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58
The computational power of population protocols
 Distributed Computing
"... We consider the model of population protocols introduced by Angluin et al. [AAD + 04], in which anonymous finitestate agents stably compute a predicate of the multiset of their inputs via twoway interactions in the allpairs family of communication networks. We prove that all predicates stably com ..."
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Cited by 59 (4 self)
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We consider the model of population protocols introduced by Angluin et al. [AAD + 04], in which anonymous finitestate agents stably compute a predicate of the multiset of their inputs via twoway interactions in the allpairs family of communication networks. We prove that all predicates stably computable in this model (and certain generalizations of it) are semilinear, answering a central open question about the power of the model. Removing the assumption of twoway interaction, we also consider several variants of the model in which agents communicate by anonymous messagepassing where the recipient of each message is chosen by an adversary and the sender is not identified to the recipient. These oneway models are distinguished by whether messages are delivered immediately or after a delay, whether a sender can record that it has sent a message, and whether a recipient can queue incoming messages, refusing to accept new messages until it has had a chance to send out messages of its own. We characterize the classes of predicates stably computable in each of these oneway models using natural subclasses of the semilinear predicates. 1
Computation with finite stochastic chemical reaction networks
 Natural Computing
, 2008
"... Abstract. A highly desired part of the synthetic biology toolbox is an embedded chemical microcontroller, capable of autonomously following a logic program specified by a set of instructions, and interacting with its cellular environment. Strategies for incorporating logic in aqueous chemistry have ..."
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Cited by 53 (17 self)
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Abstract. A highly desired part of the synthetic biology toolbox is an embedded chemical microcontroller, capable of autonomously following a logic program specified by a set of instructions, and interacting with its cellular environment. Strategies for incorporating logic in aqueous chemistry have focused primarily on implementing components, such as logic gates, that are composed into larger circuits, with each logic gate in the circuit corresponding to one or more molecular species. With this paradigm, designing and producing new molecular species is necessary to perform larger computations. An alternative approach begins by noticing that chemical systems on the small scale are fundamentally discrete and stochastic. In particular, the exact molecular counts of each molecular species present, is an intrinsically available form of information. This might appear to be a very weak form of information, perhaps quite difficult for computations to utilize. Indeed, it has been shown that errorfree Turing universal computation is impossible in this setting. Nevertheless, we show a design of a chemical computer that achieves fast and reliable Turinguniversal computation using molecular counts. Our scheme uses only a small number of different molecular species to do computation of arbitrary complexity. The total probability of error of the computation can be made arbitrarily small (but not zero) by adjusting the initial molecular counts of certain species. While physical implementations would be difficult, these results demonstrate that molecular counts can be a useful form of information for small molecular systems such as those operating within cellular environments. Key words. stochastic chemical kinetics; molecular counts; Turinguniversal computation; probabilistic computation 1. Introduction. Many
A simple population protocol for fast robust approximate majority
 Distributed Computing, 21st International Symposium, DISC 2007
, 2008
"... Abstract We describe and analyze a 3state oneway population protocol to compute approximate majority in the model in which pairs of agents are drawn uniformly at random to interact. Given an initial configuration of x’s, y’s and blanks that contains at least one nonblank, the goal is for the agen ..."
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Cited by 34 (2 self)
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Abstract We describe and analyze a 3state oneway population protocol to compute approximate majority in the model in which pairs of agents are drawn uniformly at random to interact. Given an initial configuration of x’s, y’s and blanks that contains at least one nonblank, the goal is for the agents to reach consensus on one of the values x or y. Additionally, the value chosen should be the majority nonblank initial value, provided it exceeds the minority by a sufficient margin. We prove that with high probability n agents reach consensus in O(n log n) interactions and the value chosen is the majority provided that its initial margin is at least ω ( √ n log n). This protocol has the additional property of tolerating Byzantine behavior in o ( √ n) of the agents, making it the first known population protocol that tolerates Byzantine agents.
Programmability of Chemical Reaction Networks
"... Summary. Motivated by the intriguing complexity of biochemical circuitry within individual cells we study Stochastic Chemical Reaction Networks (SCRNs), a formal model that considers a set of chemical reactions acting on a finite number of molecules in a wellstirred solution according to standard c ..."
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Cited by 25 (6 self)
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Summary. Motivated by the intriguing complexity of biochemical circuitry within individual cells we study Stochastic Chemical Reaction Networks (SCRNs), a formal model that considers a set of chemical reactions acting on a finite number of molecules in a wellstirred solution according to standard chemical kinetics equations. SCRNs have been widely used for describing naturally occurring (bio)chemical systems, and with the advent of synthetic biology they become a promising language for the design of artificial biochemical circuits. Our interest here is the computational power of SCRNs and how they relate to more conventional models of computation. We survey known connections and give new connections between SCRNs and
Deterministic Function Computation with Chemical Reaction Networks ∗
"... We study the deterministic computation of functions on tuples of natural numbers by chemical reaction networks (CRNs). CRNs have been shown to be efficiently Turinguniversal when allowing for a small probability of error. CRNs that are guaranteed to converge on a correct answer, on the other hand, ..."
