Fully abstract models for effectful λ-calculi via category-theoretic logical relations

We present a construction which, under suitable assumptions, takes a model of Moggi’s computational λ-calculus with sum types, effect operations and primitives, and yields a model that is adequate and fully abstract. The construction, which uses the theory of fibrations, categorical glueing, ⊤⊤-lifting, and ⊤⊤-closure, takes inspiration from O’Hearn & Riecke’s fully abstract model for PCF. Our construction can be applied in the category of sets and functions, as well as the category of diffeological spaces and smooth maps and the category of quasi-Borel spaces, which have been studied as semantics for differentiable and probabilistic programming

Extensional Collapse Situations I: non-termination and unrecoverable errors

We consider a simple model of higher order, functional computation over the booleans. Then, we enrich the model in order to encompass non-termination and unrecoverable errors, taken separately or jointly. We show that the models so defined form a lattice when ordered by the extensional collapse situation relation, introduced in order to compare models with respect to the amount of "intensional information" that they provide on computation. The proofs are carried out by exhibiting suitable applied {\lambda}-calculi, and by exploiting the fundamental lemma of logical relations

A type-theory for higher-order amortized analysis

Die Verifikation von "Worst-Case" Schranken für Ressourcennutzung ist ein wichtiges Problem in der Informatik. Der Nutzen einer solchen Verifikation hängt von der Präzision der Analyse ab. Aus Gründen der Präzision ist es manchmal nützlich, die durchschnittlichen Kosten einer Folge von Operationen zu berücksichtigen, statt die Kosten jeder einzelnen Operation getrennt zu betrachten. Diese Art von Analyse wird oft als amortisierte Ressourcenanalyse bezeichnet. Typischerweise profitieren Programme, die ihren Zustand optimieren, um die Kosten zukünftiger Ausführungen zu reduzieren, von solchen Ansätzen. Die Analyse der Ressourcennutzung einer mit zwei (LIFO) Listen implementierten funktionalen (FIFO) Schlange ist ein klassisches Beispiel für eine amortisierte Analyse. In dieser Arbeit präsentieren wir λamor, eine Typentheorie für die amortisierte Analyse der Ressourcennutzung höherstufiger Programme. Eine typische amortisierte Analyse speichert einen "ghost state", der als Potenzial bezeichnet wird, zusammen mit den Datenstrukturen. Die Kernidee der amortisierten Analyse ist es, zu zeigen, dass das dem Programm zur Verfügung stehende Potenzial ausreicht, um die Ressourcennutzung des Programms zu erfassen. Die Verifikation in λamor basiert auf der Realisierung dieser Idee in einer Typentheorie. Wir erreichen dies indem wir ein allgemeines typentheoretisches Konstrukt zur Darstellung des Potenzials auf der Ebene von Typen definieren und anschließend eine affine Typentheorie aufbauen. Mit λamor zeigen wir, dass eine typentheoretische amortisierte Analyse mit gut verstandenen Konzepten aus substrukturellen und modalen Typentheorien durchgeführt werden kann. Trotzdem ergibt sich ein äußerst aussagekräftiges Framework, das für die Ressourcenanalyse von höherstufigen Programmen, sowohl ein einem "strikten", als auch in einem "lazy" Setting, verwendet werden kann. Wir präsentieren Einbettungen zweier stark verschiedener Arten von typentheoretischen Ressourcenanalyseframeworks (eines basiert auf Effekten, das andere auf Koeffekten) in λamor. Wir zeigen, dass λamor korrekt (sound) ist (mithilfe eines "Logical relations" Modells) und, dass es vollständig für PCF-Programme ist (unter Verwendung einer der Einbettungen). Als nächstes verwenden wir Ideen von λamor, um eine andere Typentheorie (genannt λcg) für einen ganz anderen Anwendungsfall - Informationsflusskontrolle (IFC) - zu entwickeln. λcg verwendet ähnliche typentheoretische Konstrukte wie λamor für das Potenzial verwendet, um die Vertraulichkeitsmarkierungen (den "ghost state" für IFC) darzustellen. Schließlich abstrahieren wir von den spezifischen "ghost states" (Potenzial und Vertraulichkeitsmarkierungen) und entwickeln eine Typentheorie für einen allgemeinen "ghost state" mit einer monoidalen Struktur.Verification of worst-case bounds (on the resource usage of programs) is an important problem in computer science. The usefulness of such verification depends on the precision of the underlying analysis. For precision, sometimes it is useful to consider the average cost over a sequence of operations, instead of separately considering the cost of each individual operation. This kind of an analysis is often referred to as amortized resource analysis. Typically, programs that optimize their internal state to reduce the cost of future executions benefit from such approaches. Analyzing resource usage of a standard functional (FIFO) queue implemented using two functional (LIFO) lists is a classic example of amortized analysis. In this thesis we present λamor, a type-theory for amortized resource analysis of higher-order functional programs. A typical amortized analysis works by storing a ghost state called the potential with data structures. The key idea underlying amortized analysis is to show that, the available potential with the program is sufficient to account for the resource usage of that program. Verification in λamor is based on internalizing this idea into a type theory. We achieve this by providing a general type-theoretic construct to represent potential at the level of types and then building an affine type-theory around it. With λamor we show that, type-theoretic amortized analysis can be performed using well understood concepts from sub-structural and modal type theories. Yet, it yields an extremely expressive framework which can be used for resource analysis of higher-order programs, both in a strict and lazy setting. We show embeddings of two very different styles (one based on effects and the other on coeffects) of type-theoretic resource analysis frameworks into λamor. We show that λamor is sound (using a logical relations model) and complete for cost analysis of PCF programs (using one of the embeddings). Next, we apply ideas from λamor to develop another type theory (called λcg) for a very different domain – Information Flow Control (IFC). λcg uses a similar typetheoretic construct (which λamor uses for the potential) to represent confidentiality label (the ghost state for IFC). Finally, we abstract away from the specific ghost states (potential and confidentiality label) and describe how to develop a type-theory for a general ghost state with a monoidal structure

