The syntax of string diagrams authorizes the definition of a framework where the. Denotationalsemanticsoflinearlogic lionelvaux i2m, universite daixmarseille, france ll2016,lyon school. 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. There is a very important property, namely the equivalence 3 between. This paper expresses more a maturation than a revision of the old program. Study of the callbyvalue mechanism for the evaluation of functional. Although syntax and semantics coexist, it is tempting to adopt an essentialist philosophical viewpoint1 and think of semantics as preexisting syntax. Introduction to linear logic and ludics, part i irif. This book is intended for students in computer science, formal linguistics, mathematical logic and to colleagues interested in categorial grammars and their logical foundations. This volume starts with the general introduction article by girard titled linear logic. In the case on linear logic we consider intuitionistic linear logic as well as classical linear logic. Pdf on mar 1, 2015, william steingartner and others published linear logic in. Introduction this paper is a survey of results on categorical modeling of linear logic, oriented towards logicians, interested in proof theory, category.
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. 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. Since there is no hope to modify the extant classical or intuitionistic. The operational semantics of logic programs was presented as resolution ave82, an inference rule optimized. Also, we give a brief introduction to some concrete models of intuitionistic linear logic. Whenever available, url links to the referenced papers are provided. 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. 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. 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. Lorenzen suggested that the constructive meaning of a proposition 91 should be. Cutelimination for classical logic is highly nondeterministic. 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.
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. According to girard, linear logic and geometry of interaction are but exercises in transcendental syntax girard b. There is a close connection between linear logic and algebra, which at its root is linguistic. Girard, is based upon a fine grain analysis of the main prooftheoretical notions of logic. Jeanyves girard, linear logic, its syntax and semantics. Semantics gives a meaning to syntax and is often viewed as a way to test if a proposition can be formulated in the system.
Formulas of classical logic are given by the grammar s 1 s. 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. In part ii, we shall go back to syntactic issues and introduce proof nets. Introduction to linear logic and ludics, part ii request pdf. This article is a gentle and readable introduction to linear logic. Computational problemstasksresources are understood as games played by a machine against the environment. 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. Linear logic does this by removing several rules, known as. There is a standard syntax for girard s linear logic, due to abramsky, and a standard semantics, due to seely. 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. On the meaning of logical rules i 3 the opposition prooftruth therefore organizes logic along one of those boulevards form vs. Therefore, in girards transcendental syntax, a proposition, also called a \dichology, is a set of elements, the \epistates5, equal to its double negation. By the way, there are two disjunctions in linear logic.
A linearlogic semantics for constraint handling rules uni ulm. All of them can be obtained by properly restricting the rules governing the exponential connectives. Linear logic is one of the outcomes of the study of semantics and the interaction between logic and computer science. Noncommutative linear logic in linguistics springerlink. It is semantics based unlike the syntax based linear logic. Relevant logic and linear logic both reject it, as opposed to intuitionistic logic, which. Furthermore, we take a look at the girard translation translating intuitionistic logic into intuitionistic linear logic. Then, p reattacks the same assertion by demanding that 0 assert the other conjunct a. A linearlogic semantics for constraint handling rules.
This volume gives an overview of linear logic in five parts. The page is about an alternative to linear logic called computability logic. Context semantics is a model of girards geometry of interaction. Section 4 will introduce our linearlogic semantics for chr, explain its bene. P can attack this assertion by demanding that 0 assert the conjunct y. As with mll the multiplicative part can be construed via the curryhoward isomorphism as an enrichment of boolean algebra. Linear logic was introduced by jeanyves girard in his seminal work girard 1987.
This assumption can make it awkward, or even impossible, to. 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. Rules chr programming language is its declarative semantics where rules are read. Originally, a semantics of linear logic in coherence. Although the logic has also been studied for its own sake, more broadly, ideas from linear logic have been influential in fields. Scott semantics, operational semantics, game semantics, continuation semantics, etc. Girards linear logic ll, 17 provides a unifying setting where this discrepancy could be solved. 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. Linear process algebra or lpa is the theory of this framework. However, no general framework existed for connecting logic and logic programming.
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. Advances in linear logic jeanyves girard, yves lafont. Theory and applications scuola di dottorato in informatica della universit a degli studi di milano curriculum di logica computazionale april 28may 11, 2011. Game semantics for linear logic 185 asserting ly a ly. Curryhoward isomorphism, and to linear logic and some of its applications in functional programming. Regnier, editors, advances in linear logic, pages 142. According to girard, linear logic and geometry of interaction are but exercises in transcendental syntax. Quantitative game semantics for linear logic ugo dal lago olivier laurenty. By carefully controlling the scope of the usual structural rules, the usual binary connectives bifurcate into two systems. 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. 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. Advances in linear logic jeanyves girard, yves lafont, laurent regnier download bok. Thematics analysis of the properties of programming languages.
Since linear logic embraces computational themes directly in its design, it often allows direct and declarative approaches to computational and resource sensitive speci. Traditionally syntax and semantics live in completely distinct worlds, one nite and accessible, the other in nite and abstract. 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. Bibliography on linear logic carnegie mellon school of. It is semanticsbased unlike the syntaxbased linear logic. The following three sections are concerned with the semantics of linear logic.
Reasoning about knowledge in linear logic oxford department of. Its basic dynamical nature has attracted computer scientists, and various promising connections have been made in the areas of optimal program. The two formats have been called additive and multiplicative, respectively, by girard. Egger lfcs, school of informatics, university of edinburgh, scotland, uk. While girards prose is notoriously demanding, exegesis may be. The subject develops along the lines of denotational semantics, proof nets and the geometry of interaction. Toward observational equivalences for linear logic. We present a game or dialogue semantics in the style of lorenzen 1959 for girards linear logic 1987.
Linear logic introduced by jeanyves girard in 1987 classical logic. Jeanyves girard, part iii of lectures on logic, european mathematical society 2011. This article presents the proof theory of linear logic using both sequent calculus and proof nets and then develops some of its semantic models. Girard in 1987, and has attractedmuch attention from computer scientists as a logical way of coping with resources and resource control. In this paper we give an alternative 2dimensional syntax for multiplicative linear logic derivations. We give phase semantics of linear logic and a phase semantic proof for the completeness and cutelimination theorems at once in 3. It is devoted to proof nets, in the limited, yet central, framework of multiplicative linear logic. 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. This is similar to the way in which representation theory provides an understanding of groups via linear maps of vector spaces. Proceedings of the workshop on linear logic, ithaca, new york, june 1993. There is a standard syntax for girards linear logic, due to abramsky, and a standard semantics, due to seely. The syntax of string diagrams authorizes the definition of.
We study the embedding of ordered monoids into quantales and then we propose general constructions and results about such an embedding. A deductive account of natural language syntax and semantics richard moot, christian retore auth. 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. Although syntax oflll is wellunderstood owing to girards careful analysis 1, semantics for lll has remained an open question. It results in a simple framework that uni es constraint programming and asynchronous process algebras. 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. 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. The book includes a general introduction to linear logic that will ensure this books use by. 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. See girard, lafont, and t aylor, proofs and types, as a more detailed. 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. This paper is the second part of an introduction to linear logic and ludics, both due to girard. Since the beginning of the nineties, the semantics foundation of lcc has been well studied.