A Modal Logic for Termgraph Rewriting

We propose a modal logic tailored to describe graph transformations and discuss some of its properties. We focus on a particular class of graphs called termgraphs. They are first-order terms augmented with sharing and cycles. Termgraphs allow one to describe classical data-structures (possibly with pointers) such as doubly-linked lists, circular lists etc. We show how the proposed logic can faithfully describe (i) termgraphs as well as (ii) the application of a termgraph rewrite rule (i.e. matching and replacement) and (iii) the computation of normal forms with respect to a given rewrite system. We also show how the proposed logic, which is more expressive than propositional dynamic logic, can be used to specify shapes of classical data-structures (e.g. binary trees, circular lists etc.)

On a Dynamic Logic for Graph Rewriting

International audienceInitially introduced by P. Balbiani, R. Echahed and A.Herzig, this dynamic logic is useful to talk about properties on termgraphs and to characterize transformations on these graphs. Also are presented the deterministic labelled graphs for which the logical framework is designed. This logic has been the starting point of a formal development, using the Coq proof assistant, to design a logical and algorithmic framework useful for verifyin and proving graph rewriting. The formalization allowed us to figure out some ambiguities in the involved concepts. This formalization is not the topic here but the clear view brought to us by the formal work, so the results will be expressed using the original mathematical objects of this logic. Some problems of this logic are demonstrated, relatively to the representation of graph rewriting. Some are minor issues but some are far more important for the adequation between the formulas about graph rewriting and the actual rewriting systems. Invalidating some resulting propositions, solutions are given to reestablish the logical characterization of graph rewriting, which was the initial purpose

Runtime verification for biochemical programs

The biochemical paradigm is well-suited for modelling autonomous systems and new programming languages are emerging from this approach. However, in order to validate such programs, we need to define precisely their semantics and to provide verification techniques. In this paper, we consider a higher-order biochemical calculus that models the structure of system states and its dynamics thanks to rewriting abstractions, namely rules and strategies. We extend this calculus with a runtime verification technique in order to perform automatic discovery of property satisfaction failure. The property specification language is a subclass of LTL safety and liveness properties

Unifying Nominal Unification

Nominal unification is proven to be quadratic in time and space. It was so by two different approaches, both inspired by the Paterson-Wegman linear unification algorithm, but dramatically different in the way nominal and first-order constraints are dealt with. To handle nominal constraints, Levy and Villaret introduced the notion of replacing while Calves and Fernandez use permutations and sets of atoms. To deal with structural constraints, the former use multi-equations in a way similar to the Martelli-Montanari algorithm while the later mimic Paterson-Wegman. In this paper we abstract over these two approaches and genralize them into the notion of modality, highlighting the general ideas behind nominal unification. We show that replacings and environments are in fact isomorphic. This isomorphism is of prime importance to prove intricate properties on both sides and a step further to the real complexity of nominal unification

Runtime Verification for Biochemical Programs

International audienceThe biochemical paradigm is well-suited for modelling autonomous systems and new programming languages are emerging from this approach. However, in order to validate such programs, we need to define precisely their semantics and to provide verification techniques. In this paper, we consider a higher-order biochemical calculus that models the structure of system states and its dynamics thanks to rewriting abstractions, namely rules and strategies. We extend this calculus with a runtime verification technique in order to perform automatic discovery of property satisfaction failure. The property specification language is a subclass of LTL safety and liveness properties

Unifying Nominal Unification

International audienceNominal unification is proven to be quadratic in time and space. It was so by two different approaches, both inspired by the Paterson-Wegman linear unification algorithm, but dramatically different in the way nominal and first-order constraints are dealt with. To handle nominal constraints, Levy and Villaret introduced the notion of replacing while Calvès and Fernández use permutations and sets of atoms. To deal with structural constraints, the former use multi-equation in a way similar to the Martelli-Montanari algorithm while the later mimic Paterson-Wegman. In this paper we abstract over these two approaches and genralize them into the notion of modality, highlighting the general ideas behind nominal unification. We show that replacings and environments are in fact isomorphic. This isomorphism is of prime importance to prove intricate properties on both sides and a step further to the real complexity of nominal unification.Il est prouvé que l'unification nominale est quadratique en temps et en espace. Elle le fût par deux approches différentes, toutes deux inspirées de l'algorithme d'unification linéaire de Paterson et Wegman, mais très différentes dans leur manière de prendre en compte les contraintes nominales et celles du premier ordre. Pour traiter les contraintes nominales, Levy et Villaret ont introduit la notion de "replacing" alors que Calvès et Fernández ont utilisés des permutations et ensembles d'atomes. Pour prendre en compte les contraintes structurelles, les premiers utilisent des multi-équations à la manière de l'algorithme de Martelli et Montanari tandis que les seconds miment celui de Paterson et Wegman. Dans ce papier nous proposons une abstraction de ces deux approches et les généralisons à travers la notion de modalité, soulignant les idées générales sous-jacentes à l'unification nominale. Nous montrons que les "replacings" et les environnements sont en fait isomorphes. Cet isomorphisme est de première importance dans la démonstrations de propriétés complexes des deux cotés et une étape supplémentaire vers la complexité réelle de l'unification nominale

