On the reaction time of some synchronous systems

This paper presents an investigation of the notion of reaction time in some synchronous systems. A state-based description of such systems is given, and the reaction time of such systems under some classic composition primitives is studied. Reaction time is shown to be non-compositional in general. Possible solutions are proposed, and applications to verification are discussed. This framework is illustrated by some examples issued from studies on real-time embedded systems.

💡 Research Summary

The paper investigates the notion of reaction time in a class of synchronous systems, providing a formal model, analyzing its properties under composition, and discussing verification implications.

Modeling framework
Synchronous systems are modeled as labeled transition systems (LTS) inspired by Moore machines. An LTS S = ⟨In, Out, Q, E, out, q₀⟩ consists of a finite set of inputs In, a finite set of outputs Out, a set of states Q, a total transition relation E ⊆ Q × In × Q, an output labeling function out : Q → Out, and an initial state q₀. The system proceeds in logical rounds; at each round the internal state is updated as a deterministic (or nondeterministic) function of the current input and the previous state, and the observable output is a function of the current state only. This abstraction captures the semantics of data‑flow languages such as Lustre and PsyC.

Reactivity and bisimilarity
The authors define a state q to be reactive if there exist two distinct inputs a₁, a₂ ∈ In such that the successors q₁ and q₂ reached by these inputs are non‑bisimilar (denoted q₁ ≁ q₂). Bisimilarity is the standard equivalence relation on LTS: two states are bisimilar if they have identical output labels and can match each other’s transitions step‑by‑step. Thus, reactivity corresponds to the existence of an observable functional dependency between inputs and future behaviour.

Separators and observable effects
When two states p and q are non‑bisimilar, a separator is a finite input word w that yields two runs from p and q producing different output traces. The pair of output traces (o₁, o₂) generated by the separator constitutes an observable effect. The first index n where o₁