Temporal graphs are graphs where the presence or properties of their vertices and edges change over time. When time is discrete, a temporal graph can be defined as a sequence of static graphs over a discrete time span, called lifetime, or as a single graph where each edge is associated with a specific set of time instants where the edge is alive. For static graphs, Courcelle’s Theorem asserts that any graph problem expressible in monadic second-order logic can be solved in linear time on graphs of bounded tree-width. We propose the first adaptation of Courcelle’s Theorem for monadic second-order logic on temporal graphs that does not explicitly rely on a parameter proportional to the lifetime, or defined as the maximum number of time-edges incident with any vertex which in the worst case is higher than the lifetime. We then introduce the notion of derivative over a sliding time window of a chosen size, and define the tree-width and twin-width of the temporal graph’s derivative. We exemplify its usefulness with meta-theorems with respect to a temporal variant of first-order logic. The resulting logic expresses a wide range of temporal graph problems including a version of temporal cliques, an important notion when querying time series databases for community structures.

Abstract Given a finite consistent set of ground literals, we present an algorithm that generates a complete first-order logic interpretation, i.e., an interpretation for all ground literals over the signature and not just those in the input set, that is also a model for the input set. The interpretation is represented by first-order linear literals. It can be effectively used to evaluate clauses. A particular application are SCL stuck states. The SCL (Simple Clause Learning) calculus always computes with respect to a finite number of ground literals. It then finds either a contradiction or a stuck state being a model with respect to the considered ground literals. Our algorithm builds a complete literal interpretation out of such a stuck state model that can then be used to evaluate the clause set. If all clauses are satisfied an overall model has been found. If it does not satisfy some clause, this information can be effectively explored to extend the scope of ground literals considered by SCL.