Abstract. In [6] J. Laird has shown that an infinitary sequential extension of PCF has a fully abstract model in his category of locally boolean domains (introduced in [8]). In this paper we introduce an extension SPCF ∞ of his language by recursive types and show that it is universal for its model in locally boolean domains. Finally we consider an infinitary target language CPS ∞ for (the) CPS translation (of [16]) and show that it is universal for a model in locally boolean domains which is constructed like Dana Scott’s D ∞ where D = 1

Game semantics is a way of characterizing programming languages and logical calculi intensionally by interpreting proofs or programs as strategies for interacting with the environment; this process of interaction can be thought of as playing a two-person game. Games can capture precisely the behaviour of higher-order programs in sequential languages, in a fashion which is very difficult or impossible for traditional forms of denotational semantics. This precision can be expressed formally via the properties of full abstraction, full completeness or universality. A remarkable feature of game semantics is its flexibility: games have been used to give fully abstract or fully complete models not only of logics such as linear logic [3] and functional languages such as PCF [1, 12, 38], but also languages with powerful imperative features such as references (state) [4, 2], continuations [32] and exceptions [13]. By contrast, in domain theory programs are modelled extensionally using sets (with certain order-theoretic structure) and functions (preserving that structure). Domains have formed the basis for much research in denotational semantics by virtue of their conceptual simplicity, and now form a wide-ranging mathematical theory. The search for order-theoretic characterizations of higher-order sequentiality has led to the development of several candidate notions, such as stability [5] and strong stability [6], and was arguably the driving force behind most work in denotational semantics for some years. However, a significant obstacle to the use of domains to reason about programming languages has been a lack of accurate models of imperative and concurrent features of the sort provided by game semantics. A possible basis for connecting the intensional world of games with the extensional world of domains was the remarkable observation of Cartwright and Felleisen [8]

Abstract. We investigate the possible ways of ordering terms of ground type in a nondeterministic (or deterministic) language that contains erroneous behaviours such as divergence, crash or deadlock. We see that the ordering at boolean type, called a “boolean precongruence”, is key: it determines the ordering at other ground types, and induces a contextual preorder. We examine the circumstances in which amb is monotone, and in which the ordering at Sierpinski type or even zero type suffices. Each boolean precongruence gives a way of lifting relations, leading to a power-poset construction. We obtain a notion of simulation, and give general conditions for when a