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Abstract. TypeScript is an extension of JavaScript intended to enable easier development of large-scale JavaScript applications. While every JavaScript program is a TypeScript program, TypeScript offers a module system, classes, interfaces, and a rich gradual type system. The intention is that TypeScript provides a smooth transition for JavaScript programmers-well-established JavaScript programming idioms are supported without any major rewriting or annotations. One interesting consequence is that the TypeScript type system is not statically sound by design. The goal of this paper is to capture the essence of TypeScript by giving a precise definition of this type system on a core set of constructs of the language. Our main contribution, beyond the familiar advantages of a robust, mathematical formalization, is a refactoring into a safe inner fragment and an additional layer of unsafe rules.

Inheritance is a fundamental concept in object-oriented programming, allowing new classes to be defined in terms of old classes. When used with care, inheritance is an essential tool for objectoriented programmers. Thus, for those interested in developing formal verification techniques, the treatment of inheritance is of paramount importance. Unfortunately, inheritance comes in a number of guises, all requiring subtle techniques.To address these subtleties, most existing verification methodologies typically adopt one of two restrictions to handle inheritance: either (1) they prevent a derived class from restricting the behaviour of its base class (typically by syntactic means) to trivialize the proof obligations; or (2) they allow a derived class to restrict the behaviour of its base class, but require that every inherited method must be reverified. Unfortunately, this means that typical inheritance-rich code either cannot be verified or results in an unreasonable number of proof obligations.In this paper, we develop a separation logic for a core objectoriented language. It allows derived classes which override the behaviour of their base class, yet supports the inheritance of methods without reverification where this is safe. For each method, we require two specifications: a static specification that is used to verify the implementation and direct method calls (in Java this would be with a super call); and a dynamic specification that is used for calls that are dynamically dispatched; along with a simple relationship between the two specifications. Only the dynamic specification is involved with behavioural subtyping. This simple separation of concerns leads to a powerful system that supports all forms of inheritance with low proof-obligation overheads. We both formalize our methodology and demonstrate its power with a series of inheritance examples.

Moggi's computational lambda calculus is a metalanguage for denotational semantics which arose from the observation that many different notions of computation have the categorical structure of a strong monad on a cartesian closed category. In this paper we show that the computational lambda calculus also arises naturally as the term calculus corresponding (by the Curry–Howard correspondence) to a novel intuitionistic modal propositional logic. We give natural deduction, sequent calculus and Hilbert-style presentations of this logic and prove strong normalisation and confluence results.

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