Untyping Typed Algebras and Colouring Cyclic Linear Logic

We prove "untyping" theorems: in some typed theories (semirings, Kleene algebras, residuated lattices, involutive residuated lattices), typed equations can be derived from the underlying untyped equations. As a consequence, the corresponding untyped decision procedures can be extended for free to the typed settings. Some of these theorems are obtained via a detour through fragments of cyclic linear logic, and give rise to a substantial optimisation of standard proof search algorithms.Comment: 21

Pure Type System conversion is always typable

International audiencePure Type Systems are usually described in two different ways, one that uses an external notion of computation like beta-reduction, and one that relies on a typed judgment of equality, directly in the typing system. For a long time, the question was open to know whether both presentations described the same theory. A first step toward this equivalence has been made by Adams for a particular class of \emph{Pure Type Systems} (PTS) called functional. Then, his result has been relaxed to all semi-full PTS in previous work. In this paper, we finally give a positive answer to the general issue, and prove that equivalence holds for any Pure Type System.Les Systèmes de Types Purs (PTS) sont habituellement présentés de deux manières différentes, une qui utilise une notion de calcul indépendante du typage, comme la béta-reduction, et une qui défini un jugement d'égalité typée au sein du système de types. La question de savoir si ces deux présentations représentaient la même théorie est restée ouverte pendant de nombreuses années. Une première réponse partielle à cette question a été apportée par Adams pour une classe particulière de PTS dit "fonctionnels". Nous avons récement étendu ce résultat à tous les PTS "semi-complets" . Dans cet article, nous pouvons finalement donner une réponse positive à la question dans toute sa généralité: l'équivalence entre les deux présentations est prouvée correcte pour n'importe quel Système de Types Purs

Towards a Logical Framework with Intersection and Union Types

International audienceWe present an ongoing implementation of a dependent-type theory (∆-framework) based on the Edinburgh Logical Framework LF, extended with Proof-functional logical connectives such as intersection , union, and strong (or minimal relevant) implication. Proof-functional connectives take into account the shape of logical proofs, thus allowing to reflect polymorphic features of proofs in formulae. This is in contrast to classical Truth-functional connec-tives where the meaning of a compound formula is only dependent on the truth value of its subformulas. Both Logical Frameworks and proof functional logics consider proofs as first class citizens. But they do it differently namely, explicitly in the former while implicitly in the latter. Their combination opens up new possibilites of formal reasoning on proof-theoretic semantics. We provide some examples in the extended type theory and we outline a type checker. The theory of the system is under investigation. Once validated in vitro, the proof-functional type theory can be successfully plugged in existing truth-functional proof assistants

Equality is typable in Semi-Full Pure Type Systems

Abstract—There are two usual ways to describe equality in a dependent typing system, one that uses an external notion of computation like beta-reduction, and one that introduces a typed judgement of beta-equality directly in the typing system. After being an open problem for some time, the general equivalence between both approaches has been solved by Adams for a class of pure type systems (PTSs) called functional. In this paper, we relax the functionality constraint and prove the equivalence for all semi-full PTSs by combining the ideas of Adams with a study of the general shape of types in PTSs. As one application, an extension of this result to systems with sub-typing would be a first step toward bringing closer the theory behind a proof assistant such as Coq to its implementation. I