Monadic augment and generalised short cut fusion

Abstract

Monads are commonplace programming devices that are used to uniformly structure computations; in particular, they are often used to mimic the effects of impure features such as state, error handling, and I/O. This paper further develops the monadic programming paradigm by investigating the extent to which monadic computations can be optimised by using generalisations of short cut fusion to eliminate monadic structures whose sole purpose is to “glue together” monadic program components. Ghani, Uustalu, and Vene have recently shown that every inductive type has an associated build combinator and an associated short cut fusion law. They have also used the notion of a parameterised monad to describe those monads that give rise to inductive types, and have shown that the standard augment combinators and cata/augment fusion rules for algebraic data types can be generalised to fixed points of all parameterised monads. We revisit these augment combinators and generalised short cut fusion rules for such types but consider them from a functional programming perspective, rather than a categorical one. In addition to making the category-theoretic ideas of Ghani, Uustalu, and Vene more easily accessible to a wider audience of functional programmers, we demonstrate their practical applicability by developing nontrivial application programs and performing modest benchmarking on them. We also show how the cata/augment rules can serve as the basis for deriving additional generic fusion laws, thus opening the way for an algebra of fusion. Finally, we offer deep theoretical insights, arguing that the augment combinators are monadic in nature, and thus that the cata/build and cata/augment rules are arguably the best generally applicable fusion rules obtainable.

A family of syntactic logical relations for the semantics of Haskell-like languages

“…For the last case, we show it by contradiction. Assume Proof: The proof uses the fact that from (13) follows that for every M ∈ Untyped ({x}):…”

“…The key result to be proved for every logical relation constructed in this way is that every function expressible in the underlying language is related to itself by the relational interpretation of its type. This parametricity theorem, or certain generalizations of it, can then be used, for example, to derive useful algebraic laws (so-called "free theorems") about polymorphic functions solely from their types [8], or to establish the semantic correctness of efficiency-improving program transformations [9][10][11][12][13]. But for all such applications, the usefulness in practice depends on a good fit between the semantics of the functional language of interest and that of the typically reduced formal calculus for which parametricity results are proved.…”

Fold-based fusion as a library: a generative programming pearl

“…Hinze et al provide a theoretical framework that unifies the above-mentioned fusion algorithms [6]. Ghani et al generalise foldr/build fusion to other inductive datatypes [4]. Although in this paper we only treat lists, sums and pairs, their work suggests that our technique can be extended to other inductive datatypes.…”

Foundations for structured programming with GADTs

“…It is well known [Ghani et al (2005), Ghani et al (2003), Takano & Meijer (1995)] that every inductive type has an associated initial algebra package. If f is a Haskell functor, then the associated inductive type M f and its associated fold and build combinators can be implemented generically in Haskell by…”