Bounded phase multi-stack pushdown systems (mpds) have been studied recently. Given a set C of configurations of a mpds M, let pre * M (C, k) be the set of configurations of M from which M can reach an element of C in at most k phases. In this paper, we show that for any mpds M, any regular set C of configurations of M and any number k, the set pre * M (C, k), is regular. We use saturation like method to construct a non-deterministic finite multi-automaton recognizing pre * M (C, k). Size of the automaton constructed is double exponential in k which is optimal as the worst case complexity measure.

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