The Complexity of Model Checking Concurrent Recursive Programs

Model Checking Concurrent Recursive and Communicating Programs via Split-Width

The work described in the following was done by Aiswarya Cyriac in collaboration with Paul Gastin and K. Narayan Kumar, and it is part of Aiswarya Cyriac's PhD thesis, which has recently been defended. It is a generalisation of our CONCUR'12 paper where split-width is introduced to address the decidability of MSO specifications for multi-pushdown systems.

We consider generic systems which incorporate shared-variable communication and communication via channels. We are considering physically distributed machines which communicate via (possibly several) reliable first-in-first-out queues. Each of these machines are capable of running potentially recursive multi-threaded programs. These programs within a machine use shared variable for communication. Such a machine consisting of a set of threads communicating by shared memory can be formally modelled as a multi-pushdown system. Thus we have a network of multi-pushdown systems communicating via FIFO queues. Moreover, these programs may use stacks and queues as data-structures to aid their local computation. We call such a system a system of concurrent processes with data-structures (CPDS).

We introduce and study a new technique called split-width for the under-approximate verification of CPDS. This parameter is based on simple shuffle and merge operations and gives us a divide-conquer-way to prove the bound of languages. When parametrised by a bound on split-width, we obtain decidability for various verification problems. We provide a uniform decision procedure for various verification problems with optimal complexities.

We expose the power of split-width in several ways. We show that our simple algebra is powerful enough to capture any class of CPDS which admits decidability for MSO model checking, and yardstick graph metrics such as tree-width and clique-width. We also show that various restrictions well-studied in the literature for obtaining decidability of reachability for the particular cases of multi-pushdown systems and message passing systems admit a bound on split-width. In fact, we propose generic controllers which subsume many of these cases.

Distributed controller design amounts to designing a controller (which is another CPDS) which, when run sychronously with a system ensures bounded split-width. These controllers are distributed in nature and are independent of the system it is controlling. Thus such a controller respects the privacy of the system (by not reading their states, for instance). Moreover, thanks to split-width such a controlled system offers efficient (in most cases optimal) decision procedures for the verification of the controlled system. We propose a generic approach to define controllable classes of CPDS in terms of quotient graphs, which admit a “suitable” acyclicity restriction. We also give a generic controller for several of the classes definable in this framework. The controllers we propose are sound and complete for the respective class, meaning that they allow all and only the behaviours of this class. Moreover, our technique for proving the bound on split-width of the controlled systems is also generic and systematic, hence may easily extend to generalisations and other classes as well.

The decidability results for the controllable classes proposed in the thesis are new while they capture, as special cases, several restrictions studied in the literature like bounded phase, bounded scope, poly-forest topology etc.