Counterexamples and Bounds for Graphs Solvable with Finite Delay

A finite directed labeled graph G with vertices {vi)~=l and labels {~)~i~ is solvable with delay d, if for every sequence of labels c h -.ok one can determine a sequence of vertices v% "" v~ such that there is a path connecting the k vertices vt ° ... v~ k labeled g~ ..~ , and for every j v~j is determined by the subsequence a~-+t ... oJ+a only. The paper presents a family of counterexamples proving a new lower bound for d. A new simple proof is given for a known upper bound for d and a new upper bound for d is given for the case where the sequences of labels are restricted to a regular event.