An Analysis of the Core-ML Language: Expressive Power and Type Reconstruction

Core-ML is a basic subset of most functional programming languages. It consists of the simply typed (or monomorphic)-calculus, simply typed equality over atomic constants, and let as the only poly-morphic construct. We present a synthesis of recent results which characterize this \toy" language's expressive power as well as its type reconstruction (or type inference) problem. More speciically: (1) Core-ML can express exactly the ELEMENTARY queries, where a program input is a database encoded as a-term and a query program is a-term whose application to the input normalizes to the output database. In addition, it is possible to express all the PTIME queries so that this normalization process is polynomial in the input size. (2) The polymorphism of let can be explained using a simple algorithmic reduction to monomorphism, and provides exibility, without aaecting expressibility. Algorithms for type reconstruction ooer the additional convenience of static typing without type declarations. Given polymorphism, the price of this convenience is an increase in complexity from linear-time in the size of the program typed (without let) to completeness in exponential-time (with let). (3) Fragments of Core-ML, based on the order of functionalities used, can be fairly expressive. For example, order 5 suuces for the PTIME queries, even without let or equality. Also, for each xed order, type reconstruction is polynomial in program size. Programming by using low order functionalities and type reconstruction is common in functional languages. Thus, our analysis partly explains the wide use and eeciency of such programming practices.