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Abstract. This paper explores the surprisingly rich design space for the simply typed lambda calculus with casts and a dynamic type. Such a calculus is the target intermediate language of the gradually typed lambda calculus but it is also interesting in its own right. In light of diverse requirements for casts, we develop a modular semantic framework, based on Henglein's Coercion Calculus, that instantiates a number of space-efficient, blame-tracking calculi, varying in what errors they detect and how they assign blame. Several of the resulting calculi extend work from the literature with either blame tracking or space efficiency, and in doing so reveal previously unknown connections. Furthermore, we introduce a new strategy for assigning blame under which casts that respect traditional subtyping are statically guaranteed to never fail. One particularly appealing outcome of this work is a novel cast calculus that is well-suited to gradual typing.

Many modern programming languages support basic generics, sufficient to implement type-safe polymorphic containers. Some languages have moved beyond this basic support, and in doing so have enabled a broader, more powerful form of generic programming. This paper reports on a comprehensive comparison of facilities for generic programming in eight programming languages: C++, Standard ML, Objective Caml, Haskell, Eiffel, Java, C# (with its proposed generics extension), and Cecil. By implementing a substantial example in each of these languages, we illustrate how the basic roles of generic programming can be represented in each language. We also identify eight language properties that support this broader view of generic programming: support for multi-type concepts, multiple constraints on type parameters, convenient associated type access, constraints on associated types, retroactive modeling, type aliases, separate compilation of algorithms and data structures, and implicit argument type deduction for generic algorithms. We find that these features are necessary to avoid awkward designs, poor maintainability, and painfully verbose code. As languages increasingly support generics, it is important that language designers understand the features necessary to enable the effective use of generics and that their absence can cause difficulties for programmers.

Language researchers and designers have extended a wide variety of type systems to support gradual typing, which enables languages to seamlessly combine dynamic and static checking. These efforts consistently demonstrate that designing a satisfactory gradual counterpart to a static type system is challenging, and this challenge only increases with the sophistication of the type system. Gradual type system designers need more formal tools to help them conceptualize, structure, and evaluate their designs. In this paper, we propose a new formal foundation for gradual typing, drawing on principles from abstract interpretation to give gradual types a semantics in terms of pre-existing static types. Abstracting Gradual Typing (AGT for short) yields a formal account of consistency---one of the cornerstones of the gradual typing approach---that subsumes existing notions of consistency, which were developed through intuition and ad hoc reasoning. Given a syntax-directed static typing judgment, the AGT approach induces a corresponding gradual typing judgment. Then the type safety proof for the underlying static discipline induces a dynamic semantics for gradual programs defined over source-language typing derivations. The AGT approach does not resort to an externally justified cast calculus: instead, run-time checks naturally arise by deducing evidence for consistent judgments during proof reduction. To illustrate the approach, we develop a novel gradually-typed counterpart for a language with record subtyping. Gradual languages designed with the AGT approach satisfy by construction the refined criteria for gradual typing set forth by Siek and colleagues.

Gradual typing is a discipline for integrating dynamic checking into a static type system. Since its introduction in functional languages, it has been adapted to a variety of type systems, including objectoriented, security, and substructural. This work studies its application to implicitly typed languages based on type inference. Siek and Vachharajani designed a gradual type inference system and algorithm that infers gradual types but still rejects ill-typed static programs. However, the type system requires local reasoning about type substitutions, an imperative inference algorithm, and a subtle correctness statement.This paper introduces a new approach to gradual type inference, driven by the principle that gradual inference should only produce static types. We present a static implicitly typed language, its gradual counterpart, and a type inference procedure. The gradual system types the same programs as Siek and Vachharajani, but has a modular structure amenable to extension. The language admits letpolymorphism, and its dynamics are defined by translation to the Polymorphic Blame Calculus (Ahmed et al. 2009(Ahmed et al. , 2011.The principal types produced by our initial type system mask the distinction between static parametric polymorphism and polymorphism that can be attributed to gradual typing. To expose this difference, we distinguish static type parameters from gradual type parameters and reinterpret gradual type consistency accordingly. The resulting extension enables programs to be interpreted using either the polymorphic or monomorphic Blame Calculi.

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Typestate reflects how the legal operations on imperative objects can change at runtime as their internal state changes. A typestate checker can statically ensure, for instance, that an object method is only called when the object is in a state for which the operation is well-defined. Prior work has shown how modular typestate checking can be achieved thanks to access permissions and state guarantees. However, typestate was not treated as a primitive language concept: typestate checkers are an additional verification layer on top of an existing language. In contrast, a typestate-oriented programming language directly supports expressing typestates. For example, in the Plaid programming language, the typestate of an object directly corresponds to its class, and that class can change dynamically. Plaid objects have not just typestate-dependent interfaces, but also typestate-dependent behaviors and runtime representations. This paper lays foundations for typestate-oriented programming by formalizing a nominal object-oriented language with mutable state that integrates typestate change and typestate checking as primitive concepts. We first describe a statically-typed language, called Featherweight Typestate (FT), where the types of object references are augmented with access permissions and state guarantees. We describe a novel flow-sensitive permission-based type system for FT. Because static typestate checking is still too rigid for some applications, we then extend this language into a gradually-typed language, called Gradual Featherweight Typestate (GFT). This language extends the notion of gradual typing to account for typestate: gradual typestate checking seamlessly combines static and dynamic checking by automatically inserting runtime checks into programs. The gradual type system of GFT allows programmers to write dynamically safe code even when the static type checker can only partly verify it.