Abstract

We consider a polymorphic type system (System F) with recursive types for the lambda calculus with constants. We use Leivant’s notion of rank to delimit the boundaries for decidable and undecidable type inference in our system. More precisely, we show that type inference in our system is undecidable at rank k ≥ 3. Similar results are known to hold for System F without recursive types. Our undecidability result is obtained by a reduction from a particular adaptation of the semi-unification problem (so-called “regular semi-unification”) whose undecidability is, interestingly, obtained by methods totally different from those used in the case of standard (or finite) semi-unification. Type inference for a rank-1 restriction of our system is known to be decidable. A simple modification to the way unification is performed (so-called “regular unification”) is all that is required. We also conjecture that type inference at rank-2 is decidable and equivalent to finding regular solutions for instances of a particular form of semi-unification known as “acyclic semi-unification”.

Citations