Automating Sized-Type Inference for Complexity Analysis

Justin Pombrio (Brown University, USA) gives the third talk in the fifth panel, Inference and Analysis on the 3rd day of the ICFP conference. Co-written by Shriram Krishnamurthi (Brown University, USA) and Mitchell Wand (Northeastern University, USA). Many languages use syntactic sugar to define parts of their surface language in terms of a smaller core. Thus some properties of the surface language, like its scoping rules, are not immediately evident. Nevertheless, IDEs, refactorers, and other tools that traffic in source code depend on these rules to present information to users and to soundly perform their operations. In this paper, we show how to lift scoping rules defined on a core language to rules on the surface, a process of scope inference. In the process we introduce a new representation of binding structure--scope as a preorder--and present a theoretical advance: proving that a desugaring system preserves alpha-equivalence even though scoping rules have been provided only for the core language. We have also implemented the system presented in this paper.

Martin Avanzini (University of Innsbruck, Austria) gives the second talk in the fifth panel, Inference and Analysis on the 3rd day of the ICFP conference. This paper introduces a new methodology for the complexity analysis of higher-order functional programs, which is based on three ingredients: a powerful type system for size analysis and a sound type inference procedure for it, a ticking monadic transformation and constraint solving. Noticeably, the presented methodology can be fully automated, and is able to analyse a series of examples which cannot be handled by most competitor methodologies. This is possible due to various key ingredients, and in particular an abstract index language and index polymorphism at higher ranks. A prototype implementation is available.

Richard A. Eisenberg (Bryn Mawr College, USA) gives the first talk in the fifth panel, Inference and Analysis, on the 3rd day of the ICFP conference. Co-written by J. Garrett Morris (University of Kansas, USA). We present an approach to support partiality in type-level computation without compromising expressiveness or type safety. Existing frameworks for type-level computation either require totality or implicitly assume it. For example, type families in Haskell provide a powerful, modular means of defining type-level computation. However, their current design implicitly assumes that type families are total, introducing nonsensical types and significantly complicating the metatheory of type families and their extensions. We propose an alternative design, using qualified types to pair type-level computations with predicates that capture their domains. Our approach naturally captures the intuitive partiality of type families, simplifying their metatheory. As evidence, we present the first complete proof of consistency for a language with closed type families.

Victor Lanvin (ENS Cachan, France) gives the third talk in the fourth panel, Integrating Static and Dynamic Typing, on the 3rd day of the ICFP conference. Co-written by Giuseppe Castagna (CNRS, University of Paris Diderot). We propose a type system for functional languages with gradual types and set-theoretic type connectives and prove its soundness. In particular, we show how to lift the definition of the domain and result type of an application from non-gradual types to gradual ones and likewise for the subtyping relation. We also show that deciding subtyping for gradual types can be reduced in linear time to deciding subtyping on non-gradual types and that the same holds true for all subtyping-related decision problems that must be solved for type inference. More generally, this work not only enriches gradual type systems with unions and intersections and with the type precision that arise from their use, but also proposes and advocates a new style of gradual types programming where union and intersection types are used by programmers to instruct the system to perform fewer dynamic checks.

Yuu Igarashi (Kyoto University, Japan) gives the second talk in the fourth panel, Integrating Static and Dynamic Typing, on the 3rd day of the ICFP conference. Co-written by Taro Sekiyama (IBM Research, Japan), Atsushi Igarashi (Kyoto University, Japan). We study an extension of gradual typing--a method to integrate dynamic typing and static typing smoothly in a single language--to parametric polymorphism and its theoretical properties, including conservativity of typing and semantics over both statically and dynamically typed languages, type safety, blame-subtyping theorem, and the gradual guarantee--the so-called refined criteria, advocated by Siek et al. We develop System F-G, which is a gradually typed extension of System F with the dynamic type and a new type consistency relation, and translation to a new polymorphic blame calculus System F-C, which is based on previous polymorphic blame calculi by Ahmed et al. The design of System F-G and System F-C, geared to the criteria, is influenced by the distinction between static and gradual type variables, first observed by Garcia and Cimini. This distinction is also useful to execute statically typed code without incurring additional overhead to manage type names as in the prior calculi. We prove that System F-G satisfies most of the criteria: all but the hardest property of the gradual guarantee on semantics. We show that a key conjecture to prove the gradual guarantee leads to the Jack-of-All-Trades property, conjectured as an important property of the polymorphic blame calculus by Ahmed et al.

