Higher-Order Bialgebraic Semantics

TL;DR

This paper extends the bialgebraic abstract GSOS framework to higher-order languages, providing a compositional semantics approach that applies to systems like the lambda calculus and combinatory logic.

Contribution

It develops a higher-order GSOS specification theory, enabling compositional semantics for higher-order languages using dinatural transformations.

Findings

Provides a general compositionality theorem for higher-order systems

Applies to lambda calculus and combinatory logic

Enables modular reasoning about higher-order language semantics

Abstract

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 $\lambda$-calculus w.r.t.\ a strong variant of Abramsky's applicative bisimilarity are obtained as instances.