A sound metalogical semantics for input/output effects

Abstract

We study the longstanding problem of semantics for input/output (I/O) expressed using side-effects. Our vehicle is a small higher-order imperative language, with operations for interactive character I/O and based on ML syntax. Unlike previous theories, we present both operational and denotational semantics for I/O effects. We use a novel labelled transition system that uniformly expresses both applicative and imperative computation. We make a standard definition of bisimilarity and prove it is a congruence using Howe's method. Next, we define a metalogical type theory M in which we may give a denotational semantics to O.M generalises Crole and Pitts' FIX-logic by adding in a parameterised recursive datatype, which is used to model I/O. M comes equipped both with judgements of equality of expressions, and an operational semantics; M itself is given a domain-theoretic semantics in the category CPPO of cppos (bottom-pointed posets with joins of ω-chains) and Scott continuous functions. We use the CPPO semantics to prove that the equational theory is computationally adequate for the operational semantics using formal approximation relations. The existence of such relations uses key ideas from Pitts' recent work. A monadic-style textual translation into M induces a denotational semantics on O. Our final result justifies metalogical reasoning: if the denotations of two O programs are equal in M then the O programs are in fact operationally equivalent.