For those of us that generally live in the world of syntax, semantic proof techniques such as realizability or logical relations/parametricity sometimes feel like magic. Why do they work? At which point in the proof is “the real work” done? Bernardy and Lasson [4] express realizability and parametricity models as syntactic models – but the abstraction/adequacy theorems are still explained as me...

Abstract In this paper, we address the problem of (reactive) realizability specifications theories richer than Booleans, including arithmetic theories. Our approach transforms theory into purely Boolean by (1) substituting literals variables, and (2) computing an additional requirement that captures dependencies between new variables imposed literals. The resulting specification can be passed t...

We employ the notions of ‘sequential function’ and ‘interrogation’ (dialogue) in order to define new partial combinatory algebra structures on sets of functions. These structures are analyzed using J. Longley’s preorder-enriched category of partial combinatory algebras and decidable applicative structures. We also investigate total combinatory algebras of partial functions. One of the results i...

We define a notion of realizability, based on a new assignment of formulas, which does not care for precise witnesses of existential statements, but only for bounds for them. The novel form of realizability supports a very general form of the FAN theorem, refutes Markov’s principle but meshes well with some classical principles, including the lesser limited principle of omniscience and weak Kön...