11 research outputs found

    Hardness of Computing and Approximating Predicates and Functions with Leaderless Population Protocols

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    Population protocols are a distributed computing model appropriate for describing massive numbers of agents with very limited computational power (finite automata in this paper), such as sensor networks or programmable chemical reaction networks in synthetic biology. A population protocol is said to require a leader if every valid initial configuration contains a single agent in a special "leader" state that helps to coordinate the computation. Although the class of predicates and functions computable with probability 1 (stable computation) is the same whether a leader is required or not (semilinear functions and predicates), it is not known whether a leader is necessary for fast computation. Due to the large number of agents n (synthetic molecular systems routinely have trillions of molecules), efficient population protocols are generally defined as those computing in polylogarithmic in n (parallel) time. We consider population protocols that start in leaderless initial configurations, and the computation is regarded finished when the population protocol reaches a configuration from which a different output is no longer reachable. In this setting we show that a wide class of functions and predicates computable by population protocols are not efficiently computable (they require at least linear time), nor are some linear functions even efficiently approximable. It requires at least linear time for a population protocol even to approximate division by a constant or subtraction (or any linear function with a coefficient outside of N), in the sense that for sufficiently small gamma > 0, the output of a sublinear time protocol can stabilize outside the interval f(m) (1 +/- gamma) on infinitely many inputs m. In a complementary positive result, we show that with a sufficiently large value of gamma, a population protocol can approximate any linear f with nonnegative rational coefficients, within approximation factor gamma, in O(log n) time. We also show that it requires linear time to exactly compute a wide range of semilinear functions (e.g., f(m)=m if m is even and 2m if m is odd) and predicates (e.g., parity, equality)

    Efficient size estimation and impossibility of termination in uniform dense population protocols

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    We study uniform population protocols: networks of anonymous agents whose pairwise interactions are chosen at random, where each agent uses an identical transition algorithm that does not depend on the population size nn. Many existing polylog(n)(n) time protocols for leader election and majority computation are nonuniform: to operate correctly, they require all agents to be initialized with an approximate estimate of nn (specifically, the exact value logn\lfloor \log n \rfloor). Our first main result is a uniform protocol for calculating log(n)±O(1)\log(n) \pm O(1) with high probability in O(log2n)O(\log^2 n) time and O(log4n)O(\log^4 n) states (O(loglogn)O(\log \log n) bits of memory). The protocol is converging but not terminating: it does not signal when the estimate is close to the true value of logn\log n. If it could be made terminating, this would allow composition with protocols, such as those for leader election or majority, that require a size estimate initially, to make them uniform (though with a small probability of failure). We do show how our main protocol can be indirectly composed with others in a simple and elegant way, based on the leaderless phase clock, demonstrating that those protocols can in fact be made uniform. However, our second main result implies that the protocol cannot be made terminating, a consequence of a much stronger result: a uniform protocol for any task requiring more than constant time cannot be terminating even with probability bounded above 0, if infinitely many initial configurations are dense: any state present initially occupies Ω(n)\Omega(n) agents. (In particular, no leader is allowed.) Crucially, the result holds no matter the memory or time permitted. Finally, we show that with an initial leader, our size-estimation protocol can be made terminating with high probability, with the same asymptotic time and space bounds.Comment: Using leaderless phase cloc

    The Dynamics and Stability of Probabilistic Population Processes

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    We study here the dynamics and stability of Probabilistic Population Processes, via the differential equations approach. We provide a quite general model following the work of Kurtz [15] for approximating discrete processes with continuous differential equations. We show that it includes the model of Angluin et al. [1], in the case of very large populations. We require that the long-term behavior of the family of increasingly large discrete processes is a good approximation to the long-term behavior of the continuous process, i.e., we exclude population protocols that are extremely unstable such as parity-dependent decision processes. For the general model, we give a sufficient condition for stability that can be checked in polynomial time. We also study two interesting sub cases: (a) Protocols whose specifications (in our terms) are configuration independent. We show that they are always stable and that their eventual subpopulation percentages are actually a Markov Chain stationary distribution. (b) Protocols that have dynamics resembling virus spread. We show that their dynamics are actually similar to the well-known Replicator Dynamics of Evolutionary Games. We also provide a sufficient condition for stability in this case

    Fast Approximate Counting and Leader Election in Populations

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    We study the problems of leader election and population size counting for population protocols: networks of finite-state anonymous agents that interact randomly under a uniform random scheduler. We show a protocol for leader election that terminates in O(logm(n)log2n)O(\log_m(n) \cdot \log_2 n) parallel time, where mm is a parameter, using O(max{m,logn})O(\max\{m,\log n\}) states. By adjusting the parameter mm between a constant and nn, we obtain a single leader election protocol whose time and space can be smoothly traded off between O(log2n)O(\log^2 n) to O(logn)O(\log n) time and O(logn)O(\log n) to O(n)O(n) states. Finally, we give a protocol which provides an upper bound n^\hat{n} of the size nn of the population, where n^\hat{n} is at most nan^a for some a>1a>1. This protocol assumes the existence of a unique leader in the population and stabilizes in Θ(logn)\Theta{(\log{n})} parallel time, using constant number of states in every node, except the unique leader which is required to use Θ(log2n)\Theta{(\log^2{n})} states

    35th Symposium on Theoretical Aspects of Computer Science: STACS 2018, February 28-March 3, 2018, Caen, France

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    A time and space optimal stable population protocol solving exact majority

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    We study population protocols, a model of distributed computing appropriate for modeling well-mixed chemical reaction networks and other physical systems where agents exchange information in pairwise interactions, but have no control over their schedule of interaction partners. The well-studied *majority* problem is that of determining in an initial population of nn agents, each with one of two opinions AA or BB, whether there are more AA, more BB, or a tie. A *stable* protocol solves this problem with probability 1 by eventually entering a configuration in which all agents agree on a correct consensus decision of A\mathsf{A}, B\mathsf{B}, or T\mathsf{T}, from which the consensus cannot change. We describe a protocol that solves this problem using O(logn)O(\log n) states (loglogn+O(1)\log \log n + O(1) bits of memory) and optimal expected time O(logn)O(\log n). The number of states O(logn)O(\log n) is known to be optimal for the class of polylogarithmic time stable protocols that are "output dominant" and "monotone". These are two natural constraints satisfied by our protocol, making it simultaneously time- and state-optimal for that class. We introduce a key technique called a "fixed resolution clock" to achieve partial synchronization. Our protocol is *nonuniform*: the transition function has the value logn\left \lceil {\log n} \right \rceil encoded in it. We show that the protocol can be modified to be uniform, while increasing the state complexity to Θ(lognloglogn)\Theta(\log n \log \log n)

    Foundations of Software Science and Computation Structures

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    This open access book constitutes the proceedings of the 24th International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2021, which was held during March 27 until April 1, 2021, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2021. The conference was planned to take place in Luxembourg and changed to an online format due to the COVID-19 pandemic. The 28 regular papers presented in this volume were carefully reviewed and selected from 88 submissions. They deal with research on theories and methods to support the analysis, integration, synthesis, transformation, and verification of programs and software systems
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