Towards a Higher-Order Mathematical Operational Semantics

TL;DR

This paper extends the abstract GSOS framework to higher-order languages, providing a new compositionality theory applicable to languages like the lambda calculus.

Contribution

It develops a higher-order abstract GSOS specification framework, enabling compositional semantics for higher-order languages.

Findings

Provides a general compositionality theorem for higher-order systems.

Applies the framework to the SKI calculus and lambda calculus.

Achieves compositionality w.r.t. a variant of Abramsky's applicative bisimilarity.

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