Population protocols are a model of computation in which an arbitrary number of indistinguishable finite-state agents interact in pairs. The goal of the agents is to decide by stable consensus whether their initial global configuration satisfies a given property, specified as a predicate on the set of configurations. The state complexity of a predicate is the number of states of a smallest protocol that computes it. Previous work by Blondin et al. has shown that the counting predicates ≥ have state complexity O (log ) for leaderless protocols and O (log log ) for protocols with leaders. We obtain the first non-trivial lower bounds: the state complexity of ≥ is Ω(log log log ) for leaderless protocols, and the inverse of a non-elementary function for protocols with leaders.
Broadcast consensus protocols (BCPs) are a model of computation, in which anonymous, identical, finite-state agents compute by sending/receiving global broadcasts. BCPs are known to compute all number predicates in $$\mathsf {NL}=\mathsf {NSPACE}(\log n)$$ NL = NSPACE ( log n ) where n is the number of agents. They can be considered an extension of the well-established model of population protocols. This paper investigates execution time characteristics of BCPs. We show that every predicate computable by population protocols is computable by a BCP with expected $$\mathcal {O}(n \log n)$$ O ( n log n ) interactions, which is asymptotically optimal. We further show that every log-space, randomized Turing machine can be simulated by a BCP with $$\mathcal {O}(n \log n \cdot T)$$ O ( n log n · T ) interactions in expectation, where T is the expected runtime of the Turing machine. This allows us to characterise polynomial-time BCPs as computing exactly the number predicates in $$\mathsf {ZPL}$$ ZPL , i.e. predicates decidable by log-space, randomised Turing machine with zero-error in expected polynomial time where the input is encoded as unary.
Population protocols are a model of distributed computation in which finitestate agents interact randomly in pairs. A protocol decides for any initial configuration whether it satisfies a fixed property, specified as a predicate on the set of configurations. The state complexity of a predicate is smallest number of states of any protocol deciding that predicate. For threshold predicates of the form x ≥ k, with k constant, prior work has shown that they have state complexity Θ(log log k) if the protocol is extended with leaders. For ordinary protocols it is only known to be in Ω(log log k) ∩ O(log k). We close this remaining gap by showing that it is Θ(log log k) as well, i.e. we construct protocols with O(n) states deciding x ≥ k with k ≥ 2 2 n .
Population protocols are a model of computation in which an arbitrary number of indistinguishable finite-state agents interact in pairs. The goal of the agents is to decide by stable consensus whether their initial global configuration satisfies a given property, specified as a predicate on the set of all initial configurations. The state complexity of a predicate is the number of states of a smallest protocol that computes it. Previous work by Blondin et al. has shown that the counting predicates x ≥ η have state complexity O(log η) for leaderless protocols and O(log log η) for protocols with leaders. We obtain the first non-trivial lower bounds: the state complexity of x ≥ η is Ω(log log log η) for leaderless protocols, and the inverse of a non-elementary function for protocols with leaders.
Esparza and Reiter have recently conducted a systematic comparative study of models of distributed computing consisting of a network of identical finite-state automata that cooperate to decide if the underlying graph of the network satisfies a given property.The study classifies models according to four criteria, and shows that twenty-four initially possible combinations collapse into seven equivalence classes with respect to their decision power, i.e. the properties that the automata of each class can decide. However, Esparza and Reiter only show (proper) inclusions between the classes, and so do not characterise their decision power. In this paper we do so for labelling properties, i.e. properties that depend only on the labels of the nodes, but not on the structure of the graph. In particular, majority (whether more nodes carry label than ) is a labelling property. Our results show that only one of the seven equivalence classes identified by Esparza and Reiter can decide majority for arbitrary networks. We then study the expressive power of the classes on bounded-degree networks, and show that three classes can. In particular, we present an algorithm for majority that works for all bounded-degree networks under adversarial schedulers, i.e. even if the scheduler must only satisfy that every node makes a move infinitely often, and prove that no such algorithm can work for arbitrary networks.
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