Factoring an Adequacy Proof (Preliminary Report)

Abstract

This paper contributes to the methodology of using metalogics for reasoning about programming languages. As a concrete example we consider a fragment of ML corresponding to call-by-value PCF and translate it into a metalogic which contains (amongst other types) computation types and a fixpoint type. The main result is a soundness property (*): if the denotations of two programs are provably equal in the metalogic, they have the same operationally observable behaviour. As usual, this follows from a computational adequacy result. In early notes, Plotkin showed how such proofs could be factored into two stages, the first non-trivial and the second (essentially) routine; our contribution is to rework his suggestion within a new framework. We define a metalogic, which incorporates computation and fixpoint types, and specify a modular translation of the ML fragment. Our proof of (*) factors into two parts. First, the term language of the metalogic is equipped with an operational semantics and a (generic) computational adequacy result obtained. Second, a simple syntactic argument establishes a correspondence between the operational behaviour of an object program and of its denotation. The first part is not routine but is proved once and for all. The second is a detailed but essentially trivial calculation that is easily adaptable to other object languages. Such a factored proof is important because it promises to scale up more easily than a monolithic one. We show that it may be adapted to an object language with call-by-name functions and one with a simple exception mechanism.