Amal Ahmed (Northeastern University, USA) gives the first talk in the fourth panel, Integrating Static and Dynamic Typing, on the 3rd day of the ICFP conference. Co-written by Dustin Jamner (Northeastern University, USA), Jeremy G. Siek (Indiana University, USA), Philip Wadler (University of Edinburgh, UK). The polymorphic blame calculus integrates static typing, including universal types, with dynamic typing. The primary challenge with this integration is preserving parametricity: even dynamically-typed code should satisfy it once it has been cast to a universal type. Ahmed et al. (2011) employ runtime type generation in the polymorphic blame calculus to preserve parametricity, but a proof that it does so has been elusive. Matthews and Ahmed (2008) gave a proof of parametricity for a closely related system that combines ML and Scheme, but later found a flaw in their proof. In this paper we present an improved version of the polymorphic blame calculus and we prove that it satisfies relational parametricity. The proof relies on a step-indexed Kripke logical relation. The step-indexing is required to make the logical relation well-defined in the case for the dynamic type. The possible worlds include the mapping of generated type names to their types and the mapping of type names to relations. We prove the Fundamental Property of this logical relation and that it is sound with respect to contextual equivalence. To demonstrate the utility of parametricity in the polymorphic blame calculus, we derive two free theorems.

Peter Thiemann (University of Freiburg, Germany) gives the fourth talk in the third panel, Contracts and Sessions, on the 3rd day of the ICFP conference. Co-written by Atsushi Igarashi (Kyoto University, Japan), Vasco Vasconcelos (University of Lisbon, Portugal), Philip Wadler (University of Edinburgh, UK). Session types are a rich type discipline, based on linear types, that lift the sort of safety claims that come with type systems to communications. However, web-based applications and micro services are often written in a mix of languages, with type disciplines in a spectrum between static and dynamic typing. Gradual session types address this mixed setting by providing a framework which grants seamless transition between statically typed handling of sessions and any required degree of dynamic typing. We propose GradualGV as an extension of the functional session type system GV with dynamic types and casts. We demonstrate type and communication safety as well as blame safety, thus extending previous results to functional languages with session-based communication. The interplay of linearity and dynamic types requires a novel approach to specifying the dynamics of the language.

Stephanie Balzer (Carnegie Mellon University, USA) gives the third talk in the third panel, Contracts and Sessions, on the 3rd day of the ICFP conference. Co-written Frank Pfenning (Carnegie Mellon University, USA). Session-typed languages building on the Curry-Howard isomorphism between linear logic and session-typed communication guarantee session fidelity and deadlock freedom. Unfortunately, these strong guarantees exclude many naturally occurring programming patterns pertaining to shared resources. In this paper, we introduce sharing into a session-typed language where types are stratified into linear and shared layers with modal operators connecting the layers. The resulting language retains session fidelity but not the absence of deadlocks, which can arise from contention for shared processes. We illustrate our language on various examples, such as the dining philosophers problem, and provide a translation of the untyped asynchronous pi-calculus into our language.

Lucas Waye (Harvard University, USA), gives the second talk in the third panel, Contracts and Sessions , on the 3rd day of the ICFP conference. Co-written by Christos Dimoulas (Harvard University, USA), Stephen Chong (Harvard University, USA). Modern service-oriented applications forgo semantically rich protocols and middleware when composing services. Instead, they embrace the loosely-coupled development and deployment of services that communicate via simple network protocols. Even though these applications do expose interfaces that are higher-order in spirit, the simplicity of the network protocols forces them to rely on brittle low-level encodings. To bridge the apparent semantic gap, programmers introduce ad-hoc and error-prone defensive code. Inspired by Design by Contract, we choose a different route to bridge this gap. We introduce Whip, a contract system for modern services. Whip (i) provides programmers with a higher-order contract language tailored to the needs of modern services; and (ii) monitors services at run time to detect services that do not live up to their advertised interfaces. Contract monitoring is local to a service. Services are treated as black boxes, allowing heterogeneous implementation languages without modification to services' code. Thus, Whip does not disturb the loosely coupled nature of modern services.

Sebastian Ullrich (KIT, Germany), gives the fourth talk in the second panel, Dependently Typed Programming, on the 3rd day of the ICFP conference. Co-written by Gabriel Exner (Vienna University of Technology, Austria), Jared Roesch (University of Washington, USA), Jeremy Avigad (Carnegie Mellon University, USA), Leonardo De Moura (Microsoft Research). Dependent type theory is a powerful framework for interactive theorem proving and automated reasoning, allowing us to encode mathematical objects, data type specifications, assertions, proofs, and programs, all in the same language. Here we show that dependent type theory can also serve as its own metaprogramming language, that is, a language in which one can write programs that assist in the construction and manipulation of terms in dependent type theory itself. Specifically, we describe the metaprogramming language currently in use in the Lean theorem prover, which extends Lean's object language with an API for accessing natively implemented procedures and provides ways of reflecting object-level expressions into the metalanguage. We provide evidence to show that our language is performant, and that it provides a convenient and flexible way of writing not only small-scale interactive tactics, but also more substantial kinds of automation.