Using modal logics to express and check global graph properties

Abstract

Graphs are among the most frequently used structures in Computer Science. Some of the properties that must be checked in many applications are connectivity, acyclicity and the Eulerian and Hamiltonian properties. In this work, we analyze how we can express these four properties with modal logics. This involves two issues: whether each of the modal languages under consideration has enough expressive power to describe these properties and how complex (computationally) it is to use these logics to actually test whether a given graph has some desired property. First, we show that these properties are not definable in a basic modal logic or in any bisimulation-invariant extension of it, like the modal µ-calculus. We then show that it is possible to express some of the above properties in a basic hybrid logic. Unfortunately, the Hamiltonian and Eulerian properties still cannot be efficiently checked. In a second attempt, we propose an extension of CTL * with nominals and show that the Hamiltonian property can be more efficiently checked in this logic than in the previous one. In a third attempt, we extend the basic hybrid logic with the ↓ operator and show that we can check the Hamiltonian property with optimal (NP) complexity in this logic. Finally, we tackle the Eulerian property in two different ways. First, we develop a generic method to express edge-related properties in hybrid logics and use it to express the Eulerian property. Second, we express a necessary and sufficient condition for the Eulerian property to hold using a graded modal logic.

Walk logic as a framework for path query languages on graph databases

“…We note that Benevides et al[11] report results on the expressive power of HCTL * that are slightly misleading, for example, they claim that Hamiltonian path and Eulerian trail are expressible, but they do not exhibit one formula that expresses the query on all inputs; instead, they need larger and larger formulas for larger and larger input graphs.…”

“…Our original reason to consider WL ∞ is that logics used in verification have a semantics based on infinite walks. As mentioned in the Introduction, a powerful such logic is hybrid CTL * , denoted here by HCTL * [37,11]. Actually we can define two variants: HCTL * under the default semantics based on infinite walks, and a finite-walk variant which we denote by HCTL * fin .…”

“…A comparison t1 ∼ t2 is expressed as t1 = t2 and a comparison t1 < t2 is expressed as t1 = t2 ∧ Reach(t1, t2). 11 The star-free regular expressions (sfre) over an alphabet Σ are defined as follows. Each a ∈ Σ is a sfre; the expression Σ * is a sfre; and if e1 and e2 are sfres, then so are e1 · e2, e1 ∪ e2, and e1 − e2.…”

“…which are typically not bisimulation-invariant. This weakness can be remedied, however, by moving to "hybrid" logics which can explicitly quantify nodes in the graph [3,22,37,11]. …”

Rule-Level Verification of Graph Transformations for Invariants Based on Edges’ Transitive Closure

“…Consequently, the proposed solution could not handle global properties that hold for the graph as a whole, thus concern possibly an infinite number of edges which are outside of the rule, as acyclicity and connectivity. Generally, global properties must be expressed and verified at a global level [3]. The challenges here are how to express those global properties in the rule's applicability condition and how to reason about a possibly infinite number of nodes and edges with a finite number of computations.…”