## Section: New Results

### Language design and type systems

#### The Mezzo programming language

Participants : Jonathan Protzenko, François Pottier.

In the past ten years, the type systems community and the separation logic community, among others, have developed highly expressive formalisms for describing ownership policies and controlling side effects in imperative programming languages. In spite of this extensive knowledge, it remains very difficult to come up with a programming language design that is simple, effective (it actually controls side effects!) and expressive (it does not force programmers to alter the design of their data structures and algorithms).

The Mezzo programming language, formerly known as HaMLet, aims to bring new answers to these questions.

We have come up with a solid design for the programming language: many features of the language have been reworked or consolidated this year, and we believe we strike a good balance between expressiveness and complexity. We wrote several flagship examples that illustrate the gains offered by Mezzo, as well as two (yet unpublished) papers discussing the design of the language. Jonathan Protzenko implemented a prototype type-checker; although it is not perfect yet, several non-trivial examples are successfully type-checked.

The current state of the Mezzo programming language is best described in [40] ; a former version of this document can be found as [39] .

François Pottier wrote a formal definition of (a slightly lower-level variant of) Mezzo, and proved that Mezzo is type-safe: that is, well-typed programs cannot crash (but they can stop abruptly if a run-time check fails). The proof, which is about 15,000 lines, has been machine-checked using Coq. A paper that describes this work is in preparation.

This work was facilitated by Pottier's experience with a similar previous proof. In particular, out of the above 15,000 lines, about 2,000 lines correspond to a re-usable library for working with de Bruijn indices, and about 3,000 lines correspond to a re-usable formalisation of “monotonic separation algebras”, which help reason about resources (memory, time, knowledge, ...) and how they evolve over time. These libraries have not yet been fully documented and released; this might be done in the future.

#### Coercion abstraction

Participants : Julien Cretin, Didier Rémy.

Expressive type systems often allow non trivial conversions between types, which may lead to complex, challenging, and sometimes ad hoc type systems. Such examples are the extension of System F with type equalities to model GADTs and type families of Haskell, or the extension of System F with explicit contracts. A useful technique to simplify the meta-theoretical studies of such systems is to make type conversions explicit as “coercions” inside terms.

Following a general approach to coercions based on System F, we introduced a language F-iota with abstraction over coercions and where all type transformations are represented as coercions. The main difficulty is dealing with coercion abstraction, as abstract coercions whose types are uninhabited cannot be erased at run-time. We proposed a restriction, called parametric F-iota, that ensures erasability of all coercions by construction. This work was presented at the POPL conference in January [22] .

We extended parametric F-iota with non-interleaved positive recursive types and with erasable isomorphisms. We generalized the presentation of the language viewing coercions as conversions between typings (pairs of a typing environment and a type) rather than between types. An extended version with full proofs will be submitted for journal publication.

We also studied a more liberal version of F-iota where coercion inhabitation is no more ensured by construction (which limits expressiveness), but instead by providing coercion witnesses in source terms. This extension requires pushing abstract coercions under redexes so that they do not block the reduction. As a consequence, coercions cannot be reified in System F, and we need a direct proof of termination of iota-reduction. We completed one such proof based on reducibility candidates.

#### Ambivalent types for principal type inference with GADTs

Participants : Jacques Garrigue [Nagoya University] , Didier Rémy.

Type inference for Generalized Abstract Data Types (GADTs) is always a matter of compromise because it is inherently non monotone: assuming more specific types for GADTs may ensure more invariants, which in turn may result in more general types. Moreover, even when types of GADTs parameters are explicitly given, they introduce equalities between types, which makes them inter-convertible but with a limited scope. This may then creates an ambiguity when leaving the scope of the equation: which representative should be used for the equivalent forms? Ideally, one should use a type disjunction, but this is not allowed—for good reasons. Hence, to avoid arbitrary choices, these situations must be rejected, forcing the user to add more annotations to resolve ambiguities.

We proposed a new approach to type inference with GADTs. While some uses of
equations are unavoidable and create real ambiguities, others are gratuitous
and create artificial ambiguities, To distinguish between the two, we
introduced *ambivalent types*: a way to trace types that have been
obtained by an unavoidable use of an equation. We then redefined
ambiguities so that only ambivalent types become ambiguous and should be
rejected or resolved by a programmer annotation.

Interestingly, the solution is fully compatible with unification-based type inference algorithms used in ML dialects. The work was presented at the ML workshop [31] and implemented in the latest version 4.00 of OCaml.

#### GADTs and Subtyping

Participants : Gabriel Scherer, Didier Rémy.

Following the addition of GADTs to the OCaml language in version 4.00 released this year, we studied the theoretical underpinnings of variance subtyping for GADTs. The question is to decide which variances should be accepted for a GADT-style type declaration that includes type equality constraints in constructor types. This question exposes a new notion of decomposability and unexpected tensions in the design of a subtyping relation. Our formalization partially reuses earlier work by François Pottier and Vincent Simonet [54] . It was presented at the ML Workshop [33] . An extended version including full proofs is available as a technical report [38] and was submitted for presentation at a conference.

#### Singleton types for code inference

Participants : Gabriel Scherer, Didier Rémy.

Inspired by tangent aspects of the PhD work of Julien Cretin, we investigated the use of singleton types for code inference. If we can prove that a type contains, in a suitably restricted pure lambda-calculus, a unique inhabitant modulo program equivalence, the compiler can infer the code of this inhabitant. This opens the way to type-directed description of boilerplate code, through type inference of finer-grained type annotations. The preliminary results seem encouraging, both on the theoretical side (identifying general situations for type-directed programming) and the practical side (mining existing OCaml code for usage situations).

#### Programming with names and binders

Participants : Nicolas Pouillard, François Pottier.

Following Nicolas Pouillard's Ph.D. defense in January 2012 [11] , Nicolas Pouillard and François Pottier produced a unified presentation of Pouillard's approach to programming with abstract syntax, in the form of a paper that was published in the Journal of Functional Programming [16] .

#### A type-and-capability calculus with hidden state

Participant : François Pottier.

During the year 2010, François Pottier developed a machine-checked proof of an expressive type-and-capability system, which can be used to type-check and prove properties of imperative ML programs. The proof is carried out in Coq and takes up roughly 20,000 lines of code. In the first half of 2011, François Pottier wrote a paper that describes the system and its proof in detail. This paper was published, after a revision, in 2012 [15] .