Formalizing Functions as Processes

We present the first formalization of Milner’s classic translation of the λ-calculus into the π-calculus. It is a challenging result with respect to variables, names, and binders, as it requires one to relate variables and binders of the λ-calculus with names and binders in the π-calculus. We formalize it in Abella, merging the set of variables and the set of names, thus circumventing the challenge and obtaining a neat formalization. About the translation, we follow Accattoli’s factoring of Milner’s result via the linear substitution calculus, which is a λ-calculus with explicit substitutions and contextual rewriting rules, mediating between the λ-calculus and the π-calculus. Another aim of the formalization is to investigate to which extent the use of contexts in Accattoli’s refinement can be formalized

Optimal Sharing Graphs for Substructural Higher-order Rewriting Systems

The notion of optimal reduction was introduced by Lévy (1980) in the context of the untyped ?-calculus, based on the concept of families of reducible expressions. It took more than a decade for an algorithm achieving this optimal reduction to be discovered, introduced by Lamping (1990) and then refined by Gonthier, Abadi & Lévy (1992). The existence of an analogous algorithm for higher-order term rewriting systems was later theorised by Van Oostrom (1996), but has of yet been unrealised. We provide such an algorithm by defining a class of higher-order rewriting systems having Intuitionistic Linear Logic (Benton, Bierman, de Paiva & Hyland 1992) as a substitution calculus, in the sense of Van Oostrom (1994), and introduce a method of translating terms and rules into equivalent Lamping-Gonthier sharing graphs. Our system thus offers a generalisation of the mechanism for optimal reduction from second- to higher-order term rewriting systems. Moreover, in the case of match-sequential systems, we provide a specific reduction strategy, as we are able to effectively identify needed redexes. Finally, we explore briefly the subtleties and complexities of applying the technique to various other term rewriting system, such as those with alternative substructural or polymorphic type systems, those with generalised patterns on the left-hand side, and those with rationally infinite terms. All these systems are built upon the same fundamental translation of ?-terms to sharing graphs

Peter, le langage qui n’existe pas...

