Counting-Based Impossibility Proofs for Renaming and Set Agreement

Renaming and set agreement are two fundamental sub-consensus tasks. In the M-renaming task, processes start with names from a large domain and must decide on distinct names in a range of size M; in the k-set agreement task, processes must decide on at most k of their input values. Renaming and set agreement are representatives of the classes of colored and colorless tasks, respectively. This paper presents simple proofs for key impossibility results for wait-free computation using only read and write operations: n processes cannot solve (n−1)-set agreement, and, if n is a prime power, n processes cannot solve (2n−2)-renaming. Our proofs consider a restricted set of executions, and combine simple operational properties of these executions with elementary counting arguments, to show the existence of an execution violating the task's requirements. This makes the proofs easier to understand, verify, and hopefully, extend.

[1]  Nancy A. Lynch,et al.  The BG distributed simulation algorithm , 2001, Distributed Computing.

[2]  Michel Raynal,et al.  Synchronous Set Agreement: a Concise Guided Tour (including a new algorithm and a list of open problems) , 2006, 2006 12th Pacific Rim International Symposium on Dependable Computing (PRDC'06).

[3]  Eli Gafni,et al.  Recursion in Distributed Computing , 2010, SSS.

[4]  Michel Raynal,et al.  The renaming problem in shared memory systems: An introduction , 2011, Comput. Sci. Rev..

[5]  Yoram Moses,et al.  A Layered Analysis of Consensus , 2002, SIAM J. Comput..

[6]  Maurice Herlihy,et al.  An Equivariance Theorem with Applications to Renaming , 2012, LATIN.

[7]  Hagit Attiya.  A direct lower bound for k-set consensus , 1998, PODC '98.

[8]  Hagit Attiya,et al.  Renaming in an asynchronous environment , 1990, JACM.

[9]  Eli Gafni.  The extended BG-simulation and the characterization of t-resiliency , 2009, STOC '09.

[10]  Maurice Herlihy,et al.  The asynchronous computability theorem for t-resilient tasks , 1993, STOC.

[11]  Maurice Herlihy,et al.  Algebraic spans , 2000 .

[12]  Maurice Herlihy,et al.  Impossibility results for asynchronous PRAM (extended abstract) , 1991, SPAA '91.

[13]  Hagit Attiya,et al.  Wiley Series on Parallel and Distributed Computing , 2004, SCADA Security: Machine Learning Concepts for Intrusion Detection and Prevention.

[14]  Maurice Herlihy,et al.  The topological structure of asynchronous computability , 1999, JACM.

[15]  Maurice Herlihy,et al.  Subconsensus Tasks: Renaming Is Weaker Than Set Agreement , 2006, DISC.

[16]  Michel Raynal,et al.  Synchronous Set Agreement: a Concise Guided Tour (with open problems) , 2006 .

[17]  Michael Henle,et al.  A combinatorial introduction to topology , 1978 .

[18]  Michael E. Saks,et al.  Wait-free k-set agreement is impossible: the topology of public knowledge , 1993, STOC.

[19]  Christian Scheideler,et al.  Stabilization, Safety, and Security of Distributed Systems , 2012, Lecture Notes in Computer Science.

[20]  Hagit Attiya,et al.  Distributed Computing: Fundamentals, Simulations and Advanced Topics , 1998 .

[21]  Soma Chaudhuri,et al.  More Choices Allow More Faults: Set Consensus Problems in Totally Asynchronous Systems , 1993, Inf. Comput..

[22]  Alan M. Frieze,et al.  The Cover Times of Random Walks on Hypergraphs , 2011, SIROCCO.

[23]  Eli Gafni,et al.  Generalized FLP impossibility result for t-resilient asynchronous computations , 1993, STOC.

[24]  Eli Gafni,et al.  Round-by-round fault detectors (extended abstract): unifying synchrony and asynchrony , 1998, PODC '98.

[25]  Sergio Rajsbaum,et al.  New combinatorial topology bounds for renaming: The upper bound , 2012, JACM.

[26]  Hagit Attiya,et al.  A Non-topological Proof for the Impossibility of k-Set Agreement , 2011, SSS.

[27]  Eli Gafni.  The 0-1-Exclusion Families of Tasks , 2008, OPODIS.

[28]  Eli Gafni.  Read-Write Reductions , 2006, ICDCN.

[29]  Eli Gafni,et al.  Round-by-Round Fault Detectors: Unifying Synchrony and Asynchrony (Extended Abstract). , 1998, PODC 1998.

[30]  Hagit Attiya,et al.  Polynominal and Adaptive Long-Lived (2k-1)-Renaming , 2000, DISC.

[31]  David Fernández-Baca.  LATIN 2012: Theoretical Informatics , 2012, Lecture Notes in Computer Science.

[32]  Eli Gafni,et al.  Immediate atomic snapshots and fast renaming , 1993, PODC '93.

[33]  Sergio Rajsbaum,et al.  New combinatorial topology upper and lower bounds for renaming , 2008, PODC '08.

[34]  Sergio Rajsbaum,et al.  New combinatorial topology bounds for renaming: the lower bound , 2010, Distributed Computing.

[35]  Hagit Attiya,et al.  The Combinatorial Structure of Wait-Free Solvable Tasks , 2002, SIAM J. Comput..

[36]  Michel Raynal,et al.  The universe of symmetry breaking tasks , 2011, PODC '11.

[37]  G. Hardy,et al.  Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work , 1978 .