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

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 in order to accept, and to a reversible GWA. This paper establishes lower bounds on the state blow-up of these transformations, as well as closely matching upper bounds. It is shown that making an n-state GWA traversing k-ary graphs halt on every input requires at most 2nk+ 1 states and at least 2(n− 1)(k− 3) states in the worst case; making a GWA return to the initial node before acceptance takes at most 2nk+n and at least 2(n−1)(k−3) states in the worst case; Automata satisfying both properties at once have at most 4nk + 1 and at least 4(n − 1)(k − 3) states in the worst case. Reversible automata have at most 4nk + 1 and at least 4(n− 1)(k − 3)− 1 states in the worst case.