The Semantics of Fair Recursion with Divergence

We recast Milner's work on SCCS, a calculus for nite but unbounded delay based on SCCS, by giving a denotational semantics for admissibility of innnite computations on a biinite domain K. Using Abramsky's SFP domain D for bisimulation we obtain a fully abstract model in D K for an operational pre-order which generalizes Milner's fortiication. Our preorder includes divergence and its restriction to nite behavior corresponds to Abramsky's nitary pre-order. By virtue of biiniteness of D K we obtain a Stone dual at the level of objects. Since Milner's delay operators and turn out to correspond to the greatest, respectively least, xed point on K, we consequently enrich SCCS with an additional recursive binding modeled as the greatest xed point. For the body 1x + p, the new recursive binding imposes nite delay, whereas the ordinary recursion admits innnite, as well as nite, delay of p. We deene a notion of admissibility and a denotational semantics for this new calculus SCCS. The canonical translation of SCCS into SCCS preserves admissibility and the denotational semantics. We discuss the fundamental diiculties in proving full abstraction of the denotational semantics for SCCS. Finally, we compare the expressive power of SCCS with that of SCCS. For certain nite state processes in SCCS the semantics on K computes !-regular languages.