From Geometric Semantics to Asynchronous Computability

We show that the protocol complex formalization of fault-tolerant protocols can be directly derived from a suitable semantics of the underlying synchronization and communication primitives, based on a geometrization of the state space. By constructing a one-to-one relationship between simplices of the protocol complex and dihomotopy classes of dipaths in the latter semantics, we describe a connection between these two geometric approaches to distributed computing: protocol complexes and directed algebraic topology. This is exemplified on atomic snapshot, iterated snapshot and layered immediate snapshot protocols, where a well-known combinatorial structure, interval orders, plays a key role. We believe that this correspondence between models will extend to proving impossibility results for much more intricate fault-tolerant distributed architectures.

[1]  Eric Goubault,et al.  A Practical Application of Geometric Semantics to Static Analysis of Concurrent Programs , 2005, CONCUR.

[2]  Eric Goubault,et al.  A Geometric View of Partial Order Reduction , 2013, MFPS.

[3]  Nancy A. Lynch,et al.  Impossibility of distributed consensus with one faulty process , 1985, JACM.

[4]  Jean Goubault-Larrecq,et al.  Natural Homology , 2015, ICALP.

[5]  Dmitry N. Kozlov,et al.  Topology of the view complex , 2013, ArXiv.

[6]  Lisbeth Fajstrup,et al.  Detecting Deadlocks in Concurrent Systems , 1996 .

[7]  Eric Goubault,et al.  Trace Spaces: An Efficient New Technique for State-Space Reduction , 2012, ESOP.

[8]  P. Fishburn.  Intransitive indifference with unequal indifference intervals , 1970 .

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

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

[11]  L. Nachbin.  Topology and order , 1965 .

[12]  Eric Goubault,et al.  Directed Algebraic Topology and Concurrency , 2016, Cambridge International Law Journal.

[13]  Maurice Herlihy,et al.  Distributed Computing Through Combinatorial Topology , 2013 .

[14]  K. Hofmann,et al.  A Compendium of Continuous Lattices , 1980 .

[15]  Eric Goubault,et al.  Iterated Chromatic Subdivisions are Collapsible , 2015, Appl. Categorical Struct..

[16]  James H. Anderson,et al.  Composite registers , 1990, PODC '90.

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

[18]  Shmuel Zaks,et al.  A combinatorial characterization of the distributed tasks which are solvable in the presence of one faulty processor , 1988, PODC '88.

[19]  Eric Goubault,et al.  Algebraic topology and concurrency , 2006, Theor. Comput. Sci..

[20]  Dmitry N. Kozlov,et al.  Chromatic subdivision of a simplicial complex , 2012 .

[21]  Eric Goubault,et al.  Homology of Higher Dimensional Automata , 1992, CONCUR.

[22]  Maurice Herlihy,et al.  Elements of Combinatorial Topology , 2014 .

[23]  M. Grandis.  Directed Algebraic Topology: Models of Non-Reversible Worlds , 2009 .

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

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

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

[27]  Jeremy Gunawardena,et al.  Homotopy and Concurrency , 2001, Bull. EATCS.

[28]  Eric Goubault,et al.  Rigorous Evidence of Freedom from Concurrency Faults in Industrial Control Software , 2011, SAFECOMP.

[29]  Eric Goubault,et al.  SOME GEOMETRIC PERSPECTIVES IN CONCURRENCY THEORY , 2003 .

[30]  Vaughan R. Pratt,et al.  Modeling concurrency with geometry , 1991, POPL '91.