We study the embedding of ordered monoids into quantales and then we propose general constructions and results about such an embedding. While the origin of the discovery of this new logic comes from a semantical analysis of the models of system f or polymorphic \\lambda\calculus, one can see the whole system of linear logic as a bold attempt to reconcile the beauty and symmetry of the systems. Section 4 will introduce our linearlogic semantics for chr, explain its bene. Its basic dynamical nature has attracted computer scientists, and various promising connections have been made in the areas of optimal program. We present a game or dialogue semantics in the style of lorenzen 1959 for girards linear logic 1987. By the way, there are two disjunctions in linear logic.
Traditionally syntax and semantics live in completely distinct worlds, one nite and accessible, the other in nite and abstract. While the origin of the discovery of this new logic comes from a semantical analysis of the models of system f or polymorphic \\lambda\calculus, one can see the whole system of linear logic as a bold attempt to reconcile the beauty and symmetry of the systems for. Introduction to linear logic and ludics, part ii request pdf. Although syntax and semantics coexist, it is tempting to adopt an essentialist philosophical viewpoint1 and think of semantics as preexisting syntax. Girards program 10 to remove the distinction between syntax and semantics, this paper describes a strict correspondence between the polarized propositional fragment of linear logic ll. Noncommutative linear logic in linguistics springerlink. In this paper we give an alternative 2dimensional syntax for multiplicative linear logic derivations. Curryhoward isomorphism, and to linear logic and some of its applications in functional programming. Although syntax oflll is wellunderstood owing to girards careful analysis 1, semantics for lll has remained an open question.
Quantitative game semantics for linear logic ugo dal lago olivier laurenty. A syntax for linear logic philip wadler, ninth international conference on the mathematical foundations of programming semantics, springer verlag lncs 802, new orleans, lousiana, april 1993. There is a very important property, namely the equivalence 3 between. Linear logic is a substructural logic proposed by jeanyves girard as a refinement of classical and intuitionistic logic, joining the dualities of the former with many of the constructive properties of the latter. The syntax of string diagrams authorizes the definition of. Advances in linear logic jeanyves girard, yves lafont, laurent regnier download bok. Proceedings of the workshop on linear logic, ithaca, new york, june 1993. This article is a gentle and readable introduction to linear logic. Girard in 1987, and has attractedmuch attention from computer scientists as a logical way of coping with resources and resource control. Idea transcendental syntax is the name of a proposal or maybe a pamphlete by jeanyves girard which means to rethink fundamental aspects of formal logic, of syntaxsemantics. Whenever available, url links to the referenced papers are provided.
Linear logic does this by removing several rules, known as. The page is about an alternative to linear logic called computability logic. Pdf on mar 1, 2015, william steingartner and others published linear logic in. By carefully controlling the scope of the usual structural rules, the usual binary connectives bifurcate into two systems.
Then, p reattacks the same assertion by demanding that 0 assert the other conjunct a. According to girard, linear logic and geometry of interaction are but exercises in transcendental syntax girard b. Introduction this paper is a survey of results on categorical modeling of linear logic, oriented towards logicians, interested in proof theory, category. This paper explores the linguistic implications of noncommutative linear logic, with particular reference to its multiplicative fragment mnll, that exhibits a direct relationship to lambeks syntactic calculus. We give phase semantics of linear logic and a phase semantic proof for the completeness and cutelimination theorems at once in 3. However, no general framework existed for connecting logic and logic programming. While girards prose is notoriously demanding, exegesis may be. A convenient lens through which one can study linear logic is its semantics.
Jeanyves girard s linear logic is resource aware, in the sense that premises represent resources that cannot be duplicated or discarded, rather than truth, which can be reused or ignored. Introduction to linear logic and ludics, part i irif. The operational semantics of logic programs was presented as resolution ave82, an inference rule optimized. A linearlogic semantics for constraint handling rules. Scott semantics, operational semantics, game semantics, continuation semantics, etc. Thematics analysis of the properties of programming languages. On the meaning of logical rules i 3 the opposition prooftruth therefore organizes logic along one of those boulevards form vs. It is semantics based unlike the syntax based linear logic. Since the beginning of the nineties, the semantics foundation of lcc has been well studied. See girard, lafont, and t aylor, proofs and types, as a more detailed.
The book includes a general introduction to linear logic that will ensure this books use by. The presentation of linear logic is simpli ed by basing it on girards logic of unity, a re nement of the concept of linear logic. We first point out some nature of linear logic, in comparison with tra ditional logics, in introduction 1, then give the syntax and the intuitive meaning of the syntax in 2. Furthermore, we take a look at the girard translation translating intuitionistic logic into intuitionistic linear logic. Theory and applications scuola di dottorato in informatica della universit a degli studi di milano curriculum di logica computazionale april 28may 11, 2011. Reasoning about knowledge in linear logic oxford department of. Bibliography on linear logic carnegie mellon school of. This article presents the proof theory of linear logic using both sequent calculus and proof nets and then develops some of its semantic models. The aim of this paper is to propose a unified analysis of the relationships between the notions of order and closure and to relate it to different semantics of intuitionistic linear logic ill. Relevant logic and linear logic both reject it, as opposed to intuitionistic logic, which. In part ii, we shall go back to syntactic issues and introduce proof nets. Advances in linear logic jeanyves girard, yves lafont.
