Free theorems in the presence of seq

Abstract

Parametric polymorphism constrains the behavior of pure functional programs in a way that allows the derivation of interesting theorems about them solely from their types, i.e., virtually for free. Unfortunately, the standard parametricity theorem fails for nonstrict languages supporting a polymorphic strict evaluation primitive like Haskell's seq. Contrary to the folklore surrounding seq and parametricity, we show that not even quantifying only over strict and bottom-reflecting relations in the ∀-clause of the underlying logical relation -and thus restricting the choice of functions with which such relations are instantiated to obtain free theorems to strict and total ones -is sufficient to recover from this failure. By addressing the subtle issues that arise when propagating up the type hierarchy restrictions imposed on a logical relation in order to accommodate the strictness primitive, we provide a parametricity theorem for the subset of Haskell corresponding to a Girard-Reynolds-style calculus with fixpoints, algebraic datatypes, and seq. A crucial ingredient of our approach is the use of an asymmetric logical relation, which leads to "inequational" versions of free theorems enriched by preconditions guaranteeing their validity in the described setting. Besides the potential to obtain corresponding preconditions for standard equational free theorems by combining some new inequational ones, the latter also have value in their own right, as is exemplified with a careful analysis of seq's impact on familiar program transformations.

Fast and loose reasoning is morally correct

“…Adding polymorphism would give us an easy way to transform free theorems [Rey83,Wad89] from the set-theoretic side (well, perhaps not settheoretic [Rey84]) to the domain-theoretic one. It should be interesting to compare those results to other work involving free theorems and seq [JV04].…”

Exploiting algebra/coalgebra duality for program fusion extensions

“…Another relevant approach is stream fusion [4]. Stream fusion represents recursive functions as stream processors 5 The laws as expressed here are only valid for programs which do not use the seq operator [18]. which can be composed and merged into a single recursive function thus achieving deforestation.…”

Parametricity, type equality, and higher-order polymorphism

“…If we wished to use these results to reason about Haskell implementations of gcast, we must extend our model to include more-in particular, general recursion and recursive types (Melliès & Vouillon, 2005;Johann & Voigtländer, 2004;Appel & McAllester, 2001;Ahmed, 2006b;Crary & Harper, 2007a). We believe that the techniques developed here are independent of those for advanced language features.…”