Familial monads and structural operational semantics

Abstract

We propose a categorical framework for structural operational semantics, in which we prove that under suitable hypotheses bisimilarity is a congruence. We then refine the framework to prove soundness of bisimulation up to context, an efficient method for reducing the size of bisimulation relations. Finally, we demonstrate the flexibility of our approach by reproving known results in three variants of the-calculus.

Cellular Monads from Positive GSOS Specifications

“…In this paper, we have introduced the familial approach to programming language theory [12] at the rather concrete level of generalised labelled transition systems ( Γ A ). Notably, we have recalled the notions of cellular monad and compositional algebra, and recalled that bisimilarity is always a congruence in a compositional algebra for a cellular monad (Lemma 4.3).…”

“…This result constitutes a first generic tool for constructing instances of the framework of [12]. However, its scope is rather limited, and we plan to refine the construction to cover other formats like tyft/tyxt [11].…”

“…In recent work [12], a new approach to abstract operational semantics was proposed, and its expressive power was demonstrated by proving for the first time an abstract soundness result for bisimulation up to context in the presence of variable binding. Bisimulation up to context is an efficient technique [23,Chapter 6] for proving program equivalences, which had previously been proved correct in the bialgebraic setting [2], but only without binding.…”

“…One of the main results [12,Corollary 4.30] is that if the considered algebra is compositional, in the sense that its structure map T (X ) → X is a functional bisimulation, and if the monad T satisfies an additional condition, then bisimilarity is indeed a congruence. The latter condition is called T s -familiality in [12], but we will here call it cellularity, because it is a specialisation of cellularity in the sense of [10] to familial functors. As mentioned above, a second main result [12,Corollary 5.15] is that under a different condition called T ∨ s -familiality, bisimulation up to context is sound.…”

“…The latter condition is called T s -familiality in [12], but we will here call it cellularity, because it is a specialisation of cellularity in the sense of [10] to familial functors. As mentioned above, a second main result [12,Corollary 5.15] is that under a different condition called T ∨ s -familiality, bisimulation up to context is sound.…”