Summary of lecture iii introducing temporal logics. In part ii, we shall go back to syntactic issues and introduce proof nets. Linear logic is a substructural logic in which the contraction rule and the weakening rule are omitted, or at least have their applicability restricted. Linear logic also provides new insights into the nature of proofs in both classical and intuitionistic logic. Its syntax and semantics is an updated and more accessible presentation, writ ten about ten years later. We give phase semantics of linear logic and a phase semantic proof for the completeness and cutelimination theorems at once in 3.
The new syntax is based on pattern matching, allowing for concise expression of programs. Pdf on mar 1, 2015, william steingartner and others published linear logic in. Pdf 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. The distinction between the two conjunctions can be explained by the possible ways of merging dialects.
A sentence or statement is a string of characters over some alphabet. True concurrency semantics for a linear logic programming language with broadcast communication. Citeseerx document details isaac councill, lee giles, pradeep teregowda. Pdf proof diagrams for multiplicative linear logic. The syntax rules of a language specify which strings of characters from the languages alphabet are in the language. Pdf file 3295 kb 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. Since there is no hope to modify the extant classical or intuitionistic. Ever since its introduction in 1987, linear logic has inspired uses as a means to. Introduction to linear logic and ludics, part i irif. Linear logic contains a fully involutive negation while maintaining a strong constructive interpretation.
Curryhoward isomorphism, and to linear logic and some of its applications in functional. The interesting part of this vector space semantics is based on. In the original definition of girard 87 linear logic is the internal logic ofhas categorical semantics in starautonomous categories seely 89, prop. Given its focus on resources, linear logic has found many applications in computer science. Relevant logic and linear logic both reject it, as opposed to intuitionistic logic, which. A linearlogic semantics for constraint handling rules uni ulm. This is made explicit by showing how to represent proofs in linear logic as linear maps between vector spaces. By carefully controlling the scope of the usual structural rules, the usual binary connectives bifurcate into two systems. Geometry of interaction also works for various calculi, for instance. If a sequent is valid, then any permutation of it created. A new solution is proposed, based on ideas taken from girards logic of unity. We give an introduction to logic tailored for algebraists, explaining how proofs in linear logic can be viewed as algorithms for constructing morphisms in symmetric closed monoidal categories with additional structure. The box structure has a deep meaning, since the nesting of boxes is ultimately responsible for cut elimination. Linear logic can also be interpreted using a semantics of games or interactions.
In the computationasdeduction approach, pieces of logics syntax such as. Reasoning about knowledge in linear logic oxford department of. Syntax and semantics provide a languages definition o users of a language definition. Jeanyves girard, linear logic, its syntax and semantics pdf.