State complexity of transforming graph-walking automata to halting, returning and reversible.

Graph-walking automata (GWA) traverse graphs by moving between the nodes following the edges, using a finite-state control to decide where to go next. It is known that every GWA can be transformed to a GWA that halts on every input, to a GWA returning to the initial node to accept, and to a reversible GWA. This paper establishes lower bounds on the state blow-up of these transformations, as well as improves the known upper bounds. It is shown that, for graphs with k labels of edge end-points, making an n-state GWA return to the initial node in the worst case requires at least 2(k - 3)n and at most 2kn + n states. Similar asymptotically tight bounds are proved for transformations ensuring other properties: for halting on every input, at least 2(k-3)(n -1) and at most 2kn +1 states; for returning and halting, at least 2(k-3)(2n -1) and at most 4kn + 1; for reversible, between 2(k - 3)(n -1) -1 and 2kn + 1; for returning and reversible, between 2(k - 3)(2n -1) -1 and 4kn + 1.(c) 2023 Elsevier Inc. All rights reserved.