Type matching, type-graphs, and the Schanuel conjecture

Abstract

This work considers type systems that are defined by type-graphs (tgraphs), which are rooted directed graphs with order among the edges leaving each node. Tgraphs are uniquely mapped into polynomials which, in turn, are each evaluated at a special point to yield an irrational number named the tgraph's magic number . This special point is chosen using the Schanuel conjecture. It is shown that each tgraph can be uniquely represented by this magic number; namely, types are equal if and only if the corresponding magic numbers are equal. Since irrational numbers require infinite precision, the algorithm for generating magic numbers is carried out using a double-precision floating-point approximation. This approximation is viewed as a hashing scheme, mapping the infinite domain of the irrational numbers into finite computer words. The proposed hashing scheme was investigated experimentally, with the conclusion that it is a good and practical hashing method. In tests involving over a million randomly chosen tgraphs, we have not encountered a single collision. We conclude that this method for representation and management of types is practical, and offers novel possibilities for enforcing strict type matching at link time among separately compiled modules.