Higher-Order Bialgebraic Semantics
This work addresses a foundational problem in programming language semantics for researchers and practitioners dealing with higher-order languages, though it is incremental as it builds on existing first-order frameworks.
The paper tackles the challenge of proving compositionality in higher-order languages by extending Turi and Plotkin's bialgebraic abstract GSOS framework to this setting, resulting in a general compositionality theorem that applies to systems like combinatory logics and the λ-calculus.
Compositionality proofs in higher-order languages are notoriously involved, and general semantic frameworks guaranteeing compositionality are hard to come by. In particular, Turi and Plotkin's bialgebraic abstract GSOS framework, which provides off-the-shelf compositionality results for first-order languages, so far does not apply to higher-order languages. In the present work, we develop a theory of abstract GSOS specifications for higher-order languages, in effect transferring the core principles of Turi and Plotkin's framework to a higher-order setting. In our theory, the operational semantics of higher-order languages is represented by certain dinatural transformations that we term \emph{(pointed) higher-order GSOS laws}. We give a general compositionality result that applies to all systems specified in this way and discuss how compositionality of combinatory logics and the $λ$-calculus w.r.t.\ a strong variant of Abramsky's applicative bisimilarity are obtained as instances.