Generalized Symmetry Breaking Tasks

Processes in a concurrent system need to coordinate using an underlying shared memory or a message-passing system in order to solve agreement tasks such as, for example, consensus or set agreement. However, coordination is often needed to "break the symmetry" of processes that are initially in the same state, for example, to get exclusive access to a shared resource, to get distinct names, or to elect a leader. This paper introduces and studies the family of generalized symmetry breaking (GSB) tasks, that includes election, renaming and many other symmetry breaking tasks. Differently from agreement tasks, a GSB task is "inputless", in the sense that processes do not propose values; the task only specifies the symmetry breaking requirement, independently of the system's initial state (where processes differ only on their identifiers). Among various results characterizing the family of GSB tasks, it is shown that perfect renaming is universal for all GSB tasks. The paper then studies the power of renaming with respect to $k$-set agreement. It shows that, in a system of $n$ processes, perfect renaming is strictly stronger than $(n-1)$-set agreement, but not stronger than $(n-2)$-set agreement. Furthermore, $(n+1)$ renaming cannot solve even $(n-1)$-set agreement. As a consequence, there are cases where set agreement and renaming are incomparable when looking at their power to implement each other. Finally, the paper shows that there is a large family of GSB tasks that are more powerful than $(n-1)$-set agreement. Some of these tasks are equivalent to $n$-renaming, while others lie strictly between $n$-renaming and $(n+1)$-renaming. Moreover, none of these GSB tasks can solve $(n-2)$-set agreement. Hence, the GSB tasks have a rich structure and are interesting in their own. The proofs of these results are based on combinatorial topology techniques and new ideas about different notions of non-determinism that can be associated with shared objects. Interestingly, this paper sheds a new light on the relations linking set agreement and symmetry breaking.

[1]  Mark Moir,et al.  Wait-Free Algorithms for Fast, Long-Lived Renaming , 1995, Sci. Comput. Program..

[2]  Shlomo Moran,et al.  Extended Impossibility Results for Asynchronous Complete Networks , 1987, Inf. Process. Lett..

[3]  Sam Toueg,et al.  Unreliable failure detectors for reliable distributed systems , 1996, JACM.

[4]  Eli Gafni,et al.  Distributed Programming with Tasks , 2010, OPODIS.

[5]  Seif Haridi,et al.  Distributed Algorithms , 1992, Lecture Notes in Computer Science.

[6]  Hagit Attiya,et al.  Computing in Totally Anonymous Asynchronous Shared Memory Systems , 1998, DISC.

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

[8]  Yehuda Afek,et al.  Tight Group Renaming on Groups of Size g Is Equivalent to g-Consensus , 2009, DISC.

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

[10]  Leslie Lamport,et al.  The part-time parliament , 1998, TOCS.

[11]  Edsger W. Dijkstra,et al.  Solution of a problem in concurrent programming control , 1965, CACM.

[12]  Maurice Herlihy,et al.  Set consensus using arbitrary objects (preliminary version) , 1994, PODC '94.

[13]  Yehuda Afek,et al.  Fast, wait-free (2k-1)-renaming , 1999, PODC '99.

[14]  Eli Gafni,et al.  Musical Benches , 2005, DISC.

[15]  Michel Raynal,et al.  Test & Set, Adaptive Renaming and Set Agreement: a Guided Visit to Asynchronous Computability , 2007, 2007 26th IEEE International Symposium on Reliable Distributed Systems (SRDS 2007).

[16]  Leslie Lamport,et al.  A fast mutual exclusion algorithm , 1987, TOCS.

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

[18]  Yehuda Afek,et al.  Group Renaming , 2008, OPODIS.

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

[20]  Nancy A. Lynch,et al.  Impossibility of distributed consensus with one faulty process , 1983, PODS '83.

[21]  Achour Mostéfaoui,et al.  From adaptive renaming to set agreement , 2009, Theor. Comput. Sci..

[22]  Eli Gafni,et al.  A simple algorithmically reasoned characterization of wait-free computation (extended abstract) , 1997, PODC '97.

[23]  Michel Raynal,et al.  The Committee Decision Problem , 2006, LATIN.

[24]  Achour Mostéfaoui,et al.  Exploring Gafni's Reduction Land: From Omegak to Wait-Free Adaptive (2p-[p/k])-Renaming Via k-Set Agreement , 2006, DISC.

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

[26]  Soma Chaudhuri,et al.  Understanding the Set Consensus Partial Order Using the Borowsky-Gafni Simulation (Extended Abstract) , 1996, WDAG.

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

[28]  Michel Raynal,et al.  Renaming Is Weaker Than Set Agreement But for Perfect Renaming: A Map of Sub-consensus Tasks , 2012, LATIN.

[29]  Nancy A. Lynch,et al.  Reaching approximate agreement in the presence of faults , 1986, JACM.

[30]  Maurice Herlihy,et al.  On the decidability of distributed decision tasks , 1996, PODC '96.

[31]  Michel Raynal,et al.  Communication and Agreement Abstractions for Fault-Tolerant Asynchronous Distributed Systems , 2010, Synthesis Lectures on Distributed Computing Theory.

[32]  Sam Toueg,et al.  Wakeup under Read/Write Atomicity , 1990, WDAG.

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

[34]  Michel Raynal,et al.  Agreement via Symmetry Breaking: On the Structure of Weak Subconsensus Tasks , 2013, 2013 IEEE 27th International Symposium on Parallel and Distributed Processing.

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

[36]  Nir Shavit,et al.  Atomic snapshots of shared memory , 1990, JACM.

[37]  Petr Kuznetsov,et al.  On set consensus numbers , 2011, Distributed Computing.

[38]  Maurice Herlihy,et al.  Wait-free synchronization , 1991, TOPL.

[39]  Marcin Paprzycki,et al.  Distributed Computing: Fundamentals, Simulations and Advanced Topics , 2001, Scalable Comput. Pract. Exp..

[40]  James E. Burns.  Symmetry in systems of asynchronous processes , 1981, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981).

[41]  Mordechai Ben-Ari,et al.  The Concept of Nondeterminism: Its Development and Implications for Teaching , 2009, SGCS.

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

[43]  Michel Raynal,et al.  The k-simultaneous consensus problem , 2010, Distributed Computing.

[44]  RajsbaumSergio,et al.  Bit complexity of breaking and achieving symmetry in chains and rings , 2008 .

[45]  Eugene Styer,et al.  Tight bounds for shared memory symmetric mutual exclusion problems , 1989, PODC '89.

[46]  Hagit Attiya,et al.  Counting-Based Impossibility Proofs for Renaming and Set Agreement , 2012, DISC.

[47]  Eli Gafni,et al.  Renaming with k-Set-Consensus: An Optimal Algorithm into n + k - 1 Slots , 2006, OPODIS.

[48]  HerlihyMaurice,et al.  A classification of wait-free loop agreement tasks , 2003 .

[49]  Maurice Herlihy,et al.  A classification of wait-free loop agreement tasks , 2003, Theor. Comput. Sci..

[50]  Michel Raynal,et al.  Concurrent Programming: Algorithms, Principles, and Foundations , 2012, Springer Berlin Heidelberg.

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

[52]  Shlomo Moran,et al.  Bit complexity of breaking and achieving symmetry in chains and rings , 2008, JACM.

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

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