Distributed Synthesis in Continuous Time

We introduce a formalism modelling communication of distributed agents strictly in continuous-time. Within this framework, we study the problem of synthesising local strategies for individual agents such that a specified set of goal states is reached, or reached with at least a given probability. The flow of time is modelled explicitly based on continuous-time randomness, with two natural implications: First, the non-determinism stemming from interleaving disappears. Second, when we restrict to a subclass of non-urgent models, the quantitative value problem for two players can be solved in EXPTIME. Indeed, the explicit continuous time enables players to communicate their states by delaying synchronisation (which is unrestricted for non-urgent models). In general, the problems are undecidable already for two players in the quantitative case and three players in the qualitative case. The qualitative undecidability is shown by a reduction to decentralized POMDPs for which we provide the strongest (and rather surprising) undecidability result so far.

💡 Research Summary

The paper introduces a formalism for modelling distributed systems that communicate strictly in continuous time. The authors build on Interactive Markov Chains (IMCs) and define Distributed IMCs as a collection of modules, each equipped with private actions, synchronisation actions (hand‑shake style), and exponentially distributed delay transitions. A key modeling choice is to restrict all delay distributions to continuous, exponential ones, which allows the use of standard continuous‑time Markov chain techniques while keeping the model expressive enough for realistic timing behaviour.

In this setting the interleaving scheduler—present in discrete‑time models to resolve the order of concurrent actions—loses its power: because the probability that two independent delay transitions fire at exactly the same instant is zero, the scheduler can only affect the order of actions that are already synchronised, not which actions occur or what the agents observe. Consequently, the synthesis problem depends solely on each agent’s local timed history and the global clock.

Two fundamental reachability problems are studied. The Existence problem asks whether there is a profile of local strategies such that, for every possible scheduler, the probability of eventually reaching a designated target set T (and staying there for a non‑zero time) is at least a given threshold p. The Value problem asks whether the supremum over strategies of the infimum over schedulers of this probability can be made arbitrarily close to p. When p = 1 the problems are called qualitative; otherwise they are quantitative.

The authors first prove that, under mild assumptions on the winning condition, the interleaving scheduler indeed has no influence in continuous‑time distributed synthesis. They then explore the decidability landscape. For non‑urgent models—where agents may postpone their decisions arbitrarily long— they show that the quantitative value problem for two players can be solved in exponential time. The non‑urgency condition enables agents to encode information in the amount of time they wait before committing to a synchronisation action; this “time‑based signalling” reduces the problem to a finite‑state MDP that can be analysed with standard EXPTIME algorithms.

In contrast, when the non‑urgency restriction is lifted, or when three or more agents are involved, the problems become undecidable. The paper proves that the quantitative existence problem is already undecidable for two players, and that the qualitative existence problem is undecidable for three or more players. The undecidability proofs rely on a reduction to decentralised partially observable Markov decision processes (DEC‑POMDPs). The authors strengthen known results by showing that even with an infinite horizon, the qualitative existence problem for DEC‑POMDPs is undecidable already with two players—a result that surpasses the decidability of the analogous problem for classic (single‑observer) POMDPs. By embedding a DEC‑POMDP into a Distributed IMC with one additional “scheduler” player, they transfer this hardness to the distributed synthesis setting.

The technical development includes a precise definition of strategies (measurable functions from local histories to distributions over local choices), schedulers (functions from global histories and choices to distributions over synchronisation labels), and the induced probability measure over plays via a cylinder construction. The paper also discusses how to avoid Zeno behaviours by forbidding cycles that involve the same synchronisation label.

Overall, the work delineates a clear boundary: continuous‑time modelling eliminates interleaving nondeterminism, but the combination of local observation limits and the number of agents re‑introduces undecidability. Only in the restricted class of non‑urgent two‑player games does the synthesis problem become tractable (EXPTIME). This insight has practical implications: designers of timed distributed protocols should either enforce non‑urgency (allowing time‑based coordination) or keep the number of cooperating agents very small if automated strategy synthesis is required. The paper’s contributions lie both in the novel continuous‑time synthesis framework and in the strengthened undecidability results for DEC‑POMDPs, which are of independent interest to the verification community.