Towards a Higher-Order Mathematical Operational Semantics
This work addresses a fundamental problem in programming language semantics for researchers and practitioners by providing a theoretical framework to simplify compositionality proofs, though it is incremental as it builds on existing first-order methods.
The paper tackles the challenge of proving compositionality for higher-order languages by extending Turi and Plotkin's abstract GSOS framework to this setting, resulting in a general compositionality theorem that applies to systems like the SKI calculus and 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 has been successfully applied to obtain 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 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 the SKI calculus and the $λ$-calculus w.r.t. a strong variant of Abramsky's applicative bisimilarity are obtained as instances.