Losing connection: the modal logic of definable link deletion

Abstract

In this article, we start with a two-player game that models communication under adverse circumstances in everyday life and study it from the perspective of a modal logic of graphs, where links can be deleted locally according to definitions available to the adversarial player. We first introduce a new language, semantics, and some typical validities. We then formulate a new type of first-order translation for this modal logic and prove its correctness. Then, a novel notion of bisimulation is proposed which leads to a characterization theorem for the logic as a fragment of first-order logic, and a further investigation is made of its expressive power against hybrid modal languages. Next, we discuss how to axiomatize this logic of link deletion, using dynamic-epistemic logics as a contrast. Finally, we show that our new modal logic lacks both the tree model property and the finite model property, and that its satisfiability problem is undecidable.

Copy and remove as dynamic operators

“…In [10] global and local graph modifiers are proposed, where the modifications can concern both the valuations of propositional variables (also known as ontic change) and the accessibility relations, as is the case with ML(cp, rm). A global graph modification a − (ϕ, ψ) [10, page 295] corresponds to our rm 1 action (ϕ; a; ψ), whereas a local graph modification of that kind would amount to a sabotage operator as the one in [35]. However, the only local graph modifiers considered there are of the ontic change kind [10, page 300].…”

The Modal Logic of Stepwise Removal

“…Other sources of undecidability. In addition to the undecidability induced by stepwise removal, there is the undecidability induced by local link-cutting or local definable point removal, taking place only at the current point of evaluation (Li, 2020). Both modifications of dynamic-epistemic logics block the usual recursion axioms, both allow for tiling encodings, but the connection remains to be clarified.…”

Graph Games and Logic Design

“…We believe that this offers a fresh, and to our knowledge, unexplored perspective on the logical analysis of graph games. 26 The match viewed from both sides The connection between logics and games advocated in this paper can be looked at from two sides. So far, we have gone from games to logics, presenting a number of ways of achieving this.…”

“…In a uniform global case, this is like link cutting in dynamic-epistemic logic. But the theory of a more realistic local variant shares many features with other complex logics discussed here,[26].17 Such variations need not be a priori, they are just as well suggested by actual board games that are reminiscent of the sabotage game, such as 'Quoridor',[20].…”

“…To see how remarkable this is, it suffices to make some obvious variations on well-known evaluation games, and observe how an immediate match with standard logics vanishes 26. After the first version of this paper started circulating, a solution to the question posed here has been found using the countable random graph, where truth coincides with truth almost everywhere in finite models [34].…”