Strictness Properties of Lazy Algebraic Datatypes

A new construction of a finite set of strictness properties for any lazy algebraic datatype is presented. The construction is based on the categorical view of the solutions to the recursive domain equations associated with such types as initial algebras. We then show how the initial algebra induction principle can be used to reason about the entailment relation on the chosen collection of properties. We examine the lattice of properties given by our construction for the type nlist of lazy lists of natural numbers and give proof rules which extend the conjunctive strictness logic of [2] to a language including the type nlist. 1 I n t r o d u c t i o n This paper concerns the problem of extending an ideal-based strictness analysis for a PCF-like language (e.g. the abstract interpretation of [4] or the strictness logic of [2, 6]) to a language which includes lazy algebraic datatypes, such as lazy lists or trees. We start by giving a brief overview of ideal-based strictness analysis, using the language of strictness logic 1. A more comprehensive account can be found in [1]. At each type a of our language AT we define a propositional theory L~ = (L~, <~) where L~ i s a set of propositions and _<oC L a x L~ is the (finitely axiomatised) entailment relation. Each proposition r is interpreted as a non-empty Scott-closed subset (i.e. an ideal) [r of the domain Da which interprets the type a in the standard denotational semantics for AT. For d E D~, r E s write d ~ r for d E [r We require soundness of s with respect to its intended interpretation, so r < r implies that [r _ [r and we can sometimes get the converse (completeness) too. For decidability, we require that the Lindenbaum algebra s of the theory s be finite. We shall assume tha t s contains at least the propositions t ~ (with It I ---D~) *Author's address: University of Cambridge, Computer Laboratory, New Museums Site, Pembroke Street, Cambridge CB2 3QG, UK. Email: Nick.Benton0cl. cam. ac .uk. Research supported by the Cambridge Philosophical Society and a SERC Fellowship. 1Most of the present work can, however, be easily reinterpreted in terms of a more traditional abstract interpretation.