This paper analyses some connectives introduced by Stephen Read, known as ‘Bullet connectives’. From the proof-theoretic semantics perspective, these connectives satisfy the requirement of harmony; however, they allow for the looping derivation of a contradiction. Namely, when the operation of detour reduction is applied to such a derivation, it leads to a non-terminating phenomenon. By appealing to Prawtiz’s notion of validity, we argue that such connectives cannot be considered as logical ones. Still, it is possible to assign a computational meaning to such connectives, as several useful computation properties are induced by their harmonious behaviour. In particular, we show that a fixed point operator can be defined by using the rules of a generalised version of the Bullet connectives. Such a generalisation is an instance of (non-terminating) recursive types. Finally, we show that type-free (i.e., untyped) lambda-calculus is interpretable in simply typed lambda-calculus extended with yet another variant of the Bullet connectives, and that the introduction rules of the Bullet connectives are interpretable by Nakano’s modality in typed lambda-calculus. We conclude that such a computational interpretation of the Bullet connectives is possible only if we separate the notion of types from that of propositions, which are meaningful and assertible linguistic entities (in fact, it is larger). Therefore, the computational interpretation we propose cannot be considered an instance of the Curry–Howard correspondence between propositions and types, and between proofs and programs.

This paper studies normalisation by evaluation for typed lambda calculus from a categorical and algebraic viewpoint. The first part of the paper analyses the lambda definability result of Jung and Tiuryn via Kripke logical relations and shows how it can be adapted to unify definability and normalisation, yielding an extensional normalisation result. In the second part of the paper, the analysis is refined further by considering intensional Kripke relations (in the form of Artin–Wraith glueing) and shown to provide a function for normalising terms, casting normalisation by evaluation in the context of categorical glueing. The technical development includes an algebraic treatment of the syntax and semantics of the typed lambda calculus that allows the definition of the normalisation function to be given within a simply typed metatheory. A normalisation-by-evaluation program in a dependently typed functional programming language is synthesised.

Mainstream object-oriented languages often fail to provide complete powerful features altogether, such as, multiple inheritance, dynamic overloading and copy semantics of inheritance. In this paper we present a core object-oriented imperative language that integrates all these features in a formal framework. We define a static type system and a translation of the language into the meta-language λ_object,, in order to account for semantic issues and prove type safety of our proposal.