Improvements for Free

"Theorems for Free!" (Wadler, FPCA 1989) is a slogan for a technique that allows to derive statements about functions just from their types. So far, the statements considered have always had a purely extensional flavor: statements relating the value semantics of program expressions, but not statements relating their runtime (or other) cost. Here we study an extension of the technique that allows precisely statements of the latter flavor, by deriving quantitative theorems for free. After developing the theory, we walk through a number of example derivations. Probably none of the statements derived in those simple examples will be particularly surprising to most readers, but what is maybe surprising, and at the very least novel, is that there is a general technique for obtaining such results on a quantitative level in a principled way. Moreover, there is good potential to bring that technique to bear on more complex examples as well. We turn our attention to short-cut fusion (Gill et al., FPCA 1993) in particular.Comment: In Proceedings QAPL 2011, arXiv:1107.074

The Sigma-Semantics: A Comprehensive Semantics for Functional Programs

A comprehensive semantics for functional programs is presented, which generalizes the well-known call-by-value and call-by-name semantics. By permitting a separate choice between call-by value and call-by-name for every argument position of every function and parameterizing the semantics by this choice we abstract from the parameter-passing mechanism. Thus common and distinguishing features of all instances of the sigma-semantics, especially call-by-value and call-by-name semantics, are highlighted. Furthermore, a property can be validated for all instances of the sigma-semantics by a single proof. This is employed for proving the equivalence of the given denotational (fixed-point based) and two operational (reduction based) definitions of the sigma-semantics. We present and apply means for very simple proofs of equivalence with the denotational sigma-semantics for a large class of reduction-based sigma-semantics. Our basis are simple first-order constructor-based functional programs with patterns

Partial type constructors: Or, making ad hoc datatypes less ad hoc

This work is licensed under a Creative Commons Attribution 4.0 International License.Functional programming languages assume that type constructors are total. Yet functional programmers know better: counterexamples range from container types that make limiting assumptions about their contents (e.g., requiring computable equality or ordering functions) to type families with defining equations only over certain choices of arguments. We present a language design and formal theory of partial type constructors, capturing the domains of type constructors using qualified types. Our design is both simple and expressive: we support partial datatypes as first-class citizens (including as instances of parametric abstractions, such as the Haskell Functor and Monad classes), and show a simple type elaboration algorithm that avoids placing undue annotation burden on programmers. We show that our type system rejects ill-defined types and can be compiled to a semantic model based on System F. Finally, we have conducted an experimental analysis of a body of Haskell code, using a proof-of-concept implementation of our system; while there are cases where our system requires additional annotations, these cases are rarely encountered in practical Haskell code

Late Data Layout: Unifying Data Representation Transformations

Values need to be represented differently when interacting with certain language features. For example, an integer has to take an object-based representation when interacting with erased generics, although, for performance reasons, the stack-based value representation is better. To abstract over these implementation details, some programming languages choose to expose a unified high-level concept (the integer) and let the compiler choose its exact representation and insert coercions where necessary. This pattern appears in multiple language features such as value classes, specialization and multi-stage programming: they all expose a unified concept which they later refine into multiple representations. Yet, the underlying compiler implementations typically entangle the core mechanism with assumptions about the alternative representations and their interaction with other language features. In this paper we present the Late Data Layout mechanism, a simple but versatile type-driven generalization that subsumes and improves the state-of-the-art representation transformations. In doing so, we make two key observations: (1) annotated types conveniently capture the semantics of using multiple representations and (2) local type inference can be used to consistently and optimally introduce coercions. We validated our approach by implementing three language features as Scala compiler extensions: value classes, specialization (using the miniboxing representation) and a simplified multi-stage programming mechanism