Certain principles are fundamental to operational semantics, regardless of the languages or idioms involved. Such principles include rule-based definitions and proof techniques for congruence results. We formulate these principles in the general context of categorical logic. From this general formulation we recover precise results for particular language idioms by interpreting the logic in particular categories. For instance, results for first-order calculi, such as CCS, arise from considering the general results in the category of sets. Results for languages involving substitution and name generation, such as the π-calculus, arise from considering the general results in categories of sheaves and group actions. As an extended example, we develop a tyft/tyxt-like rule format for open bisimulation in the π-calculus.

We propose a simple order-theoretic generalization of set-theoretic inductive definitions. This generalization covers inductive, co-inductive and bi-inductive definitions and is preserved by abstraction. This allows the structural operational semantics to describe simultaneously the finite/terminating and infinite/diverging behaviors of programs. This is illustrated on the structural bifinitary small/big-step trace/relational/operational semantics of the call-by-value λ-calculus.

Information and Computation journal homepage: www.elsevier.com/locate/ic

A novel, general approach is proposed to proving the compositionality of process equivalences on languages defined by Structural Operational Semantics (SOS). The approach, based on modal logic, is inspired by the simple observation that if the set of formulas satisfied by a process can be derived from the corresponding sets for its subprocesses, then the logical equivalence is a congruence. Striving for generality, SOS rules are modeled categorically as bialgebraic distributive laws for some notions of process syntax and behaviour, and modal logics are modeled via coalgebraic polyadic modal logic. Compositionality is proved by providing a suitable notion of behaviour for the logic together with a dual distributive law, reflecting the one modeling the SOS specification. Concretely, the dual laws may appear as SOS-like rules where logical formulas play the role of processes, and their behaviour models logical decomposition over process syntax. The approach can be used either to proving compositionality for specific languages or for defining SOS congruence formats.