Four imprints of Belnap's useful four-valued logic in computer science

Abstract

We review four areas of theoretical computer science which share technical or philosophical ideas with the work of Belnap on his useful four-valued logic. Perhaps surprisingly, the inspiration by Belnap-Dunn logic is acknowledged only in the study of d-frames. The connections of Belnap's work and linear logic, Blame Calculus or the study of LVars are not openly admitted. The key to three of these connections with Belnap's work go via the twist-product representation of bilattices. On the one hand, it allows us to view a large class of models of linear logic as based on Belnap-Dunn logic. On the other hand, d-frames admit two twist-product representation theorems and, also, the key theorem of Blame Calculus is essentially a twist-product representation theorem too, albeit with a strong proof-theoretic flavour.