Inferring Recursive Data Types

This paper contains ve results on the problem of inferring types. The rst is that type inference over recursive types with unions and data constructors can be done in cubic time using a ow analysis. The second is a general theorem characterizing the time complexity of bottom-up logic programs. The O(n 3) running time of the ow analysis is a corollary of this bottom-up run time theorem. The third is that for shallow case statements typability by the semantic types of Aiken, Wim-mers and Lakshman is equivalent to typability by recursive types and hence can be determined by ow analysis. The fourth is that, even for rst order programs of arity one, typability by recursive types is PSPACE hard for polymorphic programs. The nal result is that for any xed bound on order and arity Hindley-Milner typability can be determined in pseudo-linear time, i.e., O(nn(n)) where is the inverse Ackerman function. The last two results suggest that let-polymorphism over simple types is fundamentally more tractable than let-polymorphism over recursive types.