A Semantic Characterization of Full Abstraction for Typed Lambda Calculi

in [Plotkin] for P C F enriched with 'parallel or' it turned out that all the finite elements of the model were definable in the enriched language. To my knowledge, this is the first model which is fully abstract and whose all finite elements need not be definable still. We leave it open to find out the properties which the ground domains should have so that all the finite elements of M are definable, (see articulate domains of [Milner]) We also leave open whether a better semantic characterization can be found when L is known to possess the addtional properties like sequent ia l ly (see [Berry]). Let us recollect the assumption on which Theorem 5.3 was proved. Basic Assumption 1. The ground domains D are algebraic, complete lattices. 2. All the finite elements of the ground domains as well as their finite projections are definable. Tha t we are using complete lattices instead of consistently complete epos is a technical problem which we would neglect at the moment. Otherwise our assumption is exactly the same as the assumption Milner used to construct a syntactic fully abstract , extensional model for £ . This surprising coincidence makes us speculate tha t the Basic Assuption might in some sense be a necessary condition to guarantee the existence of a fully abstract , extensional model for £ . We can justify it as follows. It seems reasonable to assume tha t if some fully abstract , extensional model for £ exists then one can construct such a model syntactically. After all the syntactic construction is the easiest one, and the whole point of Milner's syntactic construction in [Milner] was to show tha t a fully abstract , extensional model for L exists. If one can show that the Basic Assumption is in some sense necessary for Milner's syntactic construction to succeed, then it follows tha t the Basic Assumption is in some sense a necessary condition for the existence of a fully abstract , extensional model for £ . It is then not surprising tha t we hit upon the same condition in the semantic characterization of full abstraction; you don' t expect to find such a characterization when a fully abstract model does not exist! What is important : this paper gives us a definite step in the opposite direction; whenever the Basic Assumption is t rue, a semantic characterization can be found. Then we can say: // there exists some fully abstract, extensional, algebraic model for t then such a model can be found semantically. It is a rather sweeping s tatement , but one can not resist the temptat ion! I would expect something of this sort, with possibly a variant of the Basic Assumption, to be true. Tha t we leave as an open question for the moment . Though we did not show it, the O predicates defined in this paper can be used to show tha t O and A are semantically equivalent. In fact such inclusive predicates were introduced in [Milne] and [Reynolds] with exactly this aim in mind: to show the semantic equivalence between operational and denotational semantics. The techniques developed in these papers were mainly meant for the cases when the domains under consideration were reflexive. It should not then come as a surprise if the technique developed in this paper could be extended to obtain a semantic characterization of full abstraction even when domains under consideration axe reflexive. In fact tha t is the case.