Recursion from Cyclic Sharing

Abstract

We have studied models of both higher-order acyclic sharing theories (Chapter 4) and (first-order) cyclic sharing theories (Chapter 6), as two orthogonal extensions of (first-order) acyclic sharing theories (Chapter 3). Now we are ready to deal with the most interesting setting in this thesis: the combination of the higher-order extension with cyclic sharing. The consequence is more than an ad hoc combination of two ideas. First, we get a wider class of models of recursive computation, which properly contains the traditional models such as domain theoretic examples but also some nontraditional examples, which are suitable for explaining recursive computation from resource-sensitive cyclic computation, i.e. recursion from cyclic sharing. Second, our theory is closely related to cyclic lambda calculi (lambda calculi with cyclic sharing) which have been studied as a foundation for practical implementation of functional recursive computation. Therefore our result relates a rich class of models and practically interesting calculi: cyclic lambda calculi serve as languages for our new class of models of recursion, while our models serve as a semantic counterpart of cyclic lambda calculi.