Lincx: A Linear Logical Framework with First-class Contexts

Linear logic provides an elegant framework for modelling stateful, imperative and con- current systems by viewing a context of assumptions as a set of resources. However, mech- anizing the meta-theory of such systems remains a challenge, as we need to manage and reason about mixed contexts of linear and intuitionistic assumptions. We present Lincx, a contextual linear logical framework with first-class mixed contexts. Lincx allows us to model (linear) abstract syntax trees as syntactic structures that may depend on intuitionistic and linear assumptions. It can also serve as a foundation for reasoning about such structures. Lincx extends the linear logical framework LLF with first-class (linear) contexts and an equational theory of context joins that can otherwise be very tedious and intricate to develop. This work may be also viewed as a generalization of contextual LF that supports both intuitionistic and linear variables, functions, and assumptions. We describe a decidable type-theoretic foundation for Lincx that only characterizes canonical forms and show that our equational theory of context joins is associative and commu- tative. Finally, we outline how Lincx may serve as a practical foundation for mechanizing the meta-theory of stateful systems.La logique linéaire represente une structure élégante pour modeler des systèmes im- pératifs, concurrents et avec des systèmes a états, en représentant un contexte d'hypothèses comme une collection de ressources. Cependant, la mécanisation de la métathéorie de ces systèmes demeure un défi, puisque nous devons gérer et raisonner à propos de contextes d'hypothèses mixtes linéaires et intuitionistiques. Nous présentons Lincx, une structure logique linéaire et contextuelle avec des contextes mixtes de première classe. Lincx nous permet d'établir des modèles (linéaires) d'arbres de syntaxe abstraits en tant que structures syntactiques qui peuvent dependre d'hypothèses intuitionistiques et linéaires. Lincx peut également servir de fondation pour raisonner à propos de telles structures. Lincx étend la structure logique linéaire LLF avec des contextes (linéaires) de premier ordre et une théorie d'equations d'assemblage de contextes qui peut autrement être très fastidieux et complexe à développer. Cet oeuvre peut également être perçu comme une généralisation du LF contextuel qui supporte les fonctions, les hypothéses et les variables intuitionistiques et linéaires. Nous décrivons une fondation de la théorie des types décidable pour Lincx qui ne décrit que les formes canoniques et montrons que notre theorie d'équations d'assemblage de contextes est associative et commutative. Finalement, nous donnons un aperçu de comment Lincx peut servir de fondation pratique pour la mécanisation de la métathéorie de systèmes à états

An Open Logical Framework

The LFP Framework is an extension of the Harper-Honsell-Plotkin's Edinburgh Logical Framework LF with external predicates, hence the name Open Logical Framework. This is accomplished by defining lock type constructors, which are a sort of \u25a1-modality constructors, releasing their argument under the condition that a possibly external predicate is satisfied on an appropriate typed judgement. Lock types are defined using the standard pattern of constructive type theory, i.e. via introduction, elimination and equality rules. Using LFP, one can factor out the complexity of encoding specific features of logical systems, which would otherwise be awkwardly encoded in LF, e.g. side-conditions in the application of rules in Modal Logics, and sub-structural rules, as in non-commutative Linear Logic. The idea of LFP is that these conditions need only to be specified, while their verification can be delegated to an external proof engine, in the style of the Poincar Principle or Deduction Modulo. Indeed such paradigms can be adequately formalized in LFP. We investigate and characterize the meta-theoretical properties of the calculus underpinning LFP: strong normalization, confluence and subject reduction. This latter property holds under the assumption that the predicates are well-behaved, i.e. closed under weakening, permutation, substitution and reduction in the arguments. Moreover, we provide a canonical presentation of LFP, based on a suitable extension of the notion of \u3b2\u3b7-long normal form, allowing for smooth formulations of adequacy statements. \ua9 The Author, 2013