Dimension Inference under Polymorphic

Numeric types can be given polymorphic dimension parameters , in order to avoid dimension errors and unit errors. The most general dimensions can be inferred automatically. It has been observed that polymorphic recursion is more important for the dimensions than for the proper types. We show that, under polymorphic recursion, type inference amounts to syntactic semi-uniication of proper types, followed by equational semi-uniication of dimensions. Syntactic semi-uniication is unfortunately undecidable, although there are procedures that work well in practice, and proper types given by the programmer can be checked. However, the dimensions form a vector space (provided that their exponents are rational numbers). We give a polynomial-time algorithm that decides if a semi-uniication problem in a vector space can be solved and, if so, returns a most general semi-uniier. 1 Introduction We will combine three good things as far as possible: dimension types, polymorphic recursion, and automatic type inference. Types with dimension parameters can be inferred in two steps: rst you infer the types proper, then you infer the dimensions. Under polymorphic recursion, the rst step is known to be undecidable, but there are partial procedures that seem to work in practice, and types that are given by the programmer can be checked. We will simply assume that for some program, the rst step has been solved (perhaps not with the most general typing), and we study only the remaining step of dimension inference. It will turn out, somewhat surprisingly, that the most general dimensions can be inferred in polynomial time. The rest of section 1 introduces and motivates the three good things. Section 2 reviews some algebraic concepts that are needed: syntactic and equational (semi-)uniica-tion. Section 3 gives an algorithm for semi-uniication in the equational theory for vector spaces. Section 4 motivates that the algorithm can be used to infer dimensions under polymorphic recursion; the reasoning mirrors Henglein's 6]. Section 5 discusses the relative merits of integers and rational numbers as dimension exponents; the integers make the dimensions form a freely generated Abelian group, whereas the rational numbers make them form a vector space. Section 6 discusses how to react to the undecidability of syntactic semi-uniication.