MSO Decidability of Multi-Pushdown Systems via Split-Width

Abstract

Abstract. Multi-threaded programs with recursion are naturally modeled as multi-pushdown systems. The behaviors are represented as multiply nested words (MNWs), which are words enriched with additional binary relations for each stack matching a push operation with the corresponding pop operation. Any MNW can be decomposed by two basic and natural operations: shuffle of two sequences of factors and merge of consecutive factors of a sequence. We say that the split-width of an MNW is k if it admits a decomposition where the number of factors in each sequence is at most k. The MSO theory of MNWs with split-width k is decidable. We introduce two very general classes of MNWs that strictly generalize known decidable classes and prove their MSO decidability via their split-width and obtain comparable or better bounds of tree-width of known classes.

Controllers for the Verification of Communicating Multi-pushdown Systems

“…The transition relation makes explicit the identity of the data-structure being accessed and the type of the operation. As observed in [1,6,12,16] it is often convenient to describe the runs of such systems as a state-labeling of words decorated with a matching relation per data-structure instead of the traditional operational semantics using configurations and moves. This will prove all the more useful when we move to the distributed setting where traditionally semantics has always been given as state-labelings of appropriate partial orders [8,10,18].…”

“…In this approach we show that every behaviour in k-Phase has split-width [5][6][7] or tree-width [16] or clique-width [4] (measures of the complexity of graphs that happen to be equivalent for our class of graphs) bounded by some function f (k). Here, we show an exponential bound on the split-width.…”

“…Here, we show an exponential bound on the split-width. Then, results from [6,7,16] imply that MSO model-checking for S is decidable and results from [5,7] imply that model-checking linear-time temporal logic formulas can be solved in double exponential time. This is optimal, since reachability of k-phase multi-pushdown systems is double exponential time hard [14].…”

“…The idea of split-width was first described in [6] for multiply nested words and subsequently extended to include behaviours of systems with stacks and queues in [5,7].…”

“…Finally, we can prove using the split-width technique [5][6][7] that our generous under-approximation class can be model-checked against a wide variety of logics.…”