A coalgebraic treatment of Structural Operational Semantics (SOS) is provided by certain natural transformations: distributive laws of endofunctors (modelling  process syntax) on other endofunctors (modelling system behaviour). Models of such laws are bialgebras, which can be seen as coalgebras for behaviour functors lifted to categories of algebras for syntax functors.

On the other hand, coalgebraic modal logic is defined by natural transformations that link system behaviour with logical syntax along a (contravariant) adjunction between two categories. These natural transformations combine behaviour and syntax in single composite endofunctors on the slice category of the adjunction, and models of logics are coalgebras for these lifted endofunctors.

In the framework of logical distributive laws, these two aspects are combined in a study of bialgebras on slice categories of adjunctions. In more elementary terms, SOS specifications are combined with modal logics for processes in a well structured manner that ensures compositionality of logical equivalence relations with respect to process syntax. Technically, this requires a suitable notion of logical behaviour, and an SOS-like distributive law where logical formulas play the role of processes.

In this talk, I briefly describe the basics of the logical distributive law approach, cast in the setting of slice categories of adjunctions. I also say how it relates to modal logic decomposition techniques used in the SOS community to prove compositionality results.