Different Types of Monotonicity for Restarting Automata

We consider several classes of rewriting automata with a restart operation and the monotonicity property of computations by such automata. It leads to three natural definitions of (right) monotonicity of automata. Besides the former monotonicity, two new types, namely a-monotonicity and g-monotonicity, for such automata are introduced. We provide a taxonomy of the relevant language classes, and answer the (un)decidability questions concerning these properties