Are there Hilbert-style Pure Type Systems?

For many a natural deduction style logic there is a Hilbert-style logic that is equivalent to it in that it has the same theorems (i.e. valid judgements with empty contexts). For intuitionistic logic, the axioms of the equivalent Hilbert-style logic can be propositions which are also known as the types of the combinators I, K and S. Hilbert-style versions of illative combinatory logic have formulations with axioms that are actual type statements for I, K and S. As pure type systems (PTSs)are, in a sense, equivalent to systems of illative combinatory logic, it might be thought that Hilbert-style PTSs (HPTSs) could be based in a similar way. This paper shows that some PTSs have very trivial equivalent HPTSs, with only the axioms as theorems and that for many PTSs no equivalent HPTS can exist. Most commonly used PTSs belong to these two classes. For some PTSs however, including lambda* and the PTS at the basis of the proof assistant Coq, there is a nontrivial equivalent HPTS, with axioms that are type statements for I, K and S.Comment: Accepted in Logical Methods in Computer Scienc

Church's Thesis and Functional Programming

David Turner's contribution to a volume published on the 70th anniversary of Church's Thesis. ERRATUM: In the published version (Ontos Verlag 2006) Wadsworth's 1976 result on Solvability and head normal form (p6 bottom) was incorrectly attributed to Böhm - this has now been corrected

A general approach to define binders using matching logic

We propose a novel shallow embedding of binders using matching logic, where the binding behavior of object-level binders is obtained for free from the behavior of the built-in existential binder of matching logic. We show that binders in various logical systems such as lambda-calculus, System F, pi-calculus, pure type systems, etc., can be defined in matching logic. We show the correctness of our definitions by proving conservative extension theorems, which state that a sequent/judgment is provable in the original system if and only if it is provable in matching logic. An appealing aspect of our embedding of binders in matching logic is that it yields models to all binders, also for free. We show that models yielded by matching logic are deductively complete to the formal reasoning in the original systems. For lambda-calculus, we further show that the yielded models are representationally complete---a desired property that is not enjoyed by many existing lambda-calculus semantics.Ope

The monitoring power of forcing program transformations

In this thesis, we are interested in semantical proofs of correctness results for complex programming languages. In particular, we advocate the need for a theoretical framework that allows one to:- design realizability semantics using basic blocks - use algebraic constructions to combine those blocks As a step towards this goal, we propose a new semantical framework, based on the composition of linear variants of Krivine realizability and Cohen forcing. The first ingredient of this framework is the Monitoring Abstract Machine: a computing environment that possesses special memory cells used to monitor the execution of programs, in the style of Miquel's KFAM. It is shown how this new machine emerges from a linear forcing program transformation. We then introduce the central notion of Monitoring Algebra and the associated realizability interpretation. Different monitoring algebras induce sound semantics of different programming languages. We then present an algebraic construction to combine different Monitoring Algebras (and the associated programming languages) based on the technique of forcing iteration. We present various results and first applications of our theory. We show that the forcing structure can be used to represent the consumption of resources, in particular time, but also step-indexing or the use of higher-order references. We finally apply our results to retrieve three complex soundness results:- we give the first semantical proof of the consistency of a contraction-free naive set theory, originally introduced by Grishin- we use our framework to obtain a polynomial time termination result for a light-logic based programming language featuring recursive types - we prove the soundness of a language with references that supports strong updates, based on a linear type system inspired by a work of Ahmed, Fluet and Morrisett.Dans cette thèse, nous nous intéressons aux preuves sémantiques de résultats de corrections pour des langages de programmation complexes. En particulier, nous mettons en évidencele besoin d'un nouveau cadre théorique permettant de:- concevoir des sémantiques de réalisabilité à partir de briques plus élémentaires.- combiner ces briques élémentaires grâce à des constructions algébriques.- prouver des théorèmes généraux réutilisables lors de preuves futures de correctionde langages de programmation. Nous proposons dans ce manuscrit un tel cadre sémantique, basé sur la composition de variantes linéaires de la réalisabilité de Krivine et du forcing de Cohen. Le premier ingrédient est la Monitoring Abstract Machine: un environnement de calcul qui utilise des cases mémoires réservées pour "surveiller" l'exécution des programmes, dans le style de la KFAM introduite par Miquel. Cette machine émerge naturellement d'une transformation de programme basée sur une variante linéaire du forcing de Cohen. Nous introduisons par la suite la notion centrale d'Algèbre de Monitoring et le modèle de réalisabilité associé. Chaque algèbre de monitoring induit une sémantique correcte pour un langage de programmation associé. Point crucial de cette thèse, nous définissons, en se basant sur la technique du forcing itéré, une construction algébrique permettant de combiner plusieurs algèbres de monitoring. Nous développons de nombreux résultats élémentaires à propos de notre théorie. En particulier, nous montrons que la structure de forcing peut être utilisée pour représenter la consommation de ressources (en particulier le temps), le step-indexing ou encore des références d'ordre supérieur. Finalement, nous appliquons notre théorie pour obtenir trois preuves de correction complexes:- nous donnons la première preuve sémantique connue de la cohérence d'une théorie des ensembles naïve sans contraction introduite originellement par Grishin dans les années 70- nous utilisons notre cadre pour obtenir un résultat de terminaison en temps polynomial pour un langage de programmation avec types récursifs basé sur une logique light- nous reprouvons la correction d'un langage avec références d'ordre supérieur et mise à jour forte, inspiré par un système de type introduit par Ahmed, Fluet et Morrisett

Inquiries into words, constraints and contexts : Festschrift in the honour of Kimmo Koskenniemi on his 60th birthday

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