Generic Top-down Discrimination∗

We introduce the notion of discrimination as a generalization of both sorting and partitioning and show that discriminators (discrimination functions) can be defined generically, by structural recursion on order and equivalence expressions denoting a rich class of total preorders and equivalence relations, respectively. Discriminators improve the asymptotic performance of generic comparison-based sorting and partitioning, yet do not expose more information than the underlying ordering relation, respectively equivalence. For a large class of order and equivalence expressions, including all standard orders for first-order recursive types, the discriminators execute in worst-case linear time. The generic discriminators can be coded compactly using list comprehensions, with order expressions specified using Generalized Algebraic Data Types (GADTs). We give some examples of the uses of discriminators, including a new most-significant-digit lexicographic sorting algorithm and type isomorphism with an associative-commutative operator. Full source code of discriminators and their applications is included. We argue discriminators should be basic operations for primitive and abstract types with equality. The basic multiset discriminator for references, originally due to Paige et al., is shown to be both efficient and fully abstract : it finds all duplicates of all references occurring in a list in linear time without leaking information about their representation. In particular, it behaves deterministically in the presence of garbage collection and nondeterministic heap allocation even when references are represented as raw machine addresses. In contrast, having only a binary equality test as in ML requires Θ(n) time, and allowing hashing for performance reasons as in Java, makes execution nondeterministic and complicates garbage collection. ∗This work has been partially supported by the Danish Research Council for Nature and Universe (FNU) under the grant Applications and Principles of Programming Languages (APPL). See http://www.diku.dk/hjemmesider/ansatte/henglein/src/gtd.zip