Minimal Typings in Atomic Subtyping

Abstract

This paper studies the problem of simplifying typings and the size-complexity of most general typings in typed programming languages with atomic subtyping. We define a notion of minimal typings relating all typings which are equivalent with respect to instantiation. The notion of instance is that of Fuh and Mishra [13], which supports many interesting simplifications. We prove that every typable term has a unique minimal typing, which is the logically most succinct among all equivalent typings. We study completeness properties, with respect to our notion of minimality, of well-known simplification techniques. Drawing upon these results, we prove a tight exponential lower bound for the worst case dag-size of constraint sets as well as of types in most general typings. To the best of our knowledge, the best previously proven lower bound was linear. 1 Introduction Subtyping is a fundamental idea in type systems for programming languages, which can in principle be integrated into standar...