“Inside every large language is a small language struggling to get out ...” [Igarashi et al. 2001]“... and inside every small language is a sharp extension looking for better expressivity ...” [Liquori & Spiwack 2008]It is my privilege and pleasure to introduce Peter, the language that does not exist... The Peter language contains almost the linguistic features I have introduced and investigated in the field of functional and object-oriented programming, plus some new features not published yet. In Peter’s Habilitation, I will try to limit as much as possible the mathematical overhead and the technicalities (e.g. full set of rules, full proofs of theorems, etc.). In my opinion, the habilitation thesis should not be a mere translation of the candidate’s most successful papers (3), nor a commented curriculum vitæ, nor a survey of all the related works in his scientific area (4), just to mention a few “classic Habilitation styles”. It is my opinion that it should be short in length since it is experienced that a very few Habilitation thesis are really downloaded, cited and read. Oftenly, habilitation thesis are not even made accessible on the Web. Peter’s Habilitation will be based on the following three points: • (Modularity) I will present a (Turing complete) kernel of Peter, called Baby Peter, and I will continue in the rest of the Habilitation to extend it in a modular fashion until the final extension, called Wise Peter. Baby Peter is a functional language with object-oriented features equipped with a sound type system. Peter bears some similarities to Atsushi, Benjamin and Phil’s Featherweight Java [IPW01] and Alonso Church’s typed lambda calculus [Chu41]. The main difference lies in an ad hoc exception-handling mechanism allowing the programmer to choose the type system according to her/his necessities and goals. Even more, it allows the programmer to write her/his own type system (see item (Type-programmable)). Some chapters will focus on operational semantics, some others on type systems, some others on both. All topics will be treated in a “lightweight fashion”. Examples of extensions are for instance mixing class-based and pure object-based features, but also improving proof languages à la LF with pattern matching facilities and including those metalanguages to Peter in order to mix algorithms and their correctness proofs. • (Verbatim-like) Instead of annoying the reader with a plain French translation of some of my most relevant papers (6), I will show, for each extension, only some key rules of the operational semantics or of the type system (every system has at least a key rule...) and some motivating examples. I do not plan to prove type soundness for each extension of Peter: the whole soundness of Wise Peter is left as a challenge for the “next” user friendly proof assistant.• (Type-programmable) Type systems for programming languages and proof languages are fixed a priori by language designers; type systems are not first class citizens. To my little knowledge, no language allows the programmer to build, choose, or mix type systems. The idea of modifying the type discipline at compile time is not completely new; a quite inspiring work has been done by the “visionary-6-pages” paper by Gilad in 2004 [Bra04] called Pluggable Type Systems. The possibility to mixing type systems and using it as a first class citizens is an interesting research strand that will constitute an original contribution in Peter’s Habilitation. With the intention of disseminating science in a simple, clear and pedagogical way, and inspired by the works of Kim [Bru99, TKB01, BDKT03, RBC+ 05, Bru02] and Gilles [Dow03, Dow07], I wish you a very nice reading of the Peter’s Habilitation. 3 Although certain parts are taken of my articles. 4 The typographic convention is that references to my papers are in “numeric” style while references to other papers are in “alphanumeric” style. 6 We provide a CD and a Web site with all my papers.C’est mon privilege et plaisir d’introduire Peter, le langage qui n’existe pas... Le langage Peter contient quasiment tous les aspects linguistiques que j’ai introduits et étudiés dans le domaine de la programmation fonctionnelle et objets, ainsi que quelques idées qui n’ont pas encore été publiées. Dans l’habilitation de Peter, la démarche que je suivrai consiste à essayer de limiter les détails concernant les aspects théoriques et techniques (c-à-d. les ensembles complets des règles de typage, suites de théorèmes abscons, etc.). Mon mémoire d’habilitation ne sera pas une traduction brutale des différents articles publiés (1), ni un curriculum vitæ commenté, ni un panorama de tous les articles dans un domaine scientifique (2), pour ne citer que quelques styles classiques de thèses d’habilitation. Tout d’abord elle sera courte car l’expérience enseigne que très peu de thèses d’habilitation sont réellement téléchargées, citées et lues. Très souvent, les thèses d’habilitation ne sont même pas accessibles sur le Web. L’Habilitation de Peter sera fondée sur les trois « dogmes » suivants: • (Modularité) Je commencerai par le plus petit fragment complet (au sens de Turing) de Peter, appelée Baby Peter et je continuerai de façon modulaire, d’extension en extension, jusqu’à l’extension finale appelée Sage Peter. Baby Peter est un langage fonctionnel avec des constructions linguistiques orientées objet et un système de types correct. Peter partage quelques similitudes avec Featherweight Java de Atsushi, Benjamin et Phil [IPW01] et le lambda calcul typé de Alonso (Church) [Chu41]. La différence principale entre Featherweight Java et Peter, est un mécanisme d’exceptions ad hoc, qui permet au programmeur de décider quel système de types sera le plus adapté à l’egard de ses nécessités et objectifs. En plus, ce mécanisme permet au programmeur d'écrire son système de types (voir point Type-programmable). Certains chapitres seront focalisés sur un nouveau système de types, tandis que, dans d’autres chapitres, l’extension sera associée à une extension de la syntaxe et du système de types. Tous les arguments seront traités d’une façon accessible au plus grand nombre de lecteurs. Comme exemples d’extensions, je citerai une forme nouvelle d'héritage multiple, une extension de Peter qui permettra à un objet de « s'échapper de sa classe », une extension de Peter avec filtrage évolué et enfin une extension de Peter qui permettra de mélanger algorithmes et preuves de correction d’algorithmes.• (Verbatim-like) Plutôt que d'asséner à mes lecteurs une traduction française mot-à-mot de mes articles scientifiques (5), j’ai privilegié une présentation simple de chaque extension, utilisant uniquement quelques règles clés de la sémantique opérationnelle ou du système de types (il y a toujours une règle clé...), en ajoutant immédiatement des exemples pour motiver et comprendre son utilisation correcte. Je ne prouverai pas la propriété de complétude de chaque système de types qui étend Peter : la complétude de Sage Peter est proposée en défi au prochain assistant à la preuve convivial. • (Type-programmable) Les systèmes de types pour les langages de programmation et pour la preuve sont fixés a priori par leurs concepteurs et ne sont pas des objets de première classe pouvant être modifiés ou simplement utilisés par le programmeur qui en subit les qualités et les faiblesses. À ma connaissance, aucun langage ne permet au programmeur de « programmer » sa discipline de types personnelle. L’idée de modifier la discipline de typage à la compilation n’est pas très nouvelle ; un article « visionnaire » de 6 pages, qui m'a eclairé, a été Pluggable Type System de Gilad [Bra04] sorti en 2004. La possibilité de permettre au programmeur d'écrire sa propre discipline de typage et de l’utiliser à la volée est par elle-même une contribution originale dans l’habilitation de Peter. Avec l’envie de diffuser la connaissance scientifique de façon simple, claire et pédagogique, inspiré par les ouvrages de Kim [Bru99,TKB01, BDKT03, RBC+ 05, Bru02] et Gilles [Dow03, Dow07], il ne me reste plus qu'à vous souhaiter une bonne lecture de l’habilitation de Peter. 1. Bien que certaines parties soient tirées de mes articles. 2. La convention typographique est que les référence à mes articles soit en style « numérique » tandis que les références à d’autres articles soit en « alphanumérique ». 5 Un CD et un site web contiendront tous mes articles. <br