ICE-Based Refinement Type Discovery for Higher-Order Functional Programs

We propose a method for automatically finding refinement types of higher-order function programs. Our method is an extension of theIceframework of Garg et al. for finding invariants. In addition to the usual positive and negative samples in machine learning, theirIceframework usesimplication constraints, which consist of pairs (x, y) such that ifxsatisfies an invariant, so doesy. From these constraints,Iceinfersinductiveinvariants effectively. We observe that the implication constraints in the originalIceframework are not suitable for finding invariants of recursive functions with multiple function calls. We thus generalize the implication constraints to those of the form({x(1),...,x(k)},y), which eans that if all of x(1),...x(k) satisfy an invariant, so doesy. We extend their algorithms for inferring likely invariants from samples, verifying the inferred invariants, and generating new samples. We have implemented our method and confirmed its effectiveness through experiments.