Strategic polymorphism requires just two combinators!

In previous work, we introduced the notion of functional strategies: first-class generic functions that can traverse terms of any type while mixing uniform and type-specific behaviour. Functional strategies transpose the notion of term rewriting strategies (with coverage of traversal) to the functional programming paradigm. Meanwhile, a number of Haskell-based models and combinator suites were proposed to support generic programming with functional strategies. In the present paper, we provide a compact and matured reconstruction of functional strategies. We capture strategic polymorphism by just two primitive combinators. This is done without commitment to a specific functional language. We analyse the design space for implementational models of functional strategies. For completeness, we also provide an operational reference model for implementing functional strategies (in Haskell). We demonstrate the generality of our approach by reconstructing representative fragments of the Strafunski library for functional strategies.Comment: A preliminary version of this paper was presented at IFL 2002, and included in the informal preproceedings of the worksho

Foundations for structured programming with GADTs

GADTs are at the cutting edge of functional programming and become more widely used every day. Nevertheless, the semantic foundations underlying GADTs are not well understood. In this paper we solve this problem by showing that the standard theory of data types as carriers of initial algebras of functors can be extended from algebraic and nested data types to GADTs. We then use this observation to derive an initial algebra semantics for GADTs, thus ensuring that all of the accumulated knowledge about initial algebras can be brought to bear on them. Next, we use our initial algebra semantics for GADTs to derive expressive and principled tools — analogous to the well-known and widely-used ones for algebraic and nested data types — for reasoning about, programming with, and improving the performance of programs involving, GADTs; we christen such a collection of tools for a GADT an initial algebra package. Along the way, we give a constructive demonstration that every GADT can be reduced to one which uses only the equality GADT and existential quantification. Although other such reductions exist in the literature, ours is entirely local, is independent of any particular syntactic presentation of GADTs, and can be implemented in the host language, rather than existing solely as a metatheoretical artifact. The main technical ideas underlying our approach are (i) to modify the notion of a higher-order functor so that GADTs can be seen as carriers of initial algebras of higher-order functors, and (ii) to use left Kan extensions to trade arbitrary GADTs for simpler-but-equivalent ones for which initial algebra semantics can be derive

Strategic polymorphism requires just two combinators!

In previous work, we introduced the notion of functional strategies: first-class generic functions that can traverse terms of any type while mixing uniform and type-specific behaviour. Functional strategies transpose the notion of term rewriting strategies (with coverage of traversal) to the functional programming paradigm. Meanwhile, a number of Haskell-based models and combinator suites were proposed to support generic programming with functional strategies. In the present paper, we provide a compact and matured reconstruction of functional strategies. We capture strategic polymorphism by just two primitive combinators. This is done without commitment to a specific functional language. We analyse the design space for implementational models of functional strategies. For completeness, we also provide an operational reference model for implementing functional strategies (in Haskell). We demonstrate the generality of our approach by reconstructing representative fragments of the Strafunski library for functional strategies

Foundations For Structured Programming With GADTs

GADTs are at the cutting edge of functional programming and be-come more widely used every day. Nevertheless, the semantic foundations underlying GADTs are not well understood. In this paper we solve this problem by showing that the standard theory of data types as carriers of initial algebras of functors can be extended from algebraic and nested data types to GADTs. We then use this observation to derive an initial algebra semantics for GADTs, thus ensuring that all of the accumulated knowledge about initial algebras can be brought to bear on them. Next, we use our initial algebra semantics for GADTs to derive expressive and principled tools —analogous to the well-known and widely-used ones for algebraic and nested data types — for reasoning about, programming with, and improving the performance of programs involving, GADTs; we christen such a collection of tools for a GADT an initial algebra package. Along the way, we give a constructive demonstration that every GADT can be reduced to one which uses only the equality GADT and existential quanti?cation. Although other such reductions exist in the literature, ours is entirely local, is independent of any particular syntactic presentation of GADTs, and can be implemented in the host language, rather than existing solely as a met theoretical artifact. The main technical ideas underlying our approach are (i) to modify the notion of a higher-order functor so that GADTs can be seen as carriers of initial algebras of higher-order functors, and (ii) to use left Kan extensions to trade arbitrary GADTs for simpler-but-equivalent ones for which initial algebra semantics can be derived

