Lower Bounds for 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, as well as to a reversible GWA. This paper establishes lower bounds on the state blow-up of these transformations: it is shown that making an n-state GWA traversing k-ary graphs return to the initial node requires at least 2(n − 1)(k − 3) states in the worst case; the same lower bound holds for the transformation to halting automata. Automata satisfying both properties at once must have at least 4(n − 1)(k − 3) states. A reversible automaton must have at least 4(n − 1)(k − 3) − 1 states. These bounds are asymptotically tight to the upper bounds proved using the methods from the literature. 2012 ACM Subject Classification Theory of computation → Formal languages and automata theory; Theory of computation → Models of computation