Symmetry Reduction for the Local Mu-Calculus

Abstract

Model checking large networks of processes is challenging due to state explosion. In many cases, individual processes are isomorphic, but there is insufficient global symmetry to simplify model checking. This work considers the verification of local properties, those defined over the neighborhood of a process. Considerably generalizing earlier results on invariance, it is shown that all local mu-calculus properties, including safety and liveness properties, are preserved by neighborhood symmetries. Hence, it suffices to check them locally over a set of representative process neighborhoods. In general, local verification approximates verification over the global state space; however, if process interactions are outward-facing, the relationship is shown to be exact. For many network topologies, even those with little global symmetry, analysis with representatives provides a significant, even exponential, reduction in the cost of verification. Moreover, it is shown that for network families generated from building-block patterns, neighborhood symmetries are easily determined, and verification over the entire family reduces to verification over a finite set of representative process neighborhoods.

Local Reasoning for Parameterized First Order Protocols

“…Local symmetry is, in general, an abstraction technique that can offer exponentially more reduction than global symmetry. In particular, ring structures are globaly rotationally symmetric, but for isomorphic processes may be fully-locally symmetric [11,12].…”

“…We have shown how analysis techniques for parametric distributed systems composed of several components running on locally symmetric topologies, introduced in [7,8,9,11,12], can be generalized and applied within a First Order Logic based theorem proving engine. The key steps are showing how the local neighbourhoods of network nodes can be encoded and reasoned about separately from FOL based reasoning about safety properties of the reachable states of a parametric family of programs.…”

“…In that work, cut off points are calculated during program analysis. As another example, in [7,11,12] the "cutoffs" are based on the program topology and the local structural symmetries amongst the nodes of the process interconnection networks.…”

“…6, we consider a modified version of the leader election protocol from Ivy [15]. This is of particular interest, since the local symmetry of [7,11,12] has not been applied to leader election. We thus extend [7,11,12] by both allowing more symmetries and infinite-state systems.…”

“…We illustrate our approach on two examples. First, as a running example, motivated by [12], is a protocol on rings of alternating red and black nodes. These rings have only rotational symmetry, however, they have substantial local symmetry [7,11,12] consisting of two equivalence classes, one of red nodes, and one of black nodes.…”