The Impact of seq on Free Theorems-Based Program Transformations

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, standard parametricity results — including so-called free theorems — fail for nonstrict languages supporting a polymorphic strict evaluation primitive such as Haskell’s seq . A folk theorem maintains that such results hold for a subset of Haskell corresponding to a GirardReynolds calculus with fixpoints and algebraic datatypes even when seq is present provided the relations which appear in their derivations are required to be bottom-reflecting and admissible. In this paper we show that this folklore is incorrect, but that parametricity results can be recovered in the presence of seq by restricting attention to left-closed, total, and admissible relations instead. The key novelty of our approach is the asymmetry introduced by left-closedness, which leads to “inequational” versions of standard parametricity results together with preconditions guaranteeing their validity even when seq is present. We use these results to derive criteria ensuring that both equational and inequational versions of short cut fusion and related program transformations based on free theorems hold in the presence of seq . This work is based on the paper Free Theorems in the Presence of seq , which appeared in 31st Symposium on Principles of Programming Languages (POPL’04), Proceedings, c ©ACM Press, 2004. http://doi.acm.org/10.1145/964001.964010 Research supported in part by the National Science Foundation under grant CCF-0429072. Research supported by the “Deutsche Forschungsgemeinschaft” under grant KU 1290/2-4. 2 P. Johann, J. Voigtländer / The Impact of seq on Free Theorems-Based Program Transformations