Kernkonzepte
The satisfiability problem for the two-variable guarded fragment logic extended with local Presburger quantifiers (GP2) is EXP-complete. This is established by a novel, deterministic graph-based algorithm that eliminates contradictory vertex or edge types until no more can be eliminated.
Zusammenfassung
The paper considers the extension of the two-variable guarded fragment logic with local Presburger quantifiers, denoted as GP2. These quantifiers can express local numerical properties, such as "the number of outgoing red edges plus twice the number of incoming green edges is at most three times the number of outgoing blue edges."
The key results are:
The satisfiability problem for GP2 is EXP-complete. The lower bound is already known, while the upper bound is established by a novel, deterministic graph-based algorithm.
The algorithm works by representing the input GP2 sentence as a graph, where vertices and edges represent the allowed types. It then successively eliminates the vertex or edge that contradicts the input sentence until there is no more vertex or edge to eliminate.
This algorithm has a different flavor from the standard tableaux method, which relies on the tree-like model property. The authors show that the tableaux method may not work well for GP2 due to the potential exponential blow-up in the branching degree.
The paper also includes a comparison with the independent work by Bednarczyk and Fiuk, who obtained the same EXP upper bound using a tableaux-based approach.
Overall, the paper presents a novel, efficient algorithm for deciding the satisfiability of the expressive GP2 logic, which captures various description logics with counting.