A categorical framework for congruence of applicative bisimilarity in higher-order languages

Tom Hirschowitz (1) , Ambroise Lafont (2)

1 LAMA - Laboratoire de Mathématiques
2 UNSW - University of New South Wales [Sydney]

Applicative bisimilarity is a coinductive characterisation of observational equivalence in call-by-name lambda-calculus, introduced by Abramsky in 1990. Howe (1989) gave a direct proof that it is a congruence. In previous work with Borthelle (2020), we abstract over this result by proposing a categorical framework for specifying operational semantics, in which we prove that (an abstract analogue of) applicative bisimilarity is automatically a congruence. However, the framework presents a few infelicities: (1) it requires a non-trivial refinement of the standard approach of Fiore, Plotkin, and Turi (1999) based on monoid algebras for specifying syntax with variable binding; (2) it relies on so-called prebisimulations instead of the more standard notion of bisimulation by lifting; (3) one of the axioms, called weak compositionality, feels ad hoc; (4) the proofs involve directed unions of relations leading to quite a few painful inductions. In this paper, we rectify all of these deficiencies. In particular, a notable novelty is that the so-called Howe closure is defined as an initial monoid algebra in a category of spans. Finally, the familiality/cellularity axiom of the previous framework is now viewed as a mere sufficient condition for the main hypothesis, preservation of functional bisimulations.

Domaines: Logique en informatique [cs.LO]

Tom Hirschowitz, Ambroise Lafont. A categorical framework for congruence of applicative bisimilarity in higher-order languages. 2020. ⟨hal-02966439v1⟩