Mœbius: metaprogramming using contextual types: the stage where system f can pattern match on itself

We describe the foundation of the metaprogramming language, Mœbius, which supports the generation of polymorphic code and, more importantly, the analysis of polymorphic code via pattern matching. Mœbius has two main ingredients: 1) we exploit contextual modal types to describe open code together with the context in which it is meaningful. In Mœbius, open code can depend on type and term variables (level 0) whose values are supplied at a later stage, as well as code variables (level 1) that stand for code templates supplied at a later stage. This leads to a multi-level modal lambda-calculus that supports System-F style polymorphism and forms the basis for polymorphic code generation. 2) we extend the multi-level modal lambda-calculus to support pattern matching on code. As pattern matching on polymorphic code may refine polymorphic type variables, we extend our type-theoretic foundation to generate and track typing constraints that arise. We also give an operational semantics and prove type preservation. Our multi-level modal foundation for Mœbius provides the appropriate abstractions for both generating and pattern matching on open code without committing to a concrete representation of variable binding and contexts. Hence, our work is a step towards building a general type-theoretic foundation for multi-staged metaprogramming that, on the one hand, enforces strong type guarantees and, on the other hand, makes it easy to generate and manipulate code. This will allow us to exploit the full potential of metaprogramming without sacrificing the reliability of and trust in the code we are producing and running.

Reasoning about “reasoning about reasoning”: semantics and contextual equivalence for probabilistic programs with nested queries and recursion

Metareasoning can be achieved in probabilistic programming languages (PPLs) using agent models that recursively nest inference queries inside inference queries. However, the semantics of this powerful, reflection-like language feature has defied an operational treatment, much less reasoning principles for contextual equivalence. We give formal semantics to a core PPL with continuous distributions, scoring, general recursion, and nested queries. Unlike prior work, the presence of nested queries and general recursion makes it impossible to stratify the definition of a sampling-based operational semantics and that of a measure-theoretic semantics—the two semantics must be defined mutually recursively. A key yet challenging property we establish is that probabilistic programs have well-defined meanings: limits exist for the step-indexed measures they induce. Beyond a semantics, we offer relational reasoning principles for probabilistic programs making nested queries. We construct a step-indexed, biorthogonal logical-relations model. A soundness theorem establishes that logical relatedness implies contextual equivalence. We demonstrate the usefulness of the reasoning principles by proving novel equivalences of practical relevance—in particular, game-playing and decisionmaking agents. We mechanize our technical developments leading to the soundness proof using the Coq proof assistant. Nested queries are an important yet theoretically underdeveloped linguistic feature in PPLs; we are first to give them semantics in the presence of general recursion and to provide them with sound reasoning principles for contextual equivalence.

A fine-grained computational interpretation of Girard’s intuitionistic proof-nets

This paper introduces a functional term calculus, called pn, that captures the essence of the operational semantics of Intuitionistic Linear Logic Proof-Nets with a faithful degree of granularity, both statically and dynamically. On the static side, we identify an equivalence relation on pn-terms which is sound and complete with respect to the classical notion of structural equivalence for proof-nets. On the dynamic side, we show that every single (exponential) step in the term calculus translates to a different single (exponential) step in the graphical formalism, thus capturing the original Girard’s granularity of proof-nets but on the level of terms. We also show some fundamental properties of the calculus such as confluence, strong normalization, preservation of β-strong normalization and the existence of a strong bisimulation that captures pairs of pn-terms having the same graph reduction.

The geometry of Bayesian programming

We give two geometry of interaction models for a typed λ-calculus with recursion endowed with operators for sampling from a continuous uniform distribution and soft conditioning, namely a paradigmatic calculus for higher-order Bayesian programming. The models are based on the category of measurable spaces and partial measurable functions, and the category of measurable spaces and s-finite kernels, respectively. The former is proved adequate with respect to both a distribution-based and a sampling-based operational semantics, while the latter is proved adequate with respect to a sampling-based operational semantics.

