From Multi to Single Stack Automata

Verification of concurrent programs modelled as multi-stack machines is an active research area. Recently decidability/complexity results have been established for powerful models such as bounded-phase visibly pushdown automata (BVMPA) [16] and ordered multi-pushdown automata (OMPA) [1]. However, the proofs of these results are quite complex and are based on different techniques and concepts for each of the considered models. We investigate in this paper the issue of reducing the verification problem of multi-stack machines to the one for single-stack machines. We believe that this is a general paradigm for understanding the expressive power and for establishing decidability results for various classes of concurrent program models. For instance, elegant (and paractically efficient) algorithms for bounded-context switch analysis of multi-pushdown systems have been recently defined based on reductions to the reachability problem of (single-stack) pushdown systems [10, 18]. We extend this view to both OMPA and BVMPA by showing that each of their emptiness problems can be reduced to the one for a class of single-stack machines. For these reductions, we introduce effective generalized pushdown automata (EGPA) where operations on stacks are (1) pop the top symbol of the stack, and (2) push a word in some (effectively) given set of words L over the stack alphabet, assuming that L is in some class of languages for which checking whether L intersects a given regular language is decidable. We show that the automata-based saturation procedure for computing the set of predecessors in standard pushdown automata can be easily extended to prove that for EGPA too the set of all predecessors of a regular set of configurations is an effectively constructible regular set. Our reductions from OMPA and BVMPA to EGPA, together with the reachability analysis procedure for EGPA, allow to provide conceptually simple algorithms for checking the emptiness problem for each of these models, and to significantly simplify the proofs for their 2ETIME upper bounds (matching their lower-bounds).