Verification of Parameterized Protocols

Recently there has been much interest in the automatic and semi-automatic veri cation of parameterized networks, i.e., veri cation of a family of systems fPi j i 2 !g, where each Pi is a network consisting of i processes. In this paper, we present a method for the veri cation of so-called universal properties of fair parameterized networks of similar processes, that is, properties of the form 8p1 : : : pn : , where is a quanti er-free LTL formula and the pi refer to processes. To prove an universal property of a parameterized network, we rst model the in nite family of networks by a single fair WS1S transition system, that is, a transition system whose variables are set (2nd-order) variables and whose transitions are described in WS1S. Then, we abstract theWS1S system into a nite state system that can be modelchecked. We present a generic abstraction relation for verifying universal properties as well as an algorithm for computing an abstract system. However, the abstract system may contain in nite computations that have no corresponding fair computations at the concrete level, and hence, in case the property of interest is a progress property, veri cation may fail because of this. Therefore, we present methods that allow to synthesize fairness conditions from the parameterized network and discuss under which conditions and how to lift fairness conditions of this network to fairness conditions on the abstract system. We implemented our methods in a tool, called pax, and applied it to several examples.