A Knowledge-Based Analysis of Global Function Computation

arXiv:0707.3435v1 [cs.DC] 23 Jul 2007 A Knowledge-Based Analysis of Global Function Computation∗ Joseph Y. Halpern Cornell University Ithaca, NY 14853 halpern@cs.cornell.edu Sabina Petride Cornell University Ithaca, NY 14853 petride@cs.cornell.edu Abstract Consider a distributed system N in which each agent has an input value and each communi- cation link has a weight. Given a global function, that is, a function f whose value depends on the whole network, the goal is for every agent to eventually compute the value f(N). We call this problem global function computation. Various solutions for instances of this problem, such as Boolean function computation, leader election, (minimum) spanning tree construction, and net- work determination, have been proposed, each under particular assumptions about what processors know about the system and how this knowledge can be acquired. We give a necessary and suf- ficient condition for the problem to be solvable that generalizes a number of well-known results [Attyia, Snir, and Warmuth 1988; Yamashita and Kameda 1996; Yamashita and Kameda 1999]. We then provide a knowledge-based (kb) program (like those of Fagin, Halpern, Moses, and Vardi [1995, 1997]) that solves global function computation whenever possible. Finally, we improve the mes- sage overhead inherent in our initial kb program by giving a counterfactual belief-based program [Halpern and Moses 2004] that also solves the global function computation whenever possible, but where agents send messages only when they believe it is necessary to do so. The latter program is shown to be implemented by a number of well-known algorithms for solving leader election. 1 Introduction Consider a distributed system N in which each agent has an input value and each communication link has a weight. Given a global function, that is, a function f whose value depends on the whole network, the goal is for every agent to eventually compute the value f(N). We call this problem global function computation. Many distributed protocols involve computing some global function of the network. This problem is typically straightforward if the network is known. For example, if the goal is to compute the spanning tree of the network, one can simply apply one of the well-known algorithms proposed by Kruskal or Prim. However, in a distributed setting, agents may have only local information, which makes the problem more difficult. For example, the algorithm proposed by Gallager, Humblet and Spira [1983] is known for its complexity.1 Moreover, the algorithm does not work for all networks, although ∗Work supported in part by NSF under grants CTC-0208535, ITR-0325453, and IIS-0534064, by ONR under grant N00014-02-1-0455, by the DoD Multidisciplinary University Research Initiative (MURI) program administered by the ONR under grants N00014-01-1-0795 and N00014-04-1-0725, and by AFOSR under grants F49620-02-1-0101 and FA9550-05-1- 0055. 1Gallager, Humblet, and Spira’s algorithm does not actually solve the minimum spanning tree as we have defined it, since agents do not compute the minimum spanning tree, but only learn relevant information about it, such as which of its edges lead in the direction of the root. 1 it is guaranteed to work correctly when agents have distinct inputs and no two edges have identical weights. Computing shortest paths between nodes in a network is another instance of global function com- putation that has been studied extensively [Ford and Fulkerson 1962; Bellman 1958]. The well-known leader election problem [Lynch 1997] can also be viewed as an instance of global computation in all systems where agents have distinct inputs: the leader is the agent with the largest (or smallest) in- put. The difficulty in solving global function computation depends on what processors know. For example, when processors know their identifiers (names) and all ids are unique, several solutions for the leader election problem have been proposed, both in the synchronous and asynchronous settings [Chang and Roberts 1979; Le Lann 1977; Peterson 1982]. On the other hand, Angluin [1980], and Johnson and Schneider [1985] proved that it is impossible to deterministically elect a leader if agents may share names. In a similar vein, Attiya, Snir and Warmuth [1988] prove that there is no deterministic algorithm that computes a non-constant Boolean global function in a ring of unknown and arbitrarily large size if agents’ names are not necessarily unique. Attiya, Gorbach, and Moran [2002] characterize what can be computed in what they call totally anonymous shared memory systems, where access to shared memory is anonymous. We aim to better understand what agents need to know to compute a global function. We do this using the framework of knowledge-based (kb) programs, proposed by Fagin, Halpern, Moses and Vardi [1995, 1997]. Intuitively, in a kb program, an agent’s actions may depend on his knowledge. To say that the agent with identity i knows some fact ϕ we simply

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