A Syntactic View of Computational Adequacy

Abstract

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.

Galois connecting call-by-value and call-by-name

“…V e e e ′ → V e to y. V e ′ to z. z ' force y The variable x is bound to a thunk of the recursive computation, so recursion is done by forcing x. (This is not the only way to add recursion to CBPV [DCL18], but is the most convenient for our purposes.) Of course, by adding recursion we lose normalization (but the semantics is still deterministic).…”

Intuitionistic fixed point logic

“…We define a predicate S(x) expressing that x has a signed digit representation. First, we define the property of being a signed digit, SD Because of (22) one easily sees that p r S(x) holds iff p is an infinite stream of signed digits that represents x, i.e. S(p, x) holds.…”

“…While Escardo works in a typed setting and concerns incremental computation on the interval domain, our result is untyped and computes arbitrary infinite data built from constructors. There exists a rich literature on computational adequacy covering, for example, typed lambda calculi with various effects [57,48], denotational semantics based on games or categories [27,66], and axiomatic approaches [22,30].…”