A Syntactic View of Computational Adequacy

When presenting a denotational semantics of a language with recursion, it is necessary to show that the semantics is computationally adequate, i.e. that every divergent term denotes the “bottom” element of a domain. We explain how to view such a theorem as a purely syntactic result. Any theory (congruence) that includes basic laws and is closed under an infinitary rule that we call “rational continuity” has the property that every divergent term is equated with the divergent constant. Therefore, to prove a model adequate, it suffices to show that it validates the basic laws and the rational continuity rule. While this approach was inspired by the categorical, ordered framework of Abramsky et. al., neither category theory nor order is needed. The purpose of the paper is to present this syntactic result for call-bypush-value extended with term-level recursion and polymorphic types. Our account begins with PCF, then includes sum types, then moves to call-by-push-value, and finally includes polymorphic types.