Step-indexed semantic interpretations of types were proposed as an alternative to purely syntactic proofs of type safety using subject reduction. The types are interpreted as sets of values indexed by the number of computation steps for which these values are guaranteed to behave like proper elements of the type. Building on work by Ahmed, Appel and others, we introduce a step-indexed semantics for the imperative object calculus of Abadi and Cardelli. Providing a semantic account of this calculus using more `traditional', domain-theoretic approaches has proved challenging due to the combination of dynamically allocated objects, higher-order store, and an expressive type system. Here we show that, using step-indexing, one can interpret a rich type discipline with object types, subtyping, recursive and bounded quantified types in the presence of state.
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