Losing Connection: Modal Logic of Definable Link Deletion
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We consider a natural variant of Sabotage Game. Different to the action of Blocker in a sabotage game, links are cut in a local, definable and uniform way, where local is used as usual; definable means that links are cut based on some considerations on the properties of the nodes in the graph; and uniform shows that it is permitted to cut more links than one in each round. As what we will see, this variant can model some interesting scenarios in everyday life. After this, a suitable modal logic is presented to characterize the game. First, we provide some preliminaries for it, including syntax, semantics and some validities. Next, we describe the standard translation and its correctness. Then, an appropriate notion of bisimulation is proposed for the logic introduced. Afterwards, we show a characterization theorem for the logic as a fragment of first-order logic that is invariant for the bisimulation, and investigate its expressive power. Further more, we prove that the logic is also a fragment of some hybrid logics, and provide some considerations on the recursion axioms. Finally, we show that the logic lacks the tree model property and the finite model property,and that the satisfiability problem for it is undecidable.
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