Computational Complexity of the Word Problem for Commutative Semigroups

Abstract

We analyze the computational complexity of some decision problems for commutative semigroups in terms of time and space on a Turing machine. The main result we present is that any decision procedure for the word problemm for commutative semigroups requires storage space at least proportional to n/logn on a multitape Turing machine. This implies that the word problem is polynomia space hard (and in particular that it is at least NP-hard). We comment on the close relation of commutative semigroups to vector addition systems and Petri nets. We also show that the lower bound of space n/logn can be extended to certain other natural algorithmic problems for commutative semigroups. Finally we show that for several other algorithmic problems for commutative semigroups there exist polynomial time algorithms.