Continuity as a computational effect

Abstract

The original purpose of component-based development was to provide techniques to master complex software, through composition, reuse and parametrisation. However, such systems are rapidly moving towards a level in which software becomes prevalently intertwined with (continuous) physical processes. A possible way to accommodate the latter in component calculi relies on a suitable encoding of continuous behaviour as (yet another) computational effect.This paper introduces such an encoding through a monad which, in the compositional development of hybrid systems, may play a role similar to the one played by 1+, powerset, and distribution monads in the characterisation of partial, nondeterministic and probabilistic components, respectively. This monad and its Kleisli category provide a universe in which the effects of continuity over (different forms of) composition can be suitably studied.

Hybrid Automata as Coalgebras

“…Finally, a coalgebraic characterisation of hybrid automata makes possible to see them as hybrid components (cf. [NBHM16]), in the spirit of [Bar03,HJ11]. Generally speaking, this sort of component reproduces the black-box perspective here adopted; and the associated calculus brings to hybrid automata several forms of composition operators (e.g., parallel, pipelining, sum), refinement techniques, and wiring mechanisms, as well as the corresponding algebraic laws.…”

Limits in categories of Vietoris coalgebras

“…With the natural transformation above, it becomes straightforwad to consider the non-deterministic bouncing ball in a topological setting. Actually, it can be shown to be a coalgebra nxt, out : S → VS × HO First, the map out : S → HO was already shown to be continuous in [NBHM16]. Then, observe that the map nxt : S → VS can be rewritten as a composite…”

“…Due to the gravitional effect (g), it falls into the ground and then bounces back up, losing, for example, half of its kinetic energy. As the documents [NBHM16,NB16] show, such a behaviour can be described coalgebraically, with the help of the functor defined below.…”

“Keep definition, change category” — A practical approach to state-based system calculi

“…This helps in some situations but is far from providing a general answer, calling for further research possibly in connection to recent advances in categorial quantum physics and monoidal categories [9]. A fully fledged, coalgebraic trace semantics for probabilistic, component oriented software systems will call for subdistributions and, more generally, measure theory [23,39] in particular if applied to hybrid systems [35].…”