Modal Expressiveness of Graph Properties

Abstract

Graphs are among the most frequently used structures in Computer Science. In this work, we analyze how we can express some important graph properties such as connectivity, acyclicity and the Eulerian and Hamiltonian properties in a modal logic. First, we show that these graph properties are not definable in a basic modal language. Second, we discuss an extension of the basic modal language with fix-point operators, the modal μ-calculus. Unfortunately, even with all its expressive power, the μ-calculus fails to express these properties. This happens because μ-calculus formulas are invariant under bisimulations. Third, we show that it is possible to express some of the above properties in a basic hybrid logic. Fourth, we propose an extension of CTL * with nominals, that we call hybrid-CTL * , and then show that it can express the Hamiltonian property in a better way than the basic hybrid logic. Finally, we introduce a promising way of expressing properties related to edges and use it to express the Eulerian property.

A logic for diffusion in social networks

“…It provides a complete axiomatization as well as a sound, complete and terminating tableau system for the new Logic for Diffusion in Social Networks. Our logic extends standard hybrid logic [2], which offers relevant additional expressive powers compared to basic modal logic: by naming agents in the network, additional structural properties of frames can be captured, as already noted by [8,9,44]. The axiomatization of our underlying static logic is very similar to the axiomatizations of [11,2] and the additional axiomatization of our full dynamic logic borrows the reduction technique from [6,46].…”

“…[8] has shown that some global properties of graphs standardly discussed in graph theory are neither definable in basic modal language (even if one adds a transitive closure modal operator to the language) nor in any bisimulation invariant extension of it, such as modal μ-calculus: connectivity, acyclicity, and Hamiltonian property (i.e., whether there is a cycle passing through each vertex of a graph exactly once), for instance. Add nominals and @ i normal modal operators and all those properties become definable, as [8] shows. While our language does not include the transitive closure operator used in [8] and therefore cannot express connectivity and acyclicity with the same succinctness, 5 it can express the Hamiltonian property in the exact same way introduced in [8].…”

A Logical Approach to Hamiltonian Graphs

“…The works presented in [5] and [6] are closely related to this one. In [5], the interest was also in how to use modal logics to express global graph properties.…”

“…Moreover, the focus of that work was on how to build axiomatizations for classes of graphs with these global properties, while our focus is on finding formulas expressing a global graph property that can be efficiently used to test whether a graph satisfies it. In [6], the goal was also to find formulas that could describe global graph properties, but the practical issue of how computationally complex it would be to use those formulas to check whether a graph satisfies the correspondent property was not addressed.…”

“…This work can be considered as a follow-up to [6]. In that work, the goal was also to find formulas that could describe global graph properties, but the practical issue of how computationally complex it would be to use those formulas to check whether a graph satisfies the correspondent property was not addressed.…”