Local Symmetry and Compositional Verification

This work considers concurrent programs formed of processes connected by an underlying network. The symmetries of the network may be used to reduce the state space of the program, by grouping together similar global states. This can result in an exponential reduction for highly symmetric networks, but it is much less effective for many networks, such as rings, which have limited global symmetry. We focus instead on the local symmetries in a network and show that they can be used to significantly reduce the complexity of compositional reasoning. Local symmetries are represented by a symmetry groupoid, a generalization of a symmetry group. Certain sub-groupoids induce quotient networks which are equivalent to the original for the purposes of compositional reasoning. We formulate a compositional reasoning principle for safety properties of process networks and define symmetry groupoids and the quotient construction. Moreover, we show how symmetry and local reasoning can be expoited to provide parameterized proofs of correctness. “Whenever you have to do with a structure-endowed entity try to determine its group of automorphisms” Hermann Weyl, Symmetry , 1952 “... there are plenty of objects which exhibit what we clearly recognize as symmetry, but which admit few or no nontrivial automorphisms. It turns out that the symmetry, and hence much of the structure, of such objects can be characterized algebraically if we use groupoids and not just groups.” Alan Weinstein, Groupoids: Unifying Internal and External Symmetry – A Tour through Some Examples, Notices of the AMS, 1996.