On probabilistic stable event structures

💡 Research Summary

The paper addresses the longstanding challenge of integrating probability with true concurrency models, focusing on stable event structures (SES) as a foundational framework. Traditional verification approaches have largely relied on interleaving semantics, which impose a global clock and linearize concurrent events into arbitrary sequences. While this simplifies reasoning, it leads to state‑space explosion and obscures the intrinsic independence of concurrently occurring actions. In contrast, true concurrency models such as Petri nets represent each component with its own local state and local time, allowing a more natural expression of probabilistic independence.

Existing probabilistic models for concurrency have typically adopted the interleaving view or have restricted themselves to conflict‑free settings. Notable prior work includes probabilistic extensions of event structures using “clusters” (mutually conflicting events) and the introduction of “branching cells” as the atomic units of nondeterministic choice. Abbes and Benveniste showed that branching cells provide the finest granularity at which probabilistic independence can be guaranteed, but they also proved that any finer decomposition inevitably loses independence. However, these approaches either ignore the phenomenon of “confusion” (situations where an event may have several possible causes that can be simultaneously enabled) or handle only very simple forms of conflict. Confusion is pervasive in realistic Petri nets and makes it difficult to assign probabilities that respect both causality and concurrency.

To overcome these limitations, the authors propose conflict‑driven stable event structures (CD‑SES), an extension of ordinary stable event structures. A CD‑SES still enforces the stability condition—each event possesses a unique minimal enabling set—but relaxes the requirement that an event have a single cause, allowing multiple alternative cause sets provided they are mutually exclusive. Moreover, the paper introduces a jump‑free restriction, which prevents abrupt changes in the set of causes when extending configurations. This restriction ensures that the causal structure evolves monotonically, a property crucial for defining a coherent probability measure.

The probabilistic semantics are built on the space of maximal configurations (\Omega(E)) of an event structure (E = (E, \le, #)). The authors define a σ‑algebra generated by “shadows” (S(v) = {\omega \in \Omega(E) \mid \omega \supseteq v}) for each (possibly infinite) configuration (v). A probability measure (P) on this σ‑algebra yields a likelihood function (p(v) = P(S(v))). The key insight is to base the notion of choice on first‑hand conflicts, i.e., immediate conflicts (#_{\mu}) that are not inherited from conflicting causes. By focusing on these immediate conflicts, the authors isolate the truly independent decision points in the system.

Next, the paper introduces stopping prefixes and R‑stopped configurations. A stopping prefix is a minimal prefix that contains all immediate conflicts; an R‑stopped configuration is obtained by iteratively extending a configuration with finite, stopped sub‑configurations. Each step of this iterative construction yields a branching cell, which is an initial stopping prefix of the residual event structure after the current configuration. Under the assumptions of pre‑regularity (finitely many events enabled by any finite configuration) and local finiteness (every event appears in some finite stopping prefix), every maximal configuration is R‑stopped. Consequently, the set of R‑stopped configurations (W(E)) provides a decomposition of the execution space into a sequence of independent choice layers.

The authors prove that the probability of an R‑stopped configuration factorises as the product of the probabilities of its constituent branching cells. This factorisation mirrors the intuitive idea that concurrent choices should be probabilistically independent, and it gives an operational method for computing probabilities without enumerating the entire configuration tree. The construction also aligns with the notion of compact unfoldings of Petri nets, previously introduced by the first author. Compact unfoldings retain a bounded amount of backward conflict information while avoiding the combinatorial blow‑up of full unfoldings. By interpreting compact unfoldings as CD‑SES, the paper bridges the gap between high‑level Petri net models and low‑level probabilistic semantics.

In the related‑work discussion, the paper situates its contribution among several strands: (1) probabilistic extensions of event structures via clusters and powerdomains; (2) domain‑theoretic approaches using continuous valuations; (3) branching‑cell based frameworks that guarantee independence at the coarsest possible granularity; and (4) recent attempts at probabilistic true‑concurrency models that either ignore confusion or lack a robust logical layer. The presented CD‑SES framework subsumes many of these earlier models while explicitly handling confusion, thereby offering a more expressive foundation for reasoning about probabilistic concurrent systems.

Finally, the paper outlines a research agenda: extending the probabilistic semantics to a full temporal logic suitable for Petri nets, exploiting the compact unfolding representation to develop model‑checking algorithms, and investigating categorical properties (e.g., adjunctions between prime and stable event structures) that could facilitate compositional reasoning. By establishing a mathematically rigorous yet practically motivated model of probabilistic stable event structures, the work paves the way for verification tools that can faithfully capture both the nondeterministic and stochastic aspects of distributed, concurrent systems.