Equational Axiomization of Bicoercibility for Polymorphic Types

Two polymorphic types σ and τ are said to be bicoercible if there is a coercion from σ to τ and conversely. We give a complete equational axiomatization of bicoercible types and prove that the relation of bicoercibility is decidable.National Science Foundation (CCR-9113196); KBN (2 P301 031 06); ESPRIT BRA7232 GENTZE

Logical relations for coherence of effect subtyping

A coercion semantics of a programming language with subtyping is typically defined on typing derivations rather than on typing judgments. To avoid semantic ambiguity, such a semantics is expected to be coherent, i.e., independent of the typing derivation for a given typing judgment. In this article we present heterogeneous, biorthogonal, step-indexed logical relations for establishing the coherence of coercion semantics of programming languages with subtyping. To illustrate the effectiveness of the proof method, we develop a proof of coherence of a type-directed, selective CPS translation from a typed call-by-value lambda calculus with delimited continuations and control-effect subtyping. The article is accompanied by a Coq formalization that relies on a novel shallow embedding of a logic for reasoning about step-indexing

Practical Datatype Specializations with Phantom Types and Recursion Schemes

Datatype specialization is a form of subtyping that captures program invariants on data structures that are expressed using the convenient and intuitive datatype notation. Of particular interest are structural invariants such as well-formedness. We investigate the use of phantom types for describing datatype specializations. We show that it is possible to express statically-checked specializations within the type system of Standard ML. We also show that this can be done in a way that does not lose useful programming facilities such as pattern matching in case expressions.Comment: 25 pages. Appeared in the Proc. of the 2005 ACM SIGPLAN Workshop on M

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

On the Power of Coercion Abstraction

International audienceErasable coercions in System F-eta, also known as retyping functions, are well-typed eta-expansions of the identity. They may change the type of terms without changing their behavior and can thus be erased before reduction. Coercions in F-eta can model subtyping of known types and some displacement of quantifiers, but not subtyping assumptions nor certain forms of delayed type instantiation. We generalize F-eta by allowing abstraction over retyping functions. We follow a general approach where computing with coercions can be seen as computing in the lambda-calculus but keeping track of which parts of terms are coercions. We obtain a language where coercions do not contribute to the reduction but may block it and are thus not erasable. We recover erasable coercions by choosing a weak reduction strategy and restricting coercion abstraction to value-forms or by restricting abstraction to coercions that are polymorphic in their domain or codomain. The latter variant subsumes F-eta, F-sub, and MLF in a unified framework

The Good, the Bad, and the Ugly: An Empirical Study of Implicit Type Conversions in JavaScript

Most popular programming languages support situations where a value of one type is converted into a value of another type without any explicit cast. Such implicit type conversions, or type coercions, are a highly controversial language feature. Proponents argue that type coercions enable writing concise code. Opponents argue that type coercions are error-prone and that they reduce the understandability of programs. This paper studies the use of type coercions in JavaScript, a language notorious for its widespread use of coercions. We dynamically analyze hundreds of programs, including real-world web applications and popular benchmark programs. We find that coercions are widely used (in 80.42% of all function executions) and that most coercions are likely to be harmless (98.85%). Furthermore, we identify a set of rarely occurring and potentially harmful coercions that safer subsets of JavaScript or future language designs may want to disallow. Our results suggest that type coercions are significantly less evil than commonly assumed and that analyses targeted at real-world JavaScript programs must consider coercions

A Bi-Directional Refinement Algorithm for the Calculus of (Co)Inductive Constructions

The paper describes the refinement algorithm for the Calculus of (Co)Inductive Constructions (CIC) implemented in the interactive theorem prover Matita. The refinement algorithm is in charge of giving a meaning to the terms, types and proof terms directly written by the user or generated by using tactics, decision procedures or general automation. The terms are written in an "external syntax" meant to be user friendly that allows omission of information, untyped binders and a certain liberal use of user defined sub-typing. The refiner modifies the terms to obtain related well typed terms in the internal syntax understood by the kernel of the ITP. In particular, it acts as a type inference algorithm when all the binders are untyped. The proposed algorithm is bi-directional: given a term in external syntax and a type expected for the term, it propagates as much typing information as possible towards the leaves of the term. Traditional mono-directional algorithms, instead, proceed in a bottom-up way by inferring the type of a sub-term and comparing (unifying) it with the type expected by its context only at the end. We propose some novel bi-directional rules for CIC that are particularly effective. Among the benefits of bi-directionality we have better error message reporting and better inference of dependent types. Moreover, thanks to bi-directionality, the coercion system for sub-typing is more effective and type inference generates simpler unification problems that are more likely to be solved by the inherently incomplete higher order unification algorithms implemented. Finally we introduce in the external syntax the notion of vector of placeholders that enables to omit at once an arbitrary number of arguments. Vectors of placeholders allow a trivial implementation of implicit arguments and greatly simplify the implementation of primitive and simple tactics