On Berry's conjectures about the stable order in PCF

PCF is a sequential simply typed lambda calculus language. There is a unique order-extensional fully abstract cpo model of PCF, built up from equivalence classes of terms. In 1979, G\'erard Berry defined the stable order in this model and proved that the extensional and the stable order together form a bicpo. He made the following two conjectures: 1) "Extensional and stable order form not only a bicpo, but a bidomain." We refute this conjecture by showing that the stable order is not bounded complete, already for finitary PCF of second-order types. 2) "The stable order of the model has the syntactic order as its image: If a is less than b in the stable order of the model, for finite a and b, then there are normal form terms A and B with the semantics a, resp. b, such that A is less than B in the syntactic order." We give counter-examples to this conjecture, again in finitary PCF of second-order types, and also refute an improved conjecture: There seems to be no simple syntactic characterization of the stable order. But we show that Berry's conjecture is true for unary PCF. For the preliminaries, we explain the basic fully abstract semantics of PCF in the general setting of (not-necessarily complete) partial order models (f-models.) And we restrict the syntax to "game terms", with a graphical representation.Comment: submitted to LMCS, 39 pages, 23 pstricks/pst-tree figures, main changes for this version: 4.1: proof of game term theorem corrected, 7.: the improved chain conjecture is made precise, more references adde

Recursion and Sequentiality in Categories of Sheaves

We present a fully abstract model of a call-by-value language with higher-order functions, recursion and natural numbers, as an exponential ideal in a topos. Our model is inspired by the fully abstract models of O'Hearn, Riecke and Sandholm, and Marz and Streicher. In contrast with semantics based on cpo's, we treat recursion as just one feature in a model built by combining a choice of modular components

10351 Abstracts Collection -- Modelling, Controlling and Reasoning About State

From 29 August 2010 to 3 September 2010, the Dagstuhl Seminar 10351 ``Modelling, Controlling and Reasoning About State \u27\u27 was held in Schloss Dagstuhl~--~Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. Links to extended abstracts or full papers are provided, if available

FICS 2010

International audienceInformal proceedings of the 7th workshop on Fixed Points in Computer Science (FICS 2010), held in Brno, 21-22 August 201

On dialogue games and coherent strategies

We explain how to see the set of positions of a dialogue game as a coherence space in the sense of Girard or as a bistructure in the sense of Curien, Plotkin and Winskel. The coherence structure on the set of positions results from a Kripke translation of tensorial logic into linear logic extended with a necessity modality. The translation is done in such a way that every innocent strategy defines a clique or a configuration in the resulting space of positions. This leads us to study the notion of configuration designed by Curien, Plotkin and Winskel for general bistructures in the particular case of a bistructure associated to a dialogue game. We show that every such configuration may be seen as an interactive strategy equipped with a backward as well as a forward dynamics based on the interplay between the stable order and the extensional order. In that way, the category of bistructures is shown to include a full subcategory of games and coherent strategies of an interesting nature