Abstract

In rendez-vous protocols an arbitrarily large number of indistinguishable finite-state agents interact in pairs. The cut-off problem asks if there exists a number B such that all initial configurations of the protocol with at least B agents in a given initial state can reach a final configuration with all agents in a given final state. In a recent paper [17], Horn and Sangnier prove that the cut-off problem is equivalent to the Petri net reachability problem for protocols with a leader, and in for leaderless protocols. Further, for the special class of symmetric protocols they reduce these bounds to and , respectively. The problem of lowering these upper bounds or finding matching lower bounds is left open. We show that the cut-off problem is -complete for leaderless protocols, -complete for symmetric protocols with a leader, and in for leaderless symmetric protocols, thereby solving all the problems left open in [17].

Highlights

  • While the existence of a reachable configuration that populates a given state with at least one agent is in P, and so well below PSPACE, the existence of a reachable configuration that populates a given state with exactly one agent is as hard as the reachability problem for Petri nets, and so non-elementary [6]

  • We show that the cut-off problem for acyclic Petri nets can be solved in polynomial time

  • [17] Horn and Sangnier introduce symmetric rendez-vous protocols, where sending and receiving a message at each state has the same effect, and show that the cut-off problem is in NP

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Summary

Introduction

Distributed systems are often designed for an unbounded number of participant agents. While the existence of a reachable configuration that populates a given state with at least one agent is in P, and so well below PSPACE, the existence of a reachable configuration that populates a given state with exactly one agent is as hard as the reachability problem for Petri nets, and so non-elementary [6] This fragility makes the analysis of parameterized questions very interesting, and much harder. A leader is a distinguished agent with its own set of states They show that in the presence of a leader the cut-off problem and the reachability problem for Petri nets problems are inter-reducible, which shows that the cut-off problem is in the Ackermannian complexity class Fω [22], and non-elementary [6]. Due to lack of space, full proofs of some of the lemmas can be found in the appendix

Preliminaries
The cut-off problem for acyclic Petri nets
Characterizing acyclic systems with cut-offs
Polynomial time algorithm
The Scaling and Insertion lemmas
The Scaling Lemma
The Insertion Lemma
Characterizing systems with cut-offs
Symmetric rendez-vous protocols
Symmetric protocols with leaders
A non-deterministic polynomial time algorithm
Full Text
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