Object Cloning for Ownership Systems

<p>Modern object-oriented programming languages frequently need the ability to clone, duplicate, and copy objects. The usual approaches taken by languages are rudimentary, primarily because these approaches operate with little understanding of the object being cloned. Deep cloning naively copies every object that has a reachable reference path from the object being cloned, even if the objects being copied have no innate relationship with that object. For more sophisticated cloning operations, languages usually only provide the capacity for programmers to define their own cloning operations for specific objects, and with no help from the type system. Sheep cloning is an automated operation that clones objects by leveraging information about those objects’ structures, which the programmer imparts into their programs with ownership types. Ownership types are a language mechanism that defines an owner for every object in the program. Ownership types create a hierarchical structure for the heap. In this thesis, we construct an extensible formal model for an object-oriented language with ownership types (Core), and use it to explore different formalisms of sheep cloning. We formalise three distinct operational semantics of sheep cloning, and for each approach we include proofs or proof outlines where appropriate, and provide a comparative analysis of each model’s benefits. Our main contribution is the descripSC formal model of sheep cloning and its proof of type soundness. The second contribution of this thesis is the formalism of Mojo-jojo, a multiple ownership system that includes existential quantification over types and context parameters, along with a constraint system for context parameters. We prove type soundness for Mojo-jojo. Multiple ownership is a mechanism which allows objects to have more than one owner. Context parameters in Mojo-jojo can use binary operators such as: intersection, union, and disjointness.</p>

Object Cloning for Ownership Systems

<p>Modern object-oriented programming languages frequently need the ability to clone, duplicate, and copy objects. The usual approaches taken by languages are rudimentary, primarily because these approaches operate with little understanding of the object being cloned. Deep cloning naively copies every object that has a reachable reference path from the object being cloned, even if the objects being copied have no innate relationship with that object. For more sophisticated cloning operations, languages usually only provide the capacity for programmers to define their own cloning operations for specific objects, and with no help from the type system. Sheep cloning is an automated operation that clones objects by leveraging information about those objects’ structures, which the programmer imparts into their programs with ownership types. Ownership types are a language mechanism that defines an owner for every object in the program. Ownership types create a hierarchical structure for the heap. In this thesis, we construct an extensible formal model for an object-oriented language with ownership types (Core), and use it to explore different formalisms of sheep cloning. We formalise three distinct operational semantics of sheep cloning, and for each approach we include proofs or proof outlines where appropriate, and provide a comparative analysis of each model’s benefits. Our main contribution is the descripSC formal model of sheep cloning and its proof of type soundness. The second contribution of this thesis is the formalism of Mojo-jojo, a multiple ownership system that includes existential quantification over types and context parameters, along with a constraint system for context parameters. We prove type soundness for Mojo-jojo. Multiple ownership is a mechanism which allows objects to have more than one owner. Context parameters in Mojo-jojo can use binary operators such as: intersection, union, and disjointness.</p>

Knowledge Dynamics and Behavioural Equivalencesin Multi-Agent Systems

We define a process calculus to describe multi-agent systems with timeouts for communication and mobility able to handle knowledge. The knowledge of an agent is represented as sets of trees whose nodes carry information; it is used to decide the interactions with other agents. The evolution of the system with exchanges of knowledge between agents is presented by the operational semantics, capturing the concurrent executions by a multiset of actions in a labelled transition system. Several results concerning the relationship between the agents and their knowledge are presented. We introduce and study some specific behavioural equivalences in multi-agent systems, including a knowledge equivalence able to distinguish two systems based on the interaction of the agents with their local knowledge.

On Correctness and Completeness of an n Queens Program

Thom Frühwirth presented a short, elegant, and efficient Prolog program for the n queens problem. However, the program may be seen as rather tricky and one may not be convinced about its correctness. This paper explains the program in a declarative way and provides proofs of its correctness and completeness. The specification and the proofs are declarative, that is they abstract from any operational semantics. The specification is approximate, it is unnecessary to describe the program’s semantics exactly. Despite the program works on non-ground terms, this work employs the standard semantics, based on logical consequence and Herbrand interpretations. Another purpose of the paper is to present an example of precise declarative reasoning about the semantics of a logic program.

Formal design and verification of system task in intelligent transportation systems based on micro-kernel architecture

The accuracy of design and implementation of an operating system in intelligent transportation systems is difficult to describe and validate because of its complexity. In this paper, we describe an OS in intelligent transportation systems with automaton theory and establish an OS state model. Based on this model, we construct an isomorphic model in Isabelle/HOL, describe the work objects and operational semantics of the system, and verify the system at the assembly level. We use a micro-kernel OS prototype (VSOS) for intelligent transportation systems as an example to illustrate our method and verify the correctness of design and implementation in VSOS with Isabelle/HOL. Verification shows that the proposed method is feasible.