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We study the deterministic computation of functions on tuples of natural numbers by chemical reaction networks (CRNs). CRNs have been shown to be efficiently Turinguniversal when allowing for a small probability of error. CRNs that are guaranteed to converge on a correct answer, on the other hand, have been shown to decide only the semilinear predicates. We introduce the notion of function, rather than predicate, computation by representing the output of a function f: N k → N l by a count of some molecular species, i.e., if the CRN starts with n1,..., nk molecules of some “input ” species X1,..., Xk, the CRN is guaranteed to converge to having f(n1,..., nk) molecules of the “output ” species Y1,..., Yl. We show that a function f: N k → N l is deterministically computed by a CRN if and only if its graph {(x, y) ∈ N k × N l  f(x) = y} is a semilinear set. Finally, we show that each semilinear function f can be computed on input x in expected time O(polylog ‖x‖1). 1
Timing in chemical reaction networks
 In SODA 2014: Proceedings of the 25th Annual ACMSIAM Symposium on Discrete Algorithms
, 2014
"... Chemical reaction networks (CRNs) formally model chemistry in a wellmixed solution. CRNs are widely used to describe information processing occurring in natural cellular regulatory networks, and with upcoming advances in synthetic biology, CRNs are a promising programming language for the design of ..."
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Cited by 15 (7 self)
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Chemical reaction networks (CRNs) formally model chemistry in a wellmixed solution. CRNs are widely used to describe information processing occurring in natural cellular regulatory networks, and with upcoming advances in synthetic biology, CRNs are a promising programming language for the design of artificial molecular control circuitry. Due to a formal equivalence between CRNs and a model of distributed computing known as population protocols, results transfer readily between the two models. We show that if a CRN respects finite density (at most O(n) additional molecules can be produced from n initial molecules), then starting from any dense initial configuration (all molecular species initially present have initial count Ω(n), where n is the initial molecular count and volume), every producible species is produced in constant time with high probability. This implies that no CRN obeying the stated constraints can function as a timer, able to produce a molecule, but doing so only after a time that is an unbounded function of the input size. This has consequences regarding an open question of Angluin, Aspnes, and Eisenstat concerning the ability of population protocols to perform fast, reliable leader election and to simulate arbitrary algorithms from a uniform initial state.
SelfStabilizing Counting in Mobile Sensor Networks
"... Distributed computing must adapt its techniques to networks of mobile agents. Indeed, we are in front of new problems like the small size of memory and the lack of computational power. In this paper, we extend the results of Angluin et al (see [1–3]) by finding selfstabilizing algorithms to count ..."
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Cited by 13 (4 self)
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Distributed computing must adapt its techniques to networks of mobile agents. Indeed, we are in front of new problems like the small size of memory and the lack of computational power. In this paper, we extend the results of Angluin et al (see [1–3]) by finding selfstabilizing algorithms to count the number of agents in the network. We focus on two different models of communication, with a fixed antenna or with pairwise interactions. In both models we decide if there exist algorithms (probabilistic, deterministic, with kfair adversary) to solve the selfstabilizing counting problem.
The Dynamics of Probabilistic Population Protocols ∗
, 807
"... We study here the dynamics (and stability) of Probabilistic Population Protocols, via the differential equations approach. We provide a quite general model and we show that it includes the model of Angluin et. al. [1], in the case of very large populations. For the general model we give a sufficient ..."
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Cited by 13 (7 self)
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We study here the dynamics (and stability) of Probabilistic Population Protocols, via the differential equations approach. We provide a quite general model and we show that it includes the model of Angluin et. al. [1], in the case of very large populations. For the general model we give a sufficient condition for stability that can be checked in polynomial time. We also study two interesting subcases: (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 wellknown Replicator Dynamics of Evolutionary Games. We also provide a sufficient condition for stability in this case. 1
Not all fair probabilistic schedulers are equivalent
 In 13th International Conference on Principles of Distributed Systems (OPODIS), volume 5923 of Lecture Notes in Computer Science
, 2009
"... Abstract. We propose a novel, generic definition of probabilistic schedulers for population protocols. We then identify the consistent probabilistic schedulers, and prove that any consistent scheduler that assigns a nonzero probability to any transition i → j, where i and j are configurations sa ..."
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Cited by 12 (7 self)
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Abstract. We propose a novel, generic definition of probabilistic schedulers for population protocols. We then identify the consistent probabilistic schedulers, and prove that any consistent scheduler that assigns a nonzero probability to any transition i → j, where i and j are configurations satisfying i 6 = j, is fair with probability 1. This is a new theoretical framework that aims to simplify proving specific probabilistic schedulers fair. In this paper we propose two new schedulers, the State Scheduler and the Transition Function Scheduler. Both possess the significant capability of being protocolaware, i.e. they can assign transition probabilities based on information concerning the underlying protocol. By using our framework we prove that the proposed schedulers, and also the Random Scheduler that was defined by Angluin et al. [2], are all fair with probability 1. Finally, we define and study equivalence between schedulers w.r.t. performance and correctness and prove that there exist fair probabilistic schedulers that are not equivalent w.r.t. to performance and others that are not equivalent w.r.t. correctness.