classical, intuitionistic and linear logics, with their seemingly unrelated or even antagonistic philosophies. The inherent incompleteness of affine or linear logics, resulting from the fundamental limitations of the underlying sequent-calculus approach, is apparently the reason why such intuitions and examples, while so heavily relied on in the popular linear-logic literature, have never really found a good explication in the form of a mathematically strict and intuitively convincing semantics. www.cis.upenn.edu /~giorgi/cl.html   (3951 words)

 Re: Linear logic semantics (Barwise) (353 lines) Linear logic combines these into one sort by incorporating the logical machinery into the nonlogical structure. Another semantic aspect of linear logic that should be emphasized is in Girard's "Geometry of Interaction" in Logic Colloquium '88 (North-Holland yellow series), in Danos' exposition of geometry of interaction (in French) in Gazette Math. Use commutative linear logic when the order in which f and g are performed makes no difference even if A and C overlap. www.cis.upenn.edu /~bcpierce/types/archives/1992/msg00047.html   (2031 words)

 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 of the connectives of intuitionistic logic (more recently also have appeared some interpretations of classical logic into linear logic). It is important to note that linear logic is not yet another exotic logic, but a canonical mathematical object, with a well structured proof-theory, even simpler than that of classical and intuitionistic logics. Let us say that linear logic acts as a kind of a looking-glass through which phenomena of the field are better understood. floc02.diku.dk /LL   (366 words)

 Linear Naming and Computation   (Site not responding. Last check: 2007-11-05) A linear name cannot be duplicated, so it requires only a small fixed amount of overhead to implement. The logic of linear naming is linear logic, which provides a different (and logic-independent) algorithmic way of thinking about procedure calling, naming, closures and thunks, and explicit control. Linear logic is a ``resource conscious'' logic that is ideal for studying issues of resource management and explicit control of evaluation. www.linearity.org   (286 words)

 Game Semantics or Linear Logic? Logical operators are understood as operations on such tasks/resources/games, atoms as variables ranging over tasks/resources/games, and validity of a logical formula as existence of a machine that always (under every particular intretpretation of atoms and against any possible behavior by the environment) successfully accomplishes/provides/wins the task/resource/game represented by the formula. The set of valid formulas in a certain fragment of the (very expressive) language of computability logic forms a logic that is similar to but by no means the same as linear logic. Computability logic starts with a mathematically strict and intuitively convincing semantics, and only after that, as a natural second step, asks the question about what the corresponding logic and its axiomatizations (syntax) are. www.csc.villanova.edu /~japaridz/CL/gsoll.html   (641 words)

 DI & CoS - Commutative/Non-commutative Linear Logic 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. This paper studies properties of the logic BV, which is an extension of multiplicative linear logic (MLL) with a self-dual non-commutative operator. For example, for linear logic it is possible to design a deductive system, in which all rules are local. alessio.guglielmi.name /res/cos/CNCLL   (1505 words)

 Introduction to Linear Logic   (Site not responding. Last check: 2007-11-05) Linear Logic was introduced by J.-Y. Girard in 1987 and it has attracted much attention from computer scientists, as it is a logical way of coping with resources and resource control. Cut-elimination for Classical Logic is highly non-deterministic; it is shown how this can be remedied either by moving to Intuitionistic Logic or to Linear Logic. A Digression - Russell's Paradox and Linear Logic www.brics.dk /LS/96/6/BRICS-LS-96-6/BRICS-LS-96-6.html   (210 words)

 Lygon Linear logic can be used to model the consumption of resources. Whereas clauses in Prolog, based on classical logic, can be used many times during the execution of a program, clauses in Lygon, based on linear logic, must (by default) be used exactly once. Linear logic has numerous applications in computing science, including natural language processing, concurrency, logic programming and resource management (for example garbage collection). www.cs.rmit.edu.au /lygon   (1027 words)