Modular and type-safe definition of Attribute Grammars with AspectAG

AspectAG is a Haskell-embedded domain-specific language (EDSL) that encodes first-class attribute grammars (AGs). AspectAG ensures the wellformedness of AGs at compile time by using extensible records and predicates encoded using old-fashioned type-level programming features, such as multiparameter type classes and functional dependencies. AspectAG suffers the usual drawbacks of EDSLs: when type errors occur they usually do not deliver error messages that refer to domain terms, but to the host language. Often, implementation details of the EDSL are leaked in those messages. The use of type-level programming techniques makes the situation worse since type-level abstraction mechanisms are quite poor. Additionally, old-fashioned type-level programs are untyped at type-level, which is inconsistent with the general approach of strongly-typed functional programming. By using modern Haskell extensions and techniques we propose a reworked version of AspectAG that tackles those weaknesses. New AG definitions are safer, both at the level of types and at the level of kinds. Furthemore, a set of identified domain-specific errors are reported with DSL-oriented messages. To achieve this, we define and use a framework for manipulating type errors that can be used in any EDSL. We show the pragmatics of AspectAG by defining languages and extending them both with new syntax and semantics. We use MateFun, a purelyfunctional language used to teach mathematics as a case study.AspectAG es un lenguaje de dominio específico embebido (EDSL) que codifica gramáticas de atributos (AGs) como ciudadanos de primera clase. AspectAG garantiza la buena formación de las AGs en tiempo de compilación por medio del uso de registros extensibles y predicados, codificados gracias al uso de características antiguas de programación a nivel de tipos, como clases multiparámetro y dependencias funcionales. AspectAG sufre las desventajas usuales de los EDSLs: cuando ocurren errores de tipado, los mensajes de error reportados no se expresan en términos del dominio, sino del lenguaje anfitrión. También es usual que detalles de implementación del EDSL se vean filtrados en estos mensajes. El uso de técnicas de programación a nivel de tipos agrava la situación porque los mecanismos de abstracción a nivel de tipos son pobres. Ademas, las técnicas de programación a nivel de tipos usadas en AspectAG son esencialmente no tipadas, lo que es inconsistente con nuestro enfoque de tipado fuerte. Usando extensiones modernas al sistema de tipos de Haskell, proponemos una nueva versión de la biblioteca AspectAG, abordando los problemas antes mencionados. Las nuevas definiciones de AGs son mas seguras tanto a nivel de tipado como a nivel de kinds (tipado a nivel de tipos). Ademas, un conjunto identificado de errores específicos del dominio son reportados con mensajes referentes al mismo. Para lograr esto, definimos y utilizamos un framework para manipular errores de tipado, que puede ser aplicado a cualquier EDSL. Mostramos la pragmática de AspectAG definiendo lenguajes y extendiéndoles con nueva sintaxis y con nueva semántica. Utilizamos el lenguaje MateFun, un lenguaje funcional puro utilizado para enseñar matemáticas como caso de estudio

System FωμF^\mu_\omega with Context-free Session Types

We study increasingly expressive type systems, from $F^\mu$ -- an extension of the polymorphic lambda calculus with equirecursive types -- to $F^{\mu;}_\omega$ -- the higher-order polymorphic lambda calculus with equirecursive types and context-free session types. Type equivalence is given by a standard bisimulation defined over a novel labelled transition system for types. Our system subsumes the contractive fragment of $F^\mu_\omega$ as studied in the literature. Decidability results for type equivalence of the various type languages are obtained from the translation of types into objects of an appropriate computational model: finite-state automata, simple grammars and deterministic pushdown automata. We show that type equivalence is decidable for a significant fragment of the type language. We further propose a message-passing, concurrent functional language equipped with the expressive type language and show that it enjoys preservation and absence of runtime errors for typable processes.Comment: 38 pages, 13 figure

Boxes Go Bananas: Encoding Higher-Order Abstract Syntax With Parametric Polymorphism (Extended Version)

Higher-order abstract syntax is a simple technique for implementing languages with functional programming. Object variables and binders are implemented by variables and binders in the host language. By using this technique, one can avoid implementing common and tricky routines dealing with variables, such as capture-avoiding substitution. However, despite the advantages this technique provides, it is not commonly used because it is difficult to write sound elimination forms (such as folds or catamorphisms) for higher-order abstract syntax. To fold over such a datatype, one must either simultaneously define an inverse operation (which may not exist) or show that all functions embedded in the datatype are parametric. In this paper, we show how first-class polymorphism can be used to guarantee the parametricity of functions embedded in higher-order abstract syntax. With this restriction, we implement a library of iteration operators over data-structures containing functionals. From this implementation, we derive fusion laws that functional programmers may use to reason about the iteration operator. Finally, we show how this use of parametric polymorphism corresponds to the Schürmann, Despeyroux and Pfenning method of enforcing parametricity through modal types. We do so by using this library to give a sound and complete encoding of their calculus into System Fω. This encoding can serve as a starting point for reasoning about higher-order structures in polymorphic languages