An Adequate While-Language for Hybrid Computation

Abstract

Hybrid computation harbours discrete and continuous dynamics in the form of an entangled mixture, inherently present in various natural phenomena and in applications ranging from control theory to microbiology. The emergent behaviours bear signs of both computational and physical processes, and thus present difficulties not only in their analysis, but also in describing them adequately in a structural, well-founded way.In order to tackle these issues and, more generally, to investigate hybridness as a dedicated computational phenomenon, we introduce a while-language for hybrid computation inspired by the fine-grain call-by-value paradigm. We equip it with operational and computationally adequate denotational semantics. The latter crucially relies on a hybrid monad supporting an (Elgot) iteration operator that we developed elsewhere. As an intermediate step, we introduce a more lightweight duration semantics furnished with analogous results and based on a new duration monad that we introduce as a lightweight counterpart to the hybrid monad. CCS CONCEPTS• Theory of computation → Timed and hybrid models.

Implementing Hybrid Semantics: From Functional to Imperative

“…the program x :" 1 ; while true do { wait x ; x :" x{2 } (zeno) to output at time instant 2 all possible values in the valuation space (the expression wait t represents a wait call of t time units). More details about Zeno points can be consulted in [18,14].…”

“…In previous work [12,14], we pursued a purely hybrid semantics via a simple deterministic functional language HybCore, with while-loops for which we used Elgot's notion of iteration [8] as the underlying semantic structure. That resulted in a semantics of finite and infinite iteration, corresponding to a refined view of divergence.…”

“…Overview and contributions. Building on our previous work [14], we devise operational and denotational semantics suitable for implementation purposes, and provide a soundness and adequacy theorem relating both these styles of semantics. Results of this kind are well-established yardsticks in the programming language theory [37], and are also beneficial from a practical perspective.…”

“…As mentioned before, in our previous work [14] we introduced a simple functional hybrid language HybCore with operational and denotational monadbased semantics. Here, we work with a similar imperative while-language, whose semantics is given in terms of a global state space of trajectories over R n , which is a commonly used carrier when working with solutions of systems of differential equations.…”

“…For the denotational model, we build on our previous work [12,14] where hybrid programs are interpreted via a suitable monad H, called the hybrid monad and capturing the computational effect of hybridness, following the seminal approach of Moggi [24,25]. Our present semantics is more lightweight and is naturally couched in terms of another monad H S , parametrized by a set S. In our case, as mentioned above, S is the set of trajectories over R n where n is the number of available program variables X .…”