Optimal algorithms for the coverability, the subword, the containment, and the equivalence problems for commutative semigroups

In this paper, we present decision procedures for the coverability, the subword, the containment, and the equivalence problems for commutative semigroups. These procedures require at most space 2^c·n, where n is the size of the problem instance, and c is some problem independent constant. Furthermore, we show that the exponential space hardness of the above problems follows from the work of Mayr and Meyer. Thus, the presented algorithms are space optimal. Our results close the gap between the 2^{c′·n·log n} space upper bound, shown by Rackoff for the coverability problem and shown by Huynh for the containment and the equivalence problems, and the exponential space lower bound resulting from the corresponding bound for the uniform word problem established by Mayr and Meyer.