 Wadler: Linear Logic   (Site not responding. Last check: 2007-11-05) 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. Call-by-name, call-by-value, call-by-need, and the linear lambda calculus The presentation of linear logic is simplified by basing it on the Logic of Unity. homepages.inf.ed.ac.uk /wadler/topics/linear-logic.html   (1065 words)

 Linear temporal logic - Wikipedia, the free encyclopedia Linear temporal logic (LTL) is a field of mathematical logic that is able to talk about the future of paths (LTL is a temporal logic). LTL can be shown to be equivalent to the first-order logic over one successor and the smaller relation, FO[S,<] as well as star-free regular expressions or deterministic finite automata with loop complexity 0. This page was last modified 14:02, 18 October 2005. en.wikipedia.org /wiki/Linear_temporal_logic   (231 words)

 ICCL Workshop - Proof Theory 2004 We investigate the LES semantics of planning problems by resorting to linear logic approach to planning within the calculus of structures presentation of linear logic. Linear logic has been discovered by observation of a semantical entity, namely the category of coherence spaces and linear maps. The best syntax we have for linear logic (multiplicative proof nets) also have problems with units; in a recent paper with Lutz we give the best solution there is so far, but this solution sill involves some amount of quotienting and its complexity is not trivial. www.iccl.tu-dresden.de /events/WPT-2004   (1976 words)

 Harry Mairson My research examines the interaction between mathematical logic and computation theory, with application to the design and analysis of functional programming languages, type systems, and database query languages. The basic technology is an implementation of Girard's `geometry of interaction' that works as well in the context of linear logic and proof nets, explaining the incremental duplication of boxes. Logic and deductive databases: It has been recognized since the early 1980s that first-order database query languages are lacking in expressive power. www.cs.brandeis.edu /~mairson   (1073 words)

 The Calculus of Structures in Maude - Ozan Kahramanogullari   (Site not responding. Last check: 2007-11-05) Although it is possible to express the existing systems for Classical Logic and systems for all the fragments of Linear Logic as Maude Modules, I have been experimenting mostly with system BV. Although multiplicative exponential linear logic is unknown to be decidable or not, NEL is shown to be undecidable. Charles Stewart and Phiniki Stouppa are investigating the modal logics in the calculus of structures. www.informatik.uni-leipzig.de /%7Eozan/maude_cos.html   (998 words)

 Lolli: A Linear Logic Programming Language   (Site not responding. Last check: 2007-11-05) Logic Programming in a Fragment of Intuitionistic Linear Logic: Extended Abstract, by Joshua S. Hodas and Dale Miller, Proceedings of the Sixth Annual Symposium on Logic in Computer Science, Amsterdam, July 1991. The logic and meta-theory are presented in detail with full proofs; more and longer examples are given, and the implementation of the language is discussed at length. This paper presents a variant of the logic of Lolli in which it is possible to directly specify the Relevant (clauses may be copied but not discarded) and Affine (clauses may be discarded but not copied) constraints. www.cse.psu.edu /~dale/lolli   (1643 words)

 Linear Logic papers of Andreas R. Blass   (Site not responding. Last check: 2007-11-05) This talk is a survey of two topics of recent interest in mathematical logic, namely linear logic and cardinal characteristics of the continuum. We discuss the extent to which game semantics is implicit in the formalism of linear logic and in the intuitions underlying linear logic. A category used by de Paiva to model linear logic also occurs in Vojtas's analysis of cardinal characteristics of the continuum. www.math.lsa.umich.edu /~ablass/ll.html   (356 words)

 UMA Forum Home Page UMA Forum is part of a research project concerning linear logic, logic programming, object orientedness and concurrency. In spite of this, UMA Forum retains a large amount of expressiveness from linear logic. Linear logic synchronous connectives are supported through second-order predicates as well. www.lcc.uma.es /~lopez/umaforum   (1369 words)

 Lutz Strassburger   (Site not responding. Last check: 2007-11-05) Now we are studying a novel kind of proof nets for classical logic, as well as a new categorical axiomatisation for proofs in classical propositional logic. Alessio Guglielmi) a noncommutative/commutative logic in the calculus of structures. The main results are an entirely local system for linear logic and several decomposition theorems which constitute a new kind of normal form for proofs. www.ps.uni-sb.de /%7Elutz   (694 words)

