A Simplicial Complex Model for Dynamic Epistemic Logic to study Distributed Task Computability

Abstract The usual S 5 n epistemic model for a multi-agent system is based on a Kripke frame, which is a graph whose edges are labeled with agents that do not distinguish between two states. We propose to uncover the higher dimensional information implicit in this structure, by considering a dual, simplicial complex model. We use dynamic epistemic logic (DEL) to study how an epistemic simplicial complex model changes after a set of agents communicate with each other. We concentrate on an action model that represents the so-called immediate snapshot communication patterns of asynchronous agents, because it is central to distributed computability (but our setting works for other communication patterns). There are topological invariants preserved from the initial epistemic complex to the one after the action model is applied, which determine the knowledge that the agents gain after communication. Finally, we describe how a distributed task specification can be modeled as a DEL action model, and show that the topological invariants determine whether the task is solvable. We thus provide a bridge between DEL and the topological theory of distributed computability, which studies task solvability in a shared memory or message passing architecture.

[1]  Eli Gafni,et al.  Three-Processor Tasks Are Undecidable , 1999, SIAM J. Comput..

[2]  Aybüke Özgün,et al.  Topo-Logic as a Dynamic-Epistemic Logic , 2017, LORI.

[3]  A. Tarski,et al.  The Algebra of Topology , 1944 .

[4]  Ariel Rubinstein,et al.  A Comment on the Logic of 'Agreeing to Disagree' Type Results , 1988 .

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

[6]  Maurice Herlihy,et al.  Simulations and reductions for colorless tasks , 2012, PODC '12.

[7]  Andreas Witzel,et al.  The synchronicity of dynamic epistemic logic , 2011, TARK XIII.

[8]  Yoram Moses,et al.  Unbeatable consensus , 2014, Distributed Computing.

[9]  Ronald Fagin,et al.  Reasoning about knowledge , 1995 .

[10]  Sergio Rajsbaum,et al.  Iterated Shared Memory Models , 2010, LATIN.

[11]  Jérémy Ledent.  Geometric semantics for asynchronous computability , 2019 .

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

[13]  Johan van Benthem,et al.  Modal Logics of Space , 2007, Handbook of Spatial Logics.

[14]  Eric Goubault,et al.  Models of fault-tolerant distributed computation via dynamic epistemic logic , 2017, ArXiv.

[15]  Emiliano Lorini,et al.  Announcements to Attentive Agents , 2016, J. Log. Lang. Inf..

[16]  John McCarthy,et al.  Artificial Intelligence, Logic and Formalizing Common Sense , 1989 .

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

[18]  Sergio Rajsbaum,et al.  Knowledge and simplicial complexes , 2020, ArXiv.

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

[20]  Jelle Gerbrandy,et al.  Dynamic epistemic logic , 1998 .

[21]  Yoram Moses,et al.  Relating Knowledge and Coordinated Action: The Knowledge of Preconditions Principle , 2016, TARK.

[22]  Shmuel Zaks,et al.  A Combinatorial Characterization of the Distributed 1-Solvable Tasks , 1990, J. Algorithms.

[23]  Mark Ryan,et al.  Knowledge in multiagent systems: initial configurations and broadcast , 1999, TOCL.

[24]  Timothy Porter,et al.  Interpreted systems and Kripke models for multiagent systems from a categorical perspective , 2004, Theor. Comput. Sci..

[25]  Sergio Rajsbaum,et al.  Collapsibility of read/write models using discrete morse theory , 2018, J. Appl. Comput. Topol..

[26]  Nancy A. Lynch,et al.  Distributed Algorithms , 1992, Lecture Notes in Computer Science.

[27]  Jan van Eijck,et al.  Logics of communication and change , 2006, Inf. Comput..

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

[29]  Maurice Herlihy,et al.  Impossibility and universality results for wait-free synchronization , 1988, PODC '88.

[30]  Hans van Ditmarsch,et al.  Semantic results for ontic and epistemic change , 2006, ArXiv.

[31]  Saul Kripke,et al.  A completeness theorem in modal logic , 1959, Journal of Symbolic Logic.

[32]  Yoichi Hirai,et al.  An Intuitionistic Epistemic Logic for Sequential Consistency on Shared Memory , 2010, LPAR.

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

[34]  Mark Ryan,et al.  On the Relation between Interpreted Systems and Kripke Models , 1997, Agents and Multi-Agent Systems Formalisms, Methodologies, and Applications.

[35]  Maurice Herlihy,et al.  The decidability of distributed decision tasks (extended abstract) , 1997, STOC '97.

[36]  J. A. Bondy,et al.  Graph Theory with Applications , 1978 .

[37]  François Schwarzentruber,et al.  Reasoning about knowledge and messages in asynchronous multi-agent systems , 2017, Mathematical Structures in Computer Science.

[38]  Yoram Moses,et al.  Unbeatable Set Consensus via Topological and Combinatorial Reasoning , 2016, PODC.

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

[40]  Lawrence S. Moss,et al.  The Logic of Public Announcements and Common Knowledge and Private Suspicions , 1998, TARK.

[41]  W. Hoek,et al.  Dynamic Epistemic Logic , 2007 .

[42]  Dmitry N. Kozlov,et al.  Combinatorial Algebraic Topology , 2007, Algorithms and computation in mathematics.

[43]  Sergio Rajsbaum,et al.  A dynamic epistemic logic analysis of the equality negation task , 2019, DaLí.

[44]  Eric Goubault,et al.  A simplicial complex model of dynamic epistemic logic for fault-tolerant distributed computing , 2017, ArXiv.

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

[46]  Hans van Ditmarsch.  Asynchronous Announcements , 2017, ArXiv.

[47]  John Havlicek.  Computable obstructions to wait-free computability , 2000, Distributed Computing.