## 8 Conclusions and Future Work  
In this paper, we first demonstrate that it is impossible to develop an asynchronous local reliable broadcast algorithm that operates under arbitrary initial system configurations where processes act solely based on local information. We then introduce the first asynchronous reliable broadcast algorithm that is able to start in any arbitrary configuration and ensures three properties: premature feedback safety, propagation reliability, and propagation 1-safety. Premature feedback safety guarantees that each neighboring process that has not yet completed the broadcast will always be in a state that only allows it to receive the broadcast from a completed process. Propagation 1-safety guarantees that a spurious message can be sent from a process to another but cannot be transmitted further. On the other hand, propagation reliability guarantees that all system processes receive each legitimate message sent by root process \(r\) and will be propagated as specified without any backtracking.  
If a new uninitialized process joins the network and becomes a neighbor of a completed process, since neighboring processes of a newly joined process cannot locally determine whether or not the newly joined process is a completed process or a non-completed process, premature feedback safety is violated. (As mentioned by Lemma 1).  
The challenge of developing multi-source asynchronous reliable broadcast algorithms that allow simultaneous broadcasts from multiple sources remains unsolved. Similarly, devising asynchronous reliable broadcast algorithms specifically designed for certain regular topologies or applications such as blockchain networks, where further restrictions are present, is another open problem.  
While our paper mentions various applications for the proposed algorithm, a comprehensive examination of the algorithm's practicality and its use in real-time applications through performance evaluations is beyond the scope of this paper. However, we view these as potential future work.  
Experimental studies are needed to assess the scalability of our proposed algorithm in comparison to distributed reset algorithms. This study will focus on the relationship between network size, number of transient faults (initialized processes vs. uninitializedprocesses), and algorithm failure rate. By simulating the scenarios, including perturbation of initialized processes by transient faults, we can evaluate the resilience of the algorithm to transient faults and its scalability and determine its compatibility with applications in the real world.  
Finally, an adaptive (to topology changes) and crash-resilient version of our proposed algorithm is highly desirable and is considered as future work.  
The work presented in this paper was evenly distributed among all authors. Each of the author contributed equally to the design, formal proof, algorithm development, and writing. We collaboratively addressed the challenges of asynchronous reliable broadcast and jointly developed the proposed algorithm and its correctness. All authors have read and agreed to the published version of the manuscript.  
This research was funded by Kuwait University Research Administration, Grant No [EO 06/19].  
No new data were created or analyzed in this study. Data sharing is not applicable to this article.  
The authors declare no conflicts of interest.