Dynamic definability

• Authors: Erich Grädel;Sebastian Siebertz
• Venue: Proceedings of the 15th International Conference on Database Theory
• Year: 2012

We investigate the logical resources required to maintain knowledge about a property of a finite structure that undergoes an ongoing series of local changes such as insertion or deletion of tuples to basic relations. Our framework is closely related to the Dyn-FO-framework of Patnaik and Immerman and the FOIES-framework of Dong, Libkin, Su and Wong, and also builds on work of Weber and Schwentick. We assume that the dynamic process starts with an arbitrary, nonempty structure, but in contrast to previous work, we assume that, in general, structures are unordered. We show how to modify known dynamic algorithms for symmetric reachability, bipartiteness, k-edge connectivity and more, to work also without an order and with dynamic processes starting at an arbitrary graph. A history independent dynamic system (also called deterministic or memoryless) is one that maintains all auxiliary information independent of the update order. In 1997, Dong and Su posed the problem whether there exist history independent dynamic systems with FO-updates for symmetric reachability or bipartiteness. We give a positive answer to this question. We further show that there is a history independent system for tree isomorphism with FO+C-updates. On the other hand we show that on unordered structures first-order logic is too weak to maintain enough information to answer the equal cardinality query and the tree isomorphism query dynamically.