Coherence and transitivity in coercive subtyping

The aim of this thesis is to study coherence and transitivity in coercive subtyping. Among other things, coherence and transitivity are key aspects for a coercive subtyping system to be consistent and for it to be implemented in a correct way. The thesis consists of three major parts. First, I prove that, for the subtyping rules of some parameterised inductive data types, coherence holds and the normal transitivity rule is admissible. Second, the notion of weak transitivity is introduced. The subtyping rules of a large class of parameterised inductive data types are suitable for weak transitivity, but not compatible with the normal transitivity rule. Third, I present a new formulation of coercive subtyping in order to combine incoherent coercions for the type of dependent pairs. There are two subtyping relations in the system and hence a further understanding of coherence and transitivity is needed. This thesis has the first case study of combining incoherent coercions in a single system. The thesis provides a clearer understanding of the subtyping rules for parameterised inductive data types and explains why the normal transitivity rule is not admissible for some natural subtyping rules. It also demonstrates that coherence and transitivity at type level can sometimes be very difficult issues in coercive subtyping. Besides providing theoretical understanding, the thesis also gives algorithms for implementing the subtyping rules for parameterised inductive data types

Coercive subtyping: Theory and implementation

International audienceCoercive subtyping is a useful and powerful framework of subtyping for type theories. The key idea of coercive subtyping is subtyping as abbreviation. In this paper, we give a new and adequate formulation of T[C], the system that extends a type theory T with coercive subtyping based on a set C of basic subtyping judgements, and show that coercive subtyping is a conservative extension and, in a more general sense, a definitional extension. We introduce an intermediate system, the star-calculus T[C]^@?, in which the positions that require coercion insertions are marked, and show that T[C]^@? is a conservative extension of T and that T[C]^@? is equivalent to T[C]. This makes clear what we mean by coercive subtyping being a conservative extension, on the one hand, and amends a technical problem that has led to a gap in the earlier conservativity proof, on the other. We also compare coercive subtyping with the 'ordinary' notion of subtyping - subsumptive subtyping, and show that the former is adequate for type theories with canonical objects while the latter is not. An improved implementation of coercive subtyping is done in the proof assistant Plastic

Definitional Functoriality for Dependent (Sub)Types

Dependently-typed proof assistant rely crucially on definitional equality, which relates types and terms that are automatically identified in the underlying type theory. This paper extends type theory with definitional functor laws, equations satisfied propositionally by a large class of container-like type constructors $F : \operatorname{Type} \to \operatorname{Type}$, equipped with a $\operatorname{map}_{F} : (A \to B) \to F\ A \to F\ B$, such as lists or trees. Promoting these equations to definitional ones strengthen the theory, enabling slicker proofs and more automation for functorial type constructors. This extension is used to modularly justify a structural form of coercive subtyping, propagating subtyping through type formers in a map-like fashion. We show that the resulting notion of coercive subtyping, thanks to the extra definitional equations, is equivalent to a natural and implicit form of subsumptive subtyping. The key result of decidability of type-checking in a dependent type system with functor laws for lists has been entirely mechanized in Coq

A theory of typed coercions and its applications

A number of important program rewriting scenarios can be recast as type-directed coercion insertion. These range from more theoretical applications such as coercive subtyping and supporting overloading in type theories, to more practical applications such as integrating static and dynamically typed code using gradual typing, and inlining code to enforce security policies such as access control and provenance tracking. In this paper we give a general theory of typedirected coercion insertion. We specifically explore the inherent tradeoff between expressiveness and ambiguity—the more powerful the strategy for generating coercions, the greater the possibility of several, semantically distinct rewritings for a given program. We consider increasingly powerful coercion generation strategies, work out example applications supported by the increased power (including those mentioned above), and identify the inherent ambiguity problems of each setting, along with various techniques to tame the ambiguities

A theory of contracts for web services

<p>Contracts are behavioural descriptions of Web services. We devise a theory of contracts that formalises the compatibility of a client to a service, and the safe replacement of a service with another service. The use of contracts statically ensures the successful completion of every possible interaction between compatible clients and services.</p> <p>The technical device that underlies the theory is the definition of filters, which are explicit coercions that prevent some possible behaviours of services and, in doing so, they make services compatible with different usage scenarios. We show that filters can be seen as proofs of a sound and complete subcontracting deduction system which simultaneously refines and extends Hennessy's classical axiomatisation of the must testing preorder. The relation is decidable and the decision algorithm is obtained via a cut-elimination process that proves the coherence of subcontracting as a logical system.</p> <p>Despite the richness of the technical development, the resulting approach is based on simple ideas and basic intuitions. Remarkably, its application is mostly independent of the language used to program the services or the clients. We also outline the possible practical impact of such a work and the perspectives of future research it opens.</p&gt

A Metatheoretic Analysis of Subtype Universes

Subtype universes were initially introduced as an expressive mechanisation of bounded quantification extending a modern type theory. In this paper, we consider a dependent type theory equipped with coercive subtyping and a generalisation of subtype universes. We prove results regarding the metatheoretic properties of subtype universes, such as consistency and strong normalisation. We analyse the causes of undecidability in bounded quantification, and discuss how coherency impacts the metatheoretic properties of theories implementing bounded quantification. We describe the effects of certain choices of subtyping inference rules on the expressiveness of a type theory, and examine various applications in natural language semantics, programming languages, and mathematics formalisation

The Essence of Nested Composition

Calculi with disjoint intersection types support an introduction form for intersections called the merge operator, while retaining a coherent semantics. Disjoint intersections types have great potential to serve as a foundation for powerful, flexible and yet type-safe and easy to reason OO languages. This paper shows how to significantly increase the expressive power of disjoint intersection types by adding support for nested subtyping and composition, which enables simple forms of family polymorphism to be expressed in the calculus. The extension with nested subtyping and composition is challenging, for two different reasons. Firstly, the subtyping relation that supports these features is non-trivial, especially when it comes to obtaining an algorithmic version. Secondly, the syntactic method used to prove coherence for previous calculi with disjoint intersection types is too inflexible, making it hard to extend those calculi with new features (such as nested subtyping). We show how to address the first problem by adapting and extending the Barendregt, Coppo and Dezani (BCD) subtyping rules for intersections with records and coercions. A sound and complete algorithmic system is obtained by using an approach inspired by Pierce\u27s work. To address the second problem we replace the syntactic method to prove coherence, by a semantic proof method based on logical relations. Our work has been fully formalized in Coq, and we have an implementation of our calculus

Type-theoretical natural language semantics: on the system F for meaning assembly

International audienceThis paper presents and extends our type theoretical framework for a compositional treatment of natural language semantics with some lexical features like coercions (e.g. of a town into a football club) and copredication (e.g. on a town as a set of people and as a location). The second order typed lambda calculus was shown to be a good framework, and here we discuss how to introduced predefined types and coercive subtyping which are much more natural than internally coded similar constructs. Linguistic applications of these new features are also exemplified