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How to integrate static and dynamic types? Recent work focuses on casts to mediate between the two. However, adding casts may degrade tail calls into a non-tail calls, increasing space consumption from constant to linear in the depth of calls.We present a new solution to this old problem, based on the notion of a threesome. A cast is specified by a source and a target type-a twosome. Any twosome factors into a downcast from the source to an intermediate type, followed by an upcast from the intermediate to the target-a threesome. Any chain of threesomes collapses to a single threesome, calculated by taking the greatest lower bound of the intermediate types. We augment this solution with blame labels to map any failure of a threesome back to the offending twosome in the source program.Herman, Tomb, and Flanagan (2007) solve the space problem by representing casts with the coercion calculus of Henglein (1994). While they provide a theoretical limit on the space overhead, there remains the practical question of how best to implement coercion reduction. The threesomes presented in this paper provide a streamlined data structure and algorithm for representing and normalizing coercions. Furthermore, threesomes provide a typed-based explanation of coercion reduction.

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.

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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.

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.

Memory bandwidth limits the performance of important kernels in many scientific applications. Such applications often use sequences of Basic Linear Algebra Subprograms (BLAS), and highly efficient implementations of those routines enable scientists to achieve high performance at little cost. However, tuning the BLAS in isolation misses opportunities for memory optimization that result from composing multiple subprograms. Because it is not practical to create a library of all BLAS combinations, we have developed a domain-specific compiler that generates them on demand. In this paper, we describe a novel algorithm for compiling linear algebra kernels and searching for the best combination of optimization choices. We also present a new hybrid analytic/empirical method for quickly evaluating the profitability of each optimization. We report experimental results showing speedups of up to 130% relative to the GotoBLAS on an AMD Opteron and up to 137% relative to MKL on an Intel Core 2.

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