13,687 research outputs found

    Towards Efficient Verification of Population Protocols

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    Population protocols are a well established model of computation by anonymous, identical finite state agents. A protocol is well-specified if from every initial configuration, all fair executions reach a common consensus. The central verification question for population protocols is the well-specification problem: deciding if a given protocol is well-specified. Esparza et al. have recently shown that this problem is decidable, but with very high complexity: it is at least as hard as the Petri net reachability problem, which is EXPSPACE-hard, and for which only algorithms of non-primitive recursive complexity are currently known. In this paper we introduce the class WS3 of well-specified strongly-silent protocols and we prove that it is suitable for automatic verification. More precisely, we show that WS3 has the same computational power as general well-specified protocols, and captures standard protocols from the literature. Moreover, we show that the membership problem for WS3 reduces to solving boolean combinations of linear constraints over N. This allowed us to develop the first software able to automatically prove well-specification for all of the infinitely many possible inputs.Comment: 29 pages, 1 figur

    On Convergence and Threshold Properties of Discrete Lotka-Volterra Population Protocols

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    In this work we focus on a natural class of population protocols whose dynamics are modelled by the discrete version of Lotka-Volterra equations. In such protocols, when an agent aa of type (species) ii interacts with an agent bb of type (species) jj with aa as the initiator, then bb's type becomes ii with probability P_ijP\_{ij}. In such an interaction, we think of aa as the predator, bb as the prey, and the type of the prey is either converted to that of the predator or stays as is. Such protocols capture the dynamics of some opinion spreading models and generalize the well-known Rock-Paper-Scissors discrete dynamics. We consider the pairwise interactions among agents that are scheduled uniformly at random. We start by considering the convergence time and show that any Lotka-Volterra-type protocol on an nn-agent population converges to some absorbing state in time polynomial in nn, w.h.p., when any pair of agents is allowed to interact. By contrast, when the interaction graph is a star, even the Rock-Paper-Scissors protocol requires exponential time to converge. We then study threshold effects exhibited by Lotka-Volterra-type protocols with 3 and more species under interactions between any pair of agents. We start by presenting a simple 4-type protocol in which the probability difference of reaching the two possible absorbing states is strongly amplified by the ratio of the initial populations of the two other types, which are transient, but "control" convergence. We then prove that the Rock-Paper-Scissors protocol reaches each of its three possible absorbing states with almost equal probability, starting from any configuration satisfying some sub-linear lower bound on the initial size of each species. That is, Rock-Paper-Scissors is a realization of a "coin-flip consensus" in a distributed system. Some of our techniques may be of independent value

    On the Necessary Memory to Compute the Plurality in Multi-Agent Systems

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    We consider the Relative-Majority Problem (also known as Plurality), in which, given a multi-agent system where each agent is initially provided an input value out of a set of kk possible ones, each agent is required to eventually compute the input value with the highest frequency in the initial configuration. We consider the problem in the general Population Protocols model in which, given an underlying undirected connected graph whose nodes represent the agents, edges are selected by a globally fair scheduler. The state complexity that is required for solving the Plurality Problem (i.e., the minimum number of memory states that each agent needs to have in order to solve the problem), has been a long-standing open problem. The best protocol so far for the general multi-valued case requires polynomial memory: Salehkaleybar et al. (2015) devised a protocol that solves the problem by employing O(k2k)O(k 2^k) states per agent, and they conjectured their upper bound to be optimal. On the other hand, under the strong assumption that agents initially agree on a total ordering of the initial input values, Gasieniec et al. (2017), provided an elegant logarithmic-memory plurality protocol. In this work, we refute Salehkaleybar et al.'s conjecture, by providing a plurality protocol which employs O(k11)O(k^{11}) states per agent. Central to our result is an ordering protocol which allows to leverage on the plurality protocol by Gasieniec et al., of independent interest. We also provide a Ω(k2)\Omega(k^2)-state lower bound on the necessary memory to solve the problem, proving that the Plurality Problem cannot be solved within the mere memory necessary to encode the output.Comment: 14 pages, accepted at CIAC 201
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