Two-Way Restarting Automata and J-Monotonicity

We introduce restarting automata as two-way automata in order to obtain a more general model which is closer to our linguistic motivations. We study the notion of j-monotonicity to show the advantages of this model. We show that the j-monotonicity can be considered as a degree of non-context-freeness and that it is a robust notion due to the considered models of restarting automata. Some other aspects concerning power and applications of two-way restarting automata are mentioned.