Egger lfcs, school of informatics, university of edinburgh, scotland, uk. Categorical semantics of linear logic paulandre mellies proof theory is the result of a short and tumultuous history, developed on the periphery of mainstream mathematics. This volume starts with the general introduction article by girard titled linear logic. There is a standard syntax for girard s linear logic, due to abramsky, and a standard semantics, due to seely. Semantics gives a meaning to syntax and is often viewed as a way to test if a proposition can be formulated in the system. Toward observational equivalences for linear logic. Cutelimination for classical logic is highly nondeterministic. The two formats have been called additive and multiplicative, respectively, by girard.
The following three sections are concerned with the semantics of linear logic. This paper is the second part of an introduction to linear logic and ludics, both due to girard. Girard, is based upon a fine grain analysis of the main prooftheoretical notions of logic. Although the logic has also been studied for its own sake, more broadly, ideas from linear logic have been influential in fields such as programming languages, game semantics. Study of the callbyvalue mechanism for the evaluation of functional. Linear logic introduced by jeanyves girard in 1987 classical logic. Game semantics for linear logic 185 asserting ly a ly.
Computational problemstasksresources are understood as games played by a machine against the environment. The subject develops along the lines of denotational semantics, proof nets and the geometry of interaction. Lorenzen suggested that the constructive meaning of a proposition 91 should be. Girards linear logic ll, 17 provides a unifying setting where this discrepancy could be solved. This assumption can make it awkward, or even impossible, to.
Since linear logic embraces computational themes directly in its design, it often allows direct and declarative approaches to computational and resource sensitive speci. Regnier, editors, advances in linear logic, pages 142. Therefore, in girards transcendental syntax, a proposition, also called a \dichology, is a set of elements, the \epistates5, equal to its double negation. Linear logic was introduced by jeanyves girard in his seminal work girard 1987. Linear logic is one of the outcomes of the study of semantics and the interaction between logic and computer science. It is devoted to proof nets, in the limited, yet central, framework of multiplicative linear logic. In the case on linear logic we consider intuitionistic linear logic as well as classical linear logic. Also, we give a brief introduction to some concrete models of intuitionistic linear logic. There is a standard syntax for girards linear logic, due to abramsky, and a standard semantics, due to seely. There is a close connection between linear logic and algebra, which at its root is linguistic. Such a framework is appealing for linguistic analysis since it allows one to develop a dynamic characterization of the notion of a function, that plays a basic role in the foundations. Jeanyves girards linear logic is resource aware, in the sense that premises represent resources that cannot be duplicated or discarded, rather than truth, which can be reused or ignored.
A very rough introduction to linear logic john wickerson, imperial college london multicore group seminar january 7, 2014 john wickerson, imperial college london linear logic. As with mll the multiplicative part can be construed via the curryhoward isomorphism as an enrichment of boolean algebra. Transcendental syntax is the name of a proposal or maybe a pamphlete by jeanyves girard which means to rethink fundamental aspects of formal logic, of syntaxsemantics. This book is intended for students in computer science, formal linguistics, mathematical logic and to colleagues interested in categorial grammars and their logical foundations. Formulas of classical logic are given by the grammar s 1 s. Denotationalsemanticsoflinearlogic lionelvaux i2m, universite daixmarseille, france ll2016,lyon school.
This volume gives an overview of linear logic in five parts. Context semantics is a model of girards geometry of interaction. Then, in section 4, we illustrate the expressivity of linear logic by sketching the proof that propositional linear logic is undecidable, a property that sharply distinguishes linear logic from classical and intuitionistic logics. Jeanyves girard, part iii of lectures on logic, european mathematical society 2011. The results we prove in this paper can be summarized in categorical terms through an equivalence of categories between the syntax and the game model. All of them can be obtained by properly restricting the rules governing the exponential connectives. Linear temporal property is a temporal logic formula that describes a set of infinite sequences for which it is true purpose translate the properties which are written using the natural languages into ltl by using special syntax. A deductive account of natural language syntax and semantics richard moot, christian retore auth. A linearlogic semantics for constraint handling rules uni ulm. The syntax of string diagrams authorizes the definition of a framework where the.
Surprisingly, an answer is suggested by another research direction, namely by the work ofthe rst author and ito on extensions oflinear logic with certain features of temporal logic 7. Since there is no hope to modify the extant classical or intuitionistic. It results in a simple framework that uni es constraint programming and asynchronous process algebras. Linear process algebra or lpa is the theory of this framework. It is semanticsbased unlike the syntaxbased linear logic. May 22, 2018 proof nets are a syntax for linear logic proofs which gives a coarser notion of proof equivalence with respect to syntactic equality together with an intuitive geometrical representation of proofs. Originally, a semantics of linear logic in coherence. This is similar to the way in which representation theory provides an understanding of groups via linear maps of vector spaces. Although the logic has also been studied for its own sake, more broadly, ideas from linear logic have been influential in fields. P can attack this assertion by demanding that 0 assert the conjunct y.
Proof nets are a syntax for linear logic proofs which gives a coarser notion of proof equivalence with respect to syntactic equality together with an intuitive geometrical representation of proofs. Jeanyves girard, linear logic, its syntax and semantics. Rules chr programming language is its declarative semantics where rules are read. According to girard, linear logic and geometry of interaction are but exercises in transcendental syntax. This paper expresses more a maturation than a revision of the old program.
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