 ICCL Summer School 2004 We express classical, linear, modal logics, and several variants thereof, and their deductive systems are as simple as those in the Sequent Calculus, but possess the desired computational properties. This course concerns the application of automated reasoning techniques to logics broadly in the substructural family, such as linear logic, affine logic, relevant logic or the like. It is naturally easy, given a cut-free sequent calculus presentation with an algorithmic flavour of a system of logic, to generate a program that searches for proofs in the logic, and given a proof of termination for some fragment of the logic this generally yields a decision procedure for that fragment. www.iccl.tu-dresden.de /events/SA-2004   (1556 words)

 BI The logic of bunched implications, BI, is a substructural system which freely combines an additive (intuitionistic) and a multiplicative (linear) implication via bunches (contexts with two combining operations, one which admits Weakening and Contraction and one which does not). Specifically, we start with the multiplicative fragment of linear logic and extend, on the one hand, to linear logic with its additives and, on the other, to the additives of the logic of bunched implications (BI). The logic of bunched implications, BI, provides a logical analysis of a basic notion of resource rich enough, for example, to form the logical basis for ``pointer logic'' and ``separation logic'' semantics for programs which manipulate mutable data structures. www.cs.bath.ac.uk /~pym/BI.html   (1345 words)

 Lutz Strassburger Find new proof nets for linear logic that are better adopted to the formalism of the calculus of structures. MELL is the multiplicative exponential fragment of linear logic, and it is not known whether provability is decidable or not. Pomset logic is a natural extension of multiplicative linear logic by a non-commutative self-dual connective. www.ps.uni-sb.de /~lutz/diploma-topics.html   (963 words)

 The undecidability of second order linear affine logic   (Site not responding. Last check: 2007-11-05) linear logic with the weakening) is decidable [Kopylov]. Moreover, we obtain the whole class of undecidability second order logics which lie between Lambek calculus (LC) and linear affine logic. The proof is based on an encoding two-counter Minsky machines in second order linear affine logic. www.cs.cornell.edu /People/kopylov/papers/mllw2   (121 words)

 Pomset Logic: A Non-Commutative Extension of Classical Linear Logic - Retore (ResearchIndex) Abstract: We extend the multiplicative fragment of linear logic with a non-commutative connective (called before), which, roughly speaking, corresponds to sequential composition. Pomset logic: a non-commutative extension of classical linear logic. 85 Linear logic: its syntax and semantics - Girard - 1995 citeseer.ist.psu.edu /retore97pomset.html   (763 words)

 Seminar on Linear Logic and Applications (CS 359) This seminar examines linear logic with particular emphasis on applications in computer science. Basic topics to be covered include classical and intuitionistic linear logic, affine and relevance logics, natural deduction and sequent calculi, and decidability and complexity results for various fragments. Applications include linear type systems for functional languages, linear type inference, linear logic programming, concurrent languages based on linear logic, linear logical frameworks and type theories. theory.stanford.edu /~iliano/courses/99-winter-linear   (294 words)

 TABLEAUX'99 A nicely formatted version of the Call for Papers and Tutorials is available in PostScript or in DVI for standard 8.5" x 11" paper; it is also available in PostScript or in DVI for a4 paper. Krysia Broda and Dov Gabbay, CLDS for Propositional Intuitionistic Logic. Heiko Mantel and Jens Otten, linTAP: A Tableau Prover for Linear Logic. www.cs.albany.edu /~nvm/tab99   (1702 words)

 Cetus Links: 18,452 Links on Objects and Components / Prolog Linear logic models causality, state transitions, and accounting for resources used in logic. Linear logic systems such as Lolli and LLP are supersets (more or less) of Prolog that define additional operations for these purposes. Inductive logic often uses a schema of classes, instances and attribute values to define a rule base. www.cetus-links.org /oo_prolog.html   (2177 words)

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