Solving Commutative Relaxations of Word Problems

We present an algebraic characterization of the standard commutative relaxation of the word problem in terms of a polynomial equality. We then consider a variant of the commutative word problem, referred to as the “Zero-to-All reachability” problem. We show that this problem is equivalent to a finite number of commutative word problems, and we use this insight to derive necessary conditions for Zero-to-All reachability. We conclude with a set of illustrative examples

The complexity of the word problems for commutative semigroups and polynomial ideals

AbstractAny decision procedure for the word problems for commutative semigroups and polynomial deals inherently requires computational storage space growing exponentially with the size of the problem instance to which the procedure is applied. This bound is achieved by a simple procedure for the semigroup problem

Abelian networks II. Halting on all inputs

Abelian networks are systems of communicating automata satisfying a local commutativity condition. We show that a finite irreducible abelian network halts on all inputs if and only if all eigenvalues of its production matrix lie in the open unit disk.Comment: Supersedes sections 5 and 6 of arXiv:1309.3445v1. To appear in Selecta Mathematic