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

TL;DR

This paper analyzes the state complexity involved in transforming graph-walking automata to halt, return, or be reversible, providing tight bounds on the number of states required for these transformations.

Contribution

It establishes precise upper and lower bounds on the state blow-up for various transformations of graph-walking automata, including halting, returning, and reversibility.

Findings

Upper bounds for state blow-up in automata transformations

Lower bounds matching the upper bounds, showing tight complexity estimates

Analysis applies to automata on k-ary graphs with n states

Abstract

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.