Philipp Rohde

Model Checking and Satisfiability for Sabotage Modal Logic

We consider the sabotage modal logic SML which was suggested by van Benthem. SML is the modal logic equipped with a ‘transition-deleting’ modality and hence a modal logic over changing models. It was shown that the problem of uniform model checking for this logic is PSPACE-complete. In this paper we show that, on the other hand, the formula complexity and the program complexity are linear, resp., polynomial time. Further we show that SML lacks nice model-theoretic properties such as bisimulation invariance, the tree model property, and the finite model property. Finally we show that the satisfiability problem for SML is undecidable. Therefore SML seems to be more related to FO than to usual modal logic.

Solving the Sabotage Game Is PSPACE-Hard

We consider the sabotage game as presented by van Benthem. In this game one player moves along the edges of a finite multi-graph and the other player takes out a link after each step. One can consider usual algorithmic tasks like reachability, Hamilton path, or complete search as winning conditions for this game. As the game definitely ends after at most the number of edges steps, it is easy to see that solving the sabotage game for the mentioned tasks takes at most PSPACE in the size of the graph. In this paper we establish the PSPACE-hardness of this problem. Furthermore, we introduce a modal logic over changing models to express tasks corresponding to the sabotage games and we show that model checking this logic is PSPACE-complete.

Moving in a Crumbling Network: The Balanced Case

In this paper we continue the study of ‘sabotage modal logic’ SML which was suggested by van Benthem. In this logic one describes the progression along edges of a transition graph in alternation with moves of a saboteur who can delete edges. A drawback of the known results on SML is the asymmetry of the two modalities of ‘moving’ and ‘deleting’: Movements are local, whereas there is a global choice for edge deletion. To balance the situation and to obtain a more realistic model for traffic and network problems, we require that also the sabotage moves (edge deletions) are subject to a locality condition. We show that the new logic, called path sabotage logic PSL, already has the same complexities as SML (model checking, satisfiability) and that it lacks the finite model property. The main effort is finding a pruned form of SML-models that can be enforced within PSL and giving appropriate reductions from SML to PSL.

Expressive power of monadic second-order logic and modal μ-calculus

We consider monadic second order logic (MSO) and the modal μ-calculus (Lμ) over transition systems (Kripke structures). It is well known that every class of transition systems which is definable by a sentence of Lμ is definable by a sentence of MSO as well. It will be shown that the converse is also true for an important fragment of MSO: every class of transition systems which is MSO-definable and which is closed under bisimulation - i.e., the sentence does not distinguish between bisimilar models - is also Lμ-definable. Hence we obtain the following expressive completeness result: the bisimulation invariant fragment of MSO and Lμ are equivalent. The result was proved by David Janin and Igor Walukiewicz. Our presentation is based on their article [91]. The main step is the development of automata-based characterizations of Lμ over arbitrary transition systems and of MSO over transition trees (see also Chapter 16). It turns out that there is a general notion of automaton subsuming both characterizations, so we obtain a common ground to compare these two logics. Moreover we need the notion of the ω-unravelling for a transition system, on the one hand to obtain a bisimilar transition tree and on the other hand to increase the possibilities of choosing successors.

On the µ-calculus augmented with sabotage

We study logics and games over dynamically changing structures. Van Benthems sabotage modal logic consists of modal logic with a cross-model modality referring to submodels from which a transition has been removed. We add constructors for forming least and greatest monadic fixed-points to that logic and obtain the sabotage μ-calculus. We introduce backup parity games as an extension of standard parity games where in addition, both players are able to delete edges of the arena and to store, resp., restore the current appearance of the arena by use of a fixed number of registers. We show that these games serve as model checking games for the sabotage μ-calculus, even if the access to registers follows a stack discipline. The problem of solving the games with restricted register access turns out to be PSPACE-complete while the more general games without a limited access become EXPTIME-complete (for at least three registers).