Towards a mathematical operational semantics

Abstract

We present a categorical theory of 'well-behaved' operational semantics which aims at complementing the established theory of domains and denotational semantics to form a coherent whole. It is shown that, if the operational rules of a programming language can be modelled as a natural transformation of a suitable general form, depending on functorial notions of syntax and behaviour, then one gets the following for free: an operational model satisfying the rules and a canonical, internally fully abstract denotational model which satisfies the operational rules. The theory is based on distributive laws and bialgebras; it specialises to the known classes of well-behaved rules for structural operational semantics, such as GSOS.

Presenting Distributive Laws

“…[34,7]). Such a ρ corresponds to a specification format of operational rules, and is sometimes referred to as a simple SOS specification.…”

On Specification Logics for Algebra-Coalgebra Structures: Reconciling Reachability and Observability

“…Such an abstraction is possible due to a compatibility between computational and observational features in the structures considered. This compatibility, which amounts to computations preserving observational indistinguishability and to observations preserving reachability, is attained using an approach similar to that of [17], where liftings of monads to categories of coalgebras are used to define operational semantics which are well-behaved w.r.t. denotational ones.…”

“…It follows from [17] that, for a combined signature (C, G, F, σ), the category Alg(T σ ) has both an initial and a final object. The initial T σ -algebra provides an observational structure on ground computations, whereas the final T σ -algebra provides a computational structure on abstract states.…”

“…In the second part of the paper, structures comprising both observational and computational features are considered. As in [5], the choice of models incorporating both coalgebraic and algebraic structure, and of the syntax used to specify these models is based on the approach in [17], and results in a compatibility between the two categories of features, in that computations preserve observational indistinguishability whereas observations preserve reachability. The sentences used for specification are the ones employed by the individual frameworks, while the associated notions of satisfaction exploit the previously-mentioned compatibility in order to abstract away unwanted detail (in the form of unreachable and respectively observationally indistinguishable behaviours).…”

Structural Operational Semantics for Continuous State Probabilistic Processes

“…To this end, we plan to apply the bialgebraic framework introduced by Turi and Plotkin [29]. However, in order to port this approach to our setting we have to solve several technical issues, due to the fact that we are working in Meas and using the Giry functor ∆.…”