On some decidability questions concerning supports of rational series

Abstract

A well-known construction of a weighted automaton over the integers, assigning zero to precisely the nonempty words from the equality set of a given Post's correspondence problem instance, is extended to the case of weights taken from an arbitrary integral domain that is not locally finite. Undecidability of problems for rational series such as universality or rationality of support thus generalises to such domains as well. In spite of being fairly simple, these findings imply that the support rationality problem can be undecidable over rings of positive characteristic, answering an open question of D. Kirsten and K. Quaas. In addition, a more general undecidability result for rational series over other than locally finite integral domains is proved, which subsumes, e.g., the undecidability of support rationality or context-freeness.