A Proof-Theoretic Assessment of Runtime Type Errors

We analyze the way in which a result concerning the absence of runtime type errors can be expressed when the semantics of a language is described using proof rules in what is sometimes called a natural semantics. We argue that the usual way of expressing such results has conceptual short-comings when compared with similar results for other methods of describing semantics. These short-comings are addressed through a form of operational semantics based on proof rules in what we call a partial proof semantics. A partial proof semantics represents steps of evaluation using proofs with logic variables and subgoals. Such a semantics allows a prooftheoretic expression of the absence of runtime type errors that addresses the problems with such results for natural semantics. We demonstrate that there is a close correspondence between partial proof semantics and a form of structural operational semantics that uses a grammar to describe evaluation contexts and rules for the evaluation of redexes that may appear in such contexts. Indeed, partial proof semantics can be seen as an intermediary between such a description and one using natural semantics. Our study is based on a case treatment for a language called RAVL for Records And Variants Language, which has a polymorphic type system that supports exible programming with records and variants. 1 Runtime Type Errors in Natural Semantics One of the primary purposes for imposing a type discipline on a programming language is to ensure the absence of certain forms of data incompatibilities that might occur at runtime in the evaluation of an ill-typed program. A characterisitic instance of such a problem occurs when a procedure call is made on an actual parameter that fails to have the proper form demanded of the formal parameter of the procedure. Such a guarantee of runtime safety can often be expressed precisely and proved rigorously for a clean language design. One of the earliest examples of such a treatment appeared in Milner's paper [Mil78] introducing the type system that forms the basis for the Standard Meta-Language (SML) [MTH90, MT91]. Similar results have been stated and proved for many subsequent language designs involving types and semantic speci cations. The goal of this paper is to study some of the relationships between various approaches to such results and use this perspective as a basis for describing a new form of operational speci cation for which it is possible to provide a proof-theoretic expression of the guarantees obtained from